{"id":269,"date":"2021-07-01T22:07:38","date_gmt":"2021-07-01T22:07:38","guid":{"rendered":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/1-7-the-real-numbers\/"},"modified":"2021-08-31T21:17:28","modified_gmt":"2021-08-31T21:17:28","slug":"1-7-the-real-numbers","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/1-7-the-real-numbers\/","title":{"raw":"1.7 The Real Numbers","rendered":"1.7 The Real Numbers"},"content":{"raw":"[latexpage]\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nBy the end of this section it is expected that you will be able to:\n<ul>\n \t<li>Identify integers, rational numbers, irrational numbers, and real numbers<\/li>\n \t<li>Locate fractions on the number line<\/li>\n \t<li>Locate decimals on the number line<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h1>Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers<\/h1>\n<p id=\"fs-id1170654983255\">We have already described numbers as <span class=\"no-emphasis\" data-type=\"term\"><em data-effect=\"italics\">counting number<\/em><em data-effect=\"italics\">s<\/em><\/span>, <span class=\"no-emphasis\" data-type=\"term\"><em data-effect=\"italics\">whole number<\/em><em data-effect=\"italics\">s<\/em><\/span>, and <span class=\"no-emphasis\" data-type=\"term\"><em data-effect=\"italics\">integers<\/em><\/span>. What is the difference between these types of numbers?<\/p>\n\\(\\begin{array}{cccccc}\\text{Counting numbers}\\hfill &amp; &amp; &amp; &amp; &amp; 1,2,3,4,\\text{\u2026}\\hfill \\\\ \\text{Whole numbers}\\hfill &amp; &amp; &amp; &amp; &amp; 0,1,2,3,4,\\text{\u2026}\\hfill \\\\ \\text{Integers}\\hfill &amp; &amp; &amp; &amp; &amp; \\text{\u2026}-3,-2,-1,0,1,2,3,\\text{\u2026}\\hfill \\end{array}\\)\n<p id=\"fs-id1170655000002\">What type of numbers would we get if we started with all the integers and then included all the fractions? The numbers we would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.<\/p>\n\n<div id=\"fs-id1170655026966\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Rational Number<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nA <strong data-effect=\"bold\">rational number<\/strong> is a number of the form \\(\\frac{p}{q}\\), where <em data-effect=\"italics\">p<\/em> and <em data-effect=\"italics\">q<\/em> are integers and \\(q\\ne 0\\).\n\nA rational number can be written as the ratio of two integers.\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\nAll signed fractions, such as \\(\\frac{4}{5},-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{8},\\frac{13}{4},-\\phantom{\\rule{0.2em}{0ex}}\\frac{20}{3}\\) are rational numbers. Each numerator and each denominator is an integer.\n<p id=\"fs-id1170654944763\">Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. Each integer can be written as a ratio of integers in many ways. For example, 3 is equivalent to \\(\\frac{3}{1},\\frac{6}{2},\\frac{9}{3},\\frac{12}{4},\\frac{15}{5}\\text{\u2026}\\)<\/p>\n<p id=\"fs-id1170654984452\">An easy way to write an integer as a ratio of integers is to write it as a fraction with denominator one.<\/p>\n\\(\\begin{array}{ccccccc}\\hfill 3=\\frac{3}{1}\\hfill &amp; &amp; &amp; \\hfill -8=-\\phantom{\\rule{0.2em}{0ex}}\\frac{8}{1}\\hfill &amp; &amp; &amp; \\hfill 0=\\frac{0}{1}\\hfill \\end{array}\\)\n<p id=\"fs-id1170655195958\">Since any integer can be written as the ratio of two integers, <em data-effect=\"italics\">all integers are rational numbers<\/em>! Remember that the counting numbers and the whole numbers are also integers, and so they, too, are rational.<\/p>\n<p id=\"fs-id1170655059465\">What about decimals? Are they rational? Let\u2019s look at a few to see if we can write each of them as the ratio of two integers.<\/p>\n<p id=\"fs-id1170654928833\">We\u2019ve already seen that integers are rational numbers. The integer \\(-8\\) could be written as the decimal \\(-8.0\\). So, clearly, some decimals are rational.<\/p>\n<p id=\"fs-id1170655083456\">Think about the decimal 7.3. Can we write it as a ratio of two integers? Because 7.3 means \\(7\\frac{3}{10}\\), we can write it as an improper fraction, \\(\\frac{73}{10}\\). So 7.3 is the ratio of the integers 73 and 10. It is a rational number.<\/p>\n<p id=\"fs-id1170655026632\">In general, any decimal that ends after a number of digits (such as 7.3 or \\(-1.2684\\) is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction.<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654943006\" data-type=\"problem\">\n<p id=\"fs-id1170655111565\">Write as the ratio of two integers: a) \\(-27\\) b) 7.31<\/p>\n\n<\/div>\n<div id=\"fs-id1170655163665\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<table id=\"eip-639\" summary=\"-\">\n<tbody>\n<tr>\n<td>a)\nWrite it as a fraction with denominator 1.<\/td>\n<td>\\(\\begin{array}{c}-27\\\\ \\frac{-27}{1}\\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td>b)\nWrite it as a mixed number. Remember, 7 is the whole number and the decimal part, 0.31, indicates hundredths.\nConvert to an improper fraction.<\/td>\n<td>\\(\\begin{array}{c}7.31\\\\ 7\\frac{31}{100}\\\\ \\frac{731}{100}\\end{array}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170655128317\">So we see that \\(-27\\) and 7.31 are both rational numbers, since they can be written as the ratio of two integers.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id1170654982461\">Write as the ratio of two integers: a) \\(-24\\) b) 3.57<\/p>\n\n<\/div>\n<div id=\"fs-id1170655098810\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1170655083207\">a) \\(\\frac{-24}{1}\\) b) \\(\\frac{357}{100}\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170654988762\">Let\u2019s look at the decimal form of the numbers we know are rational.<\/p>\n<p id=\"fs-id1170654905314\">We have seen that <em data-effect=\"italics\">every<\/em> <em data-effect=\"italics\">integer is a rational number<\/em>, since \\(a=\\frac{a}{1}\\) for any integer, <em data-effect=\"italics\">a<\/em>. We can also change any integer to a decimal by adding a decimal point and a zero.<\/p>\n\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<th style=\"width: 14.2857%;\" scope=\"row\">Integer<\/th>\n<td style=\"width: 14.2857%;\">-2<\/td>\n<td style=\"width: 14.2857%;\">-1<\/td>\n<td style=\"width: 14.2857%;\">0<\/td>\n<td style=\"width: 14.2857%;\">1<\/td>\n<td style=\"width: 14.2857%;\">2<\/td>\n<td style=\"width: 14.2857%;\">3<\/td>\n<\/tr>\n<tr>\n<th style=\"width: 14.2857%;\" scope=\"row\">Decimal form<\/th>\n<td style=\"width: 14.2857%;\">-2.0<\/td>\n<td style=\"width: 14.2857%;\">-1.0<\/td>\n<td style=\"width: 14.2857%;\">0.0<\/td>\n<td style=\"width: 14.2857%;\">1.0<\/td>\n<td style=\"width: 14.2857%;\">2.0<\/td>\n<td style=\"width: 14.2857%;\">3.0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nThese decimal numbers stop.\n<p id=\"fs-id1170655108063\">We have also seen that <em data-effect=\"italics\">every<\/em> <em data-effect=\"italics\">fraction is a rational number<\/em>. Look at the decimal form of the fractions we considered above.<\/p>\n\n<table style=\"border-collapse: collapse; width: 100%; height: 128px;\" border=\"0\">\n<tbody>\n<tr style=\"height: 55px;\">\n<th style=\"width: 20%; height: 55px;\" scope=\"row\">Ratio of integers<\/th>\n<td style=\"width: 20%; height: 55px;\">\\(\\frac{4}{5}\\)<\/td>\n<td style=\"width: 20%; height: 55px;\">-\\(\\frac{7}{8}\\)<\/td>\n<td style=\"width: 20%; height: 55px;\">\\(\\frac{13}{4}\\)<\/td>\n<td style=\"width: 20%; height: 55px;\">-\\(\\frac{20}{3}\\)<\/td>\n<\/tr>\n<tr style=\"height: 55px;\">\n<th style=\"width: 20%; height: 55px;\" scope=\"row\">The decimal form<\/th>\n<td style=\"width: 20%; height: 55px;\">\\(0.8\\)<\/td>\n<td style=\"width: 20%; height: 55px;\">\\(-0.875\\)<\/td>\n<td style=\"width: 20%; height: 55px;\">\\(3.25\\)<\/td>\n<td style=\"width: 20%; height: 55px;\">\\(-6.666..\\).<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nThese decimals either stop or repeat.\n<p id=\"fs-id1170654986576\">What do these examples tell us?<\/p>\n<p id=\"fs-id1170655196916\"><em data-effect=\"italics\">Every rational number can be written both as a ratio of integers<\/em>, \\(\\frac{p}{q}\\),<em data-effect=\"italics\">where p and q are integers and <\/em>\\(q\\ne 0\\),<em data-effect=\"italics\">and as a decimal that either stops or repeats.<\/em><\/p>\n<p id=\"fs-id1170654942363\">Here are the numbers we looked at above expressed as a ratio of integers and as a decimal:<\/p>\n\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Fractions<\/caption>\n<tbody>\n<tr valign=\"top\">\n<th style=\"width: 5.77386%;\" scope=\"row\" data-valign=\"top\" data-align=\"left\">Number<\/th>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\">\\(\\frac{4}{5}\\)<\/td>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{8}\\)<\/td>\n<td style=\"width: 6.89655%;\" data-valign=\"top\" data-align=\"left\">\\(\\frac{13}{4}\\)<\/td>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{20}{3}\\)<\/td>\n<\/tr>\n<tr valign=\"top\">\n<th style=\"width: 5.77386%;\" scope=\"row\" data-valign=\"top\" data-align=\"left\">Ratio of Integers<\/th>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\">\\(\\frac{4}{5}\\)<\/td>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{8}\\)<\/td>\n<td style=\"width: 6.89655%;\" data-valign=\"top\" data-align=\"left\">\\(\\frac{13}{4}\\)<\/td>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{20}{3}\\)<\/td>\n<\/tr>\n<tr valign=\"top\">\n<th style=\"width: 5.77386%;\" scope=\"row\" data-valign=\"top\" data-align=\"left\">Decimal Form<\/th>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\">\\(0.8\\)<\/td>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\">\\(-0.875\\)<\/td>\n<td style=\"width: 6.89655%;\" data-valign=\"top\" data-align=\"left\">\\(3.25\\)<\/td>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\">\\(-6.\\stackrel{\\text{\u2013}}{6}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Integers<\/caption>\n<tbody>\n<tr valign=\"top\">\n<th style=\"width: 5.77386%;\" scope=\"row\" data-valign=\"top\" data-align=\"left\">Number<\/th>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\">\\(-2\\)<\/td>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\">\\(-1\\)<\/td>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\">\\(0\\)<\/td>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\">\\(1\\)<\/td>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\">\\(2\\)<\/td>\n<td style=\"width: 0.801925%;\" data-valign=\"top\" data-align=\"left\">\\(3\\)<\/td>\n<\/tr>\n<tr valign=\"top\">\n<th style=\"width: 5.77386%;\" scope=\"row\" data-valign=\"top\" data-align=\"left\">Ratio of Integers<\/th>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{2}{1}\\)<\/td>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{1}\\)<\/td>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\">\\(\\frac{0}{1}\\)<\/td>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\">\\(\\frac{1}{1}\\)<\/td>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\">\\(\\frac{2}{1}\\)<\/td>\n<td style=\"width: 0.801925%;\" data-valign=\"top\" data-align=\"left\">\\(\\frac{3}{1}\\)<\/td>\n<\/tr>\n<tr valign=\"top\">\n<th style=\"width: 5.77386%;\" scope=\"row\" data-valign=\"top\" data-align=\"left\">Decimal Form<\/th>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\">\\(-2.0\\)<\/td>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\">\\(-1.0\\)<\/td>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\">\\(0.0\\)<\/td>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\">\\(1.0\\)<\/td>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\">\\(2.0\\)<\/td>\n<td style=\"width: 0.801925%;\" data-valign=\"top\" data-align=\"left\">\\(3.0\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1170655007226\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Rational Number<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nA <strong data-effect=\"bold\">rational number<\/strong> is a number of the form \\(\\frac{p}{q}\\), where <em data-effect=\"italics\">p<\/em> and <em data-effect=\"italics\">q<\/em> are integers and \\(q\\ne 0\\).\n\nIts decimal form stops or repeats.\n\n<\/div>\n<\/div>\nAre there any decimals that do not stop or repeat? Yes!\n\n<\/div>\n<\/div>\n<p id=\"fs-id1170655027582\">The number \\(\\pi \\) (the Greek letter <em data-effect=\"italics\">pi<\/em>, pronounced \u201cpie\u201d), which is very important in describing circles, has a decimal form that does not stop or repeat.<\/p>\n\\(\\pi =3.141592654..\\).\n<p id=\"fs-id1170655162537\">We can even create a decimal pattern that does not stop or repeat, such as<\/p>\n\\(2.01001000100001\\dots \\)\n<p id=\"fs-id1170655097911\">Numbers whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call these numbers irrational. More on irrational numbers later on is this course.<\/p>\n\n<div id=\"fs-id1170655121425\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Irrational Number<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nAn irrational number is a number that cannot be written as the ratio of two integers.\n\nIts decimal form does not stop and does not repeat.\n\n<\/div>\n<\/div>\nLet\u2019s summarize a method we can use to determine whether a number is rational or irrational.\n\n<\/div>\n<\/div>\n<div id=\"fs-id1170655353858\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Rational or Irrational?<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655353858\" data-type=\"note\">\n<p id=\"fs-id1170655353986\">If the decimal form of a number<\/p>\n\n<ul id=\"fs-id1170655107719\" data-bullet-style=\"bullet\">\n \t<li><em data-effect=\"italics\">repeats or stops<\/em>, the number is <strong data-effect=\"bold\">rational<\/strong>.<\/li>\n \t<li><em data-effect=\"italics\">does not repeat and does not stop<\/em>, the number is irrational<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170655128145\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1170655161217\" data-type=\"exercise\">\n<div id=\"fs-id1170655161219\" data-type=\"problem\">\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654967985\" data-type=\"problem\">\n<p id=\"fs-id1170654967987\">Given the numbers \\(0.58\\stackrel{\\text{-}}{3},0.47,3.605551275..\\). list the a) rational numbers b) irrational numbers.<\/p>\n\n<\/div>\n<div id=\"fs-id1170655025175\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-798\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td>a)\nLook for decimals that repeat or stop.<\/td>\n<td>The 3 repeats in \\(0.58\\stackrel{\\text{\u2013}}{3}\\).\nThe decimal 0.47 stops after the 7.\nSo \\(0.58\\stackrel{\\text{-}}{3}\\) and 0.47 are rational.<\/td>\n<\/tr>\n<tr>\n<td>b)\nLook for decimals that neither stop nor repeat.<\/td>\n<td>\\(3.605551275\\text{\u2026}\\) has no repeating block of digits and it does not stop.\nSo \\(3.605551275\\text{\u2026}\\) is irrational.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655161219\" data-type=\"problem\">\n<p id=\"fs-id1170654943758\">For the given numbers list the a) rational numbers b) irrational numbers: \\(0.29,0.81\\stackrel{\\text{-}}{6},2.515115111\\text{\u2026}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1170654963540\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1170655117899\">a) \\(0.29,0.81\\stackrel{\\text{-}}{6}\\) b) \\(2.515115111\\text{\u2026}\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170655128145\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1170655161217\" data-type=\"exercise\">\n<div id=\"fs-id1170655161219\" data-type=\"problem\">\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 3<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655218302\" data-type=\"problem\">\n<p id=\"fs-id1170655083134\">For each number given, identify whether it is rational or irrational: a) \\(\\sqrt{36}\\) b) \\(\\sqrt{44}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1170655154779\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p style=\"padding-left: 40px;\">a) Recognize that 36 is a perfect square, since \\({6}^{2}=36.\\) So \\(\\sqrt{36}=6,\\) therefore \\(\\sqrt{36}\\) is rational.<\/p>\n<p style=\"padding-left: 40px;\">b) Remember that \\({6}^{2}=36\\) and \\({7}^{2}=49,\\) so 44 is not a perfect square. Therefore, the decimal form of \\(\\sqrt{44}\\) will never repeat and never stop, so \\(\\sqrt{44}\\) is irrational.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 3<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655083580\" data-type=\"problem\">\n<p id=\"fs-id1170655202060\">For each number given, identify whether it is rational or irrational: a) \\(\\sqrt{81}\\) b) \\(\\sqrt{17}.\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1170655216149\"><span class=\"token\">a)<\/span> rational b) irrational<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\nWe have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. The irrational numbers are numbers whose decimal form does not stop and does not repeat. When we put together the rational numbers and the irrational numbers, we get the set of <span data-type=\"term\">real number<\/span><strong data-effect=\"bold\">s<\/strong>.\n<div id=\"fs-id1170655121069\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Real Number<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nA <strong data-effect=\"bold\">real number<\/strong> is a number that is either rational or irrational.\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"CNX_ElemAlg_Figure_01_08_001\" class=\"bc-figure figure\">\n<div>\n<p id=\"fs-id1170655000486\">All the numbers we use in algebra are real numbers. <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_001\">\u00a0Figure 1<\/a> illustrates how the number sets we\u2019ve discussed in this section fit together.<\/p>\n\n<div id=\"CNX_ElemAlg_Figure_01_08_001\" class=\"bc-figure figure\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"654\"]<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_01_08_001_new.jpg\" alt=\"This figure consists of a Venn diagram. To start there is a large rectangle marked Real Numbers. The right half of the rectangle consists of Irrational Numbers. The left half consists of Rational Numbers. Within the Rational Numbers rectangle, there are Integers \u2026, negative 2, negative 1, 0, 1, 2, \u2026. Within the Integers rectangle, there are Whole Numbers 0, 1, 2, 3, \u2026 Within the Whole Numbers rectangle, there are Counting Numbers 1, 2, 3, \u2026\" width=\"654\" height=\"396\" data-media-type=\"image\/jpeg\"> Figure 1 This chart shows the number sets that make up the set of real numbers. Does the term \u201creal numbers\u201d seem strange to you? Are there any numbers that are not \u201creal,\u201d and, if so, what could they be?[\/caption]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1170655082021\">Do you remember that the square root of a negative number was not a real number?<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 4<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654925513\" data-type=\"problem\">\n<p id=\"fs-id1170654925515\">For each number given, identify whether it is a real number or not a real number: <span class=\"token\">\u24d0<\/span> \\(\\sqrt{-169}\\) <span class=\"token\">\u24d1<\/span> \\(\\-\\sqrt{64}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1170654944094\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n&nbsp;\n\n<span class=\"token\">a)<\/span> There is no real number whose square is \\(-169.\\) Therefore, \\(\\sqrt{-169}\\) is not a real number.\n\nb) Since the negative is in front of the radical, \\(-\\sqrt{64}\\) is \\(-8,\\) Since \\(-8\\) is a real number, \\(-\\sqrt{64}\\) is a real number.\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 4<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655062079\" data-type=\"problem\">\n<p id=\"fs-id1170655062081\">For each number given, identify whether it is a real number or not a real number: a) \\(\\sqrt{-196}\\) b) \\(-\\sqrt{81}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1170655164781\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1170655164783\"><span class=\"token\">a)<\/span> not a real number b) real number<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170655062462\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1170655062465\" data-type=\"exercise\">\n<div id=\"fs-id1170655083274\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 5<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655162698\" data-type=\"problem\">\n<p id=\"fs-id1170655162700\">Given the numbers \\(-7,\\frac{14}{5},8,\\sqrt{5},5.9,\\text{-}\\sqrt{64},\\) list the a) whole numbers b) integers c) rational numbers d) irrational numbers e) real numbers.<\/p>\n\n<\/div>\n<div id=\"fs-id1170654943557\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n&nbsp;\n<p id=\"fs-id1167835374083\"><span class=\"token\">a)<\/span> Remember, the whole numbers are 0, 1, 2, 3, \u2026 and 8 is the only whole number given.<span data-type=\"newline\">\n<\/span><\/p>\n<span class=\"token\">b)<\/span> The integers are the whole numbers, their opposites, and 0. So the whole number 8 is an integer, and \\(-7\\) is the opposite of a whole number so it is an integer, too. Also, notice that 64 is the square of 8 so \\(-\\sqrt{64}=-8.\\) So the integers are \\(-7,8,-\\sqrt{64}.\\)<span data-type=\"newline\">\n<\/span>\n\n<span class=\"token\">c)<\/span> Since all integers are rational, then \\(-7,8,-\\sqrt{64}\\) are rational. Rational numbers also include fractions and decimals that repeat or stop, so \\(\\frac{14}{5}\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}5.9\\) are rational. So the list of rational numbers is \\(-7,\\frac{14}{5},8,5.9,-\\sqrt{64}.\\)<span data-type=\"newline\">\n<\/span>\n\nd) Remember that 5 is not a perfect square, so \\(\\sqrt{5}\\) is irrational.<span data-type=\"newline\">\n<\/span>\n\ne) All the numbers listed are real numbers.\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 5<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655166461\" data-type=\"problem\">\n<p id=\"fs-id1170655166463\">For the given numbers, list the a) whole numbers b) integers c) rational numbers d) irrational numbers e) real numbers: \\(-3,-\\sqrt{2},0.\\stackrel{\\text{-}}{3},\\frac{9}{5},4,\\sqrt{49}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1170655098112\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1170655106839\"><span class=\"token\">a) <\/span>\\(4,\\sqrt{49}\\) b) \\(-3,4,\\sqrt{49}\\) c) \\(-3,0.\\stackrel{\\text{-}}{3},\\frac{9}{5},4,\\sqrt{49}\\) d) \\(\\text{-}\\sqrt{2}\\) e) \\(-3,\\text{-}\\sqrt{2},0.\\stackrel{\\text{-}}{3},\\frac{9}{5},4,\\sqrt{49}\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"CNX_ElemAlg_Figure_01_08_001\" class=\"bc-figure figure\"><\/div>\n<h1>Locate Fractions on the Number Line<\/h1>\nThe last time we looked at the <span class=\"no-emphasis\" data-type=\"term\">number line<\/span>, it only had positive and negative integers on it. We now want to include <span class=\"no-emphasis\" data-type=\"term\">fraction<\/span>s and decimals on it.\n<p id=\"fs-id1170655091189\">Let\u2019s start with fractions and locate \\(\\frac{1}{5},-\\phantom{\\rule{0.2em}{0ex}}\\frac{4}{5},3,\\frac{7}{4},-\\phantom{\\rule{0.2em}{0ex}}\\frac{9}{2},-5,\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}\\frac{8}{3}\\) on the number line.<\/p>\n<p id=\"fs-id1170655203491\">We\u2019ll start with the whole numbers \\(3\\) and \\(-5\\). because they are the easiest to plot. See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_002\">Figure 2<\/a>.<\/p>\n<p id=\"fs-id1170655353090\">The <span class=\"no-emphasis\" data-type=\"term\">proper fractions<\/span> listed are \\(\\frac{1}{5}\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}-\\phantom{\\rule{0.2em}{0ex}}\\frac{4}{5}\\). We know the proper fraction \\(\\frac{1}{5}\\) has value less than one and so would be located between \\(\\text{0 and 1.}\\) The denominator is 5, so we divide the unit from 0 to 1 into 5 equal parts \\(\\frac{1}{5},\\frac{2}{5},\\frac{3}{5},\\frac{4}{5}\\). We plot \\(\\frac{1}{5}\\). See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_002\">Figure 2<\/a>.<\/p>\n<p id=\"fs-id1170655188953\">Similarly, \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{4}{5}\\) is between 0 and \\(-1\\). After dividing the unit into 5 equal parts we plot \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{4}{5}\\). See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_002\">Figure 2<\/a>.<\/p>\n<p id=\"fs-id1170655215831\">Finally, look at the <span class=\"no-emphasis\" data-type=\"term\">improper fractions<\/span> \\(\\frac{7}{4},-\\phantom{\\rule{0.2em}{0ex}}\\frac{9}{2},\\frac{8}{3}\\). These are fractions in which the numerator is greater than the denominator. Locating these points may be easier if you change each of them to a mixed number. See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_002\">Figure 2<\/a>.<\/p>\n\\(\\begin{array}{ccccccc}\\hfill \\frac{7}{4}=1\\frac{3}{4}\\hfill &amp; &amp; &amp; \\hfill -\\phantom{\\rule{0.2em}{0ex}}\\frac{9}{2}=-4\\frac{1}{2}\\hfill &amp; &amp; &amp; \\hfill \\frac{8}{3}=2\\frac{2}{3}\\hfill \\end{array}\\)\n<p id=\"fs-id1170654944810\"><a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_002\">Figure 2<\/a> shows the number line with all the points plotted.<\/p>\n\n<div id=\"CNX_ElemAlg_Figure_01_08_002\" class=\"bc-figure figure\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"656\"]<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_002_new.jpg\" alt=\"There is a number line shown that runs from negative 6 to positive 6. From left to right, the numbers marked are negative 5, negative 9\/2, negative 4\/5, 1\/5, 4\/5, 8\/3, and 3. The number negative 9\/2 is halfway between negative 5 and negative 4. The number negative 4\/5 is slightly to the right of negative 1. The number 1\/5 is slightly to the right of 0. The number 4\/5 is slightly to the left of 1. The number 8\/3 is between 2 and 3, but a little closer to 3.\" width=\"656\" height=\"75\" data-media-type=\"image\/jpeg\"> Figure 2[\/caption]\n\n<\/div>\n<div>\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 6<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655166545\" data-type=\"problem\">\n<p id=\"fs-id1170655166548\">Locate and label the following on a number line: \\(4,\\frac{3}{4},-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{4},-3,\\frac{6}{5},-\\phantom{\\rule{0.2em}{0ex}}\\frac{5}{2},\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{3}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1170654905871\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1170654905873\">Locate and plot the integers, \\(4,-3\\).<\/p>\n<p id=\"fs-id1170655098863\">Locate the proper fraction \\(\\frac{3}{4}\\) first. The fraction \\(\\frac{3}{4}\\) is between 0 and 1. Divide the distance between 0 and 1 into four equal parts then, we plot \\(\\frac{3}{4}\\). Similarly plot \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{4}\\).<\/p>\n<p id=\"fs-id1170654941093\">Now locate the improper fractions \\(\\frac{6}{5},-\\phantom{\\rule{0.2em}{0ex}}\\frac{5}{2},\\frac{7}{3}\\). It is easier to plot them if we convert them to mixed numbers and then plot them as described above: \\(\\frac{6}{5}=1\\frac{1}{5},-\\phantom{\\rule{0.2em}{0ex}}\\frac{5}{2}=-2\\frac{1}{2},\\frac{7}{3}=2\\frac{1}{3}\\).<\/p>\n<span id=\"fs-id1170655098989\" data-type=\"media\" data-alt=\"There is a number line shown that runs from negative 6 to positive 6. From left to right, the numbers marked are negative 3, negative 5\/2, negative 1\/4, 3\/4, 6\/5, 7\/3, and 4. The number negative 5\/2 is halfway between negative 3 and negative 2. The number negative 1\/4 is slightly to the left of 0. The number 3\/4 is slightly to the left of 1. The number 6\/5 is slightly to the right of 1. The number 7\/3 is between 2 and 3, but a little closer to 2.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_003_img_new.jpg\" alt=\"There is a number line shown that runs from negative 6 to positive 6. From left to right, the numbers marked are negative 3, negative 5\/2, negative 1\/4, 3\/4, 6\/5, 7\/3, and 4. The number negative 5\/2 is halfway between negative 3 and negative 2. The number negative 1\/4 is slightly to the left of 0. The number 3\/4 is slightly to the left of 1. The number 6\/5 is slightly to the right of 1. The number 7\/3 is between 2 and 3, but a little closer to 2.\" data-media-type=\"image\/jpeg\"><\/span>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id1170655133229\">Locate and label the following on a number line: \\(-1,\\frac{1}{3},\\frac{6}{5},-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{4},\\frac{9}{2},5,-\\phantom{\\rule{0.2em}{0ex}}\\frac{8}{3}\\).<\/p>\n\n<\/div>\n<details><summary>Show answer<\/summary>\n<div id=\"fs-id1170654914689\" data-type=\"solution\"><span data-type=\"media\" data-alt=\"There is a number line shown that runs from negative 4 to positive 5. From left to right, the numbers marked are negative 8\/3, negative 7\/4, negative 1, 1\/3, 6\/5, 9\/2, and 5. The number negative 8\/3 is between negative 3 and negative 2 but slightly closer to negative 3. The number negative 7\/4 is slightly to the right of negative 2. The number 1\/3 is slightly to the right of 0. The number 6\/5 is slightly to the right of 1. The number 9\/2 is halfway between 4 and 5.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_004_img_new.jpg\" alt=\"There is a number line shown that runs from negative 4 to positive 5. From left to right, the numbers marked are negative 8\/3, negative 7\/4, negative 1, 1\/3, 6\/5, 9\/2, and 5. The number negative 8\/3 is between negative 3 and negative 2 but slightly closer to negative 3. The number negative 7\/4 is slightly to the right of negative 2. The number 1\/3 is slightly to the right of 0. The number 6\/5 is slightly to the right of 1. The number 9\/2 is halfway between 4 and 5.\" data-media-type=\"image\/jpeg\"><\/span><\/div>\n<\/details><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170655353172\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1170655353176\" data-type=\"exercise\">\n<div data-type=\"problem\">\n\nIn <a href=\"#fs-id1170655150929\">Example 5<\/a>, we\u2019ll use the inequality symbols to order fractions. In previous chapters we used the number line to order numbers.\n\n<\/div>\n<\/div>\n<\/div>\n<ul id=\"fs-id1166425210298\" data-bullet-style=\"bullet\">\n \t<li><em data-effect=\"italics\">a &lt; b<\/em> \u201c<em data-effect=\"italics\">a<\/em> is less than <em data-effect=\"italics\">b<\/em>\u201d when <em data-effect=\"italics\">a<\/em> is to the left of <em data-effect=\"italics\">b<\/em> on the number line<\/li>\n \t<li><em data-effect=\"italics\">a &gt; b<\/em> \u201c<em data-effect=\"italics\">a<\/em> is greater than <em data-effect=\"italics\">b<\/em>\u201d when <em data-effect=\"italics\">a<\/em> is to the right of <em data-effect=\"italics\">b<\/em> on the number line<\/li>\n<\/ul>\n<p id=\"fs-id1170655121376\">As we move from left to right on a number line, the values increase.<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 7<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id1170655150929\">Order each of the following pairs of numbers, using &lt; or &gt;. It may be helpful to refer <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_006\">Figure 3<\/a>.<\/p>\na) -\\(\\frac{2}{3}\\)____\\(-1\\) b) -3\\(\\frac{1}{2}\\)____\\(-3\\) c) \\(\\frac{3}{4}\\)____ -\\(\\frac{1}{4}\\) d) \\(-2\\)____ -\\(\\frac{8}{3}\\)\n<div id=\"CNX_ElemAlg_Figure_01_08_006\" class=\"bc-figure figure\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"656\"]<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_006_img_new.jpg\" alt=\"There is a number line shown that runs from negative 4 to positive 4. From left to right, the numbers marked are negative 3 and 1\/2, negative 3, negative 8\/3, negative 2, negative 1, negative 3\/4, negative 2\/3, and negative 1\/4. The number negative 3 and 1\/2 is between negative 4 and negative 3 The number negative 8\/3 is between negative 3 and negative 2, but closer to negative 3. The numbers negative 3\/4, negative 2\/3, and negative 1\/4 are all between negative 1 and 0.\" width=\"656\" height=\"75\" data-media-type=\"image\/jpeg\"> Figure 3[\/caption]\n\n<\/div>\n<div><\/div>\n<\/div>\n<div>\\(\\begin{array}{c} \\text{-3} \\frac{1}{2} \\rule{2em}{0.4pt} -3 \\\\ -3 \\frac{1}{2} &lt; -3 \\end{array}\\)<\/div>\n<div>\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<table id=\"eip-313\" summary=\"\/\">\n<tbody>\n<tr>\n<td style=\"width: 513.406px;\">a)\n-\\(\\frac{2}{3}\\) is to the right of \\(-1\\) on the number line.<\/td>\n<td style=\"width: 834.406px;\">-\\(\\frac{2}{3}\\)___-1\n\n-\\(\\frac{2}{3}\\) &gt; -1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 513.406px;\">b)\n-\\(3\\frac{1}{2}\\) is to the right of \\(-3\\) on the number line.<\/td>\n<td style=\"width: 834.406px;\">\\(\\begin{array}{c}-3\\frac{1}{2}\\rule{2em}{0.4pt}-3 \\\\ -3\\frac{1}{2}&lt;-3\\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 513.406px;\">c)\n\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{3}{4}\\) is to the right of \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{4}\\) on the number line.<\/td>\n<td style=\"width: 834.406px;\">\\(\\begin{array}{c}-\\phantom{\\rule{0.2em}{0ex}}\\frac{3}{4}\\rule{2em}{0.4pt}-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{4}\\\\ -\\phantom{\\rule{0.2em}{0ex}}\\frac{3}{4}&lt;-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{4}\\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 513.406px;\">d)\n\\(-2\\) is to the right of \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{8}{3}\\) on the number line.<\/td>\n<td style=\"width: 834.406px;\"><img class=\"alignnone wp-image-13457\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/Screenshot-2021-02-22-at-9.35.54-AM-e1614015412311.png\" alt=\"negative 2 is greater than negative 8 divided by 3.\" width=\"117\" height=\"59\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 7<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654941144\" data-type=\"problem\">\n<p id=\"fs-id1170654941146\">Order each of the following pairs of numbers, using &lt; or &gt;:<\/p>\n<p id=\"fs-id1166422659176\">a) \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{3}\\rule{2em}{0.4pt}-1\\) b) \\(-1\\frac{1}{2}\\rule{2em}{0.4pt}-2\\) c) \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{2}{3}\\rule{2em}{0.4pt}-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{3}\\) d) \\(-3\\rule{2em}{0.4pt}-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{3}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1170654940483\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1170654940485\">a) &gt; b) &gt; c) &lt; d) &lt;<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<h1>Locate Decimals on the Number Line<\/h1>\n<p id=\"fs-id1170655098623\">Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 8<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655029934\" data-type=\"problem\">\n<p id=\"fs-id1170655029936\">Locate 0.4 on the number line.<\/p>\n\n<\/div>\n<div id=\"fs-id1170655029941\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1170655081279\">A proper fraction has value less than one. The decimal number 0.4 is equivalent to \\(\\frac{4}{10}\\), a proper fraction, so 0.4 is located between 0 and 1. On a number line, divide the interval between 0 and 1 into 10 equal parts. Now label the parts 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. We write 0 as 0.0 and 1 and 1.0, so that the numbers are consistently in tenths. Finally, mark 0.4 on the number line. See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_007\">Figure 4<\/a>.<\/p>\n\n<div id=\"CNX_ElemAlg_Figure_01_08_007\" class=\"bc-figure figure\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"667\"]<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_007_new.jpg\" alt=\"There is a number line shown that runs from 0.0 to 1. The only point given is 0.4, which is between 0.3 and 0.5.\" width=\"667\" height=\"42\" data-media-type=\"image\/jpeg\"> Figure 4[\/caption]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 8<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654962328\" data-type=\"problem\">\n<p id=\"fs-id1170654962330\">Locate on the number line: 0.6<\/p>\n\n<\/div>\n<details><summary>Show answer<\/summary>\n<div id=\"fs-id1170654968605\" data-type=\"solution\"><span id=\"fs-id1170654968607\" data-type=\"media\" data-alt=\"There is a number line shown that runs from 0.0 to 1. The only point given is 0.6, which is between 0.5 and 0.7.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_008_img_new.jpg\" alt=\"There is a number line shown that runs from 0.0 to 1. The only point given is 0.6, which is between 0.5 and 0.7.\" data-media-type=\"image\/jpeg\"><\/span><\/div>\n<\/details><\/div>\n<\/div>\n<div class=\"try\" data-type=\"note\">\n<div id=\"fs-id1170654962326\" data-type=\"exercise\">\n<div id=\"fs-id1170654962328\" data-type=\"problem\">\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 9<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655195763\" data-type=\"problem\">\n<p id=\"fs-id1170655195765\">Locate \\(-0.74\\) on the number line.<\/p>\n\n<\/div>\n<div id=\"fs-id1170655090205\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1170655090207\">The decimal \\(-0.74\\) is equivalent to \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{74}{100}\\), so it is located between 0 and \\(-1\\). On a number line, mark off and label the hundredths in the interval between 0 and \\(-1\\). See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_010\">Figure 5<\/a>.<\/p>\n\n<div id=\"CNX_ElemAlg_Figure_01_08_010\" class=\"bc-figure figure\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"676\"]<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_010_new.jpg\" alt=\"There is a number line shown that runs from negative 1.00 to 0.00. The only point given is negative 0.74, which is between negative 0.8 and negative 0.7.\" width=\"676\" height=\"61\" data-media-type=\"image\/jpeg\"> Figure 5[\/caption]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 9<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655057804\" data-type=\"problem\">\n<p id=\"fs-id1170655057806\">Locate on the number line: \\(-0.6\\).<\/p>\n\n<\/div>\n<details><summary>Show answer<\/summary>\n<div id=\"fs-id1170655003781\" data-type=\"solution\"><span id=\"fs-id1170655003783\" data-type=\"media\" data-alt=\"There is a number line shown that runs from negative 1.00 to 0.00. The only point given is negative 0.6, which is between negative 0.8 and negative 0.4.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_011_img_new.jpg\" alt=\"There is a number line shown that runs from negative 1.00 to 0.00. The only point given is negative 0.6, which is between negative 0.8 and negative 0.4.\" data-media-type=\"image\/jpeg\"><\/span><\/div>\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170655082976\">Which is larger, 0.04 or 0.40? If you think of this as money, you know that ?0.40 (forty cents) is greater than ?0.04 (four cents). So,<\/p>\n<p id=\"fs-id1170654940800\">\\(0.40\\) &gt; \\(0.04\\)<\/p>\n<p id=\"fs-id1170655061503\">Again, we can use the number line to order numbers.<\/p>\n\n<ul id=\"fs-id1166422832290\" data-bullet-style=\"bullet\">\n \t<li><em data-effect=\"italics\">a &lt; b<\/em> \u201c<em data-effect=\"italics\">a<\/em> is less than <em data-effect=\"italics\">b<\/em>\u201d when <em data-effect=\"italics\">a<\/em> is to the left of <em data-effect=\"italics\">b<\/em> on the number line<\/li>\n \t<li><em data-effect=\"italics\">a &gt; b<\/em> \u201c<em data-effect=\"italics\">a<\/em> is greater than <em data-effect=\"italics\">b<\/em>\u201d when <em data-effect=\"italics\">a<\/em> is to the right of <em data-effect=\"italics\">b<\/em> on the number line<\/li>\n<\/ul>\n<p id=\"fs-id1170655060667\">Where are 0.04 and 0.40 located on the number line? See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_013\">Figure 6<\/a>.<\/p>\n\n<div id=\"CNX_ElemAlg_Figure_01_08_013\" class=\"bc-figure figure\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"676\"]<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_013_new.jpg\" alt=\"There is a number line shown that runs from negative 0.0 to 1.0. From left to right, there are points 0.04 and 0.4 marked. The point 0.04 is between 0.0 and 0.1. The point 0.4 is between 0.3 and 0.5.\" width=\"676\" height=\"58\" data-media-type=\"image\/jpeg\"> Figure 6[\/caption]\n\n<\/div>\n<p id=\"fs-id1170654934978\">We see that 0.40 is to the right of 0.04 on the number line. This is another way to demonstrate that 0.40 &gt; 0.04<\/p>\n<p id=\"fs-id1170654934983\">How does 0.31 compare to 0.308? This doesn\u2019t translate into money to make it easy to compare. But if we convert 0.31 and 0.308 into fractions, we can tell which is larger.<\/p>\n\n<table id=\"eip-id1169746116939\" style=\"width: 100%;\" summary=\"Two numbers are given: 0.31 and 0.308. There is a table with instructions on the left and mathematical steps on the right. In the first row we have \u201cConvert to fractions.\u201d To the right of this we have 31\/100 and 308\/1000. In the following row, it says \u201cWe need a common denominator to compare them. To the right of this we have the quantity (31 times 10) over the quantity (100 times 10). The fraction 308\/1000 remains the same. Below this, we have the fractions 310\/1000 and 308 over 1000.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>0.31<\/td>\n<td>0.308<\/td>\n<\/tr>\n<tr>\n<td>Convert to fractions.<\/td>\n<td>\\(\\frac{31}{100}\\)<\/td>\n<td>\\(\\frac{308}{1000}\\)<\/td>\n<\/tr>\n<tr>\n<td>We need a common denominator to compare them.<\/td>\n<td><span id=\"eip-id1169746116992\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_014a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<td><span id=\"eip-id1169746117002\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_014b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>\\(\\frac{310}{1000}\\)<\/td>\n<td>\\(\\frac{308}{1000}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170654943714\">Because 310 &gt; 308, we know that \\(\\frac{310}{1000}\\) &gt; \\(\\frac{308}{1000}\\). Therefore, 0.31 &gt; 0.308<\/p>\n<p id=\"fs-id1170654935899\">Notice what we did in converting 0.31 to a fraction\u2014we started with the fraction \\(\\frac{31}{100}\\) and ended with the equivalent fraction \\(\\frac{310}{1000}\\). Converting \\(\\frac{310}{1000}\\) back to a decimal gives 0.310. So 0.31 is equivalent to 0.310. Writing zeros at the end of a decimal does not change its value!<\/p>\n\n<div id=\"fs-id1166420392202\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\frac{31}{100}=\\frac{310}{1000}\\phantom{\\rule{1em}{0ex}}\\text{and}\\phantom{\\rule{1em}{0ex}}0.31=0.310\\)<\/div>\nWe say 0.31 and 0.310 are equivalent decimals.\n<div data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Equivalent Decimals<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nTwo decimals are equivalent if they convert to equivalent fractions.\n\n<\/div>\n<\/div>\nWe use equivalent decimals when we order decimals.\n\n<\/div>\n<\/div>\n<p id=\"fs-id1170654944395\">The steps we take to order decimals are summarized here.<\/p>\n\n<div id=\"fs-id1170654944399\" class=\"howto\" data-type=\"note\">\n<div data-type=\"title\">\n<div>\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">HOW TO: Order Decimals.<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<ol id=\"fs-id1166422524060\" class=\"stepwise\" type=\"1\">\n \t<li>Write the numbers one under the other, lining up the decimal points.<\/li>\n \t<li>Check to see if both numbers have the same number of digits. If not, write zeros at the end of the one with fewer digits to make them match.<\/li>\n \t<li>Compare the numbers as if they were whole numbers.<\/li>\n \t<li>Order the numbers using the appropriate inequality sign.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 10<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id1170655083061\">Order \\(0.64\\rule{2em}{0.4pt}0.6\\) using &lt; or &gt;.<\/p>\n\n<\/div>\n<div id=\"fs-id1170654943856\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<table id=\"eip-61\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td>Write the numbers one under the other, lining up the decimal points.<\/td>\n<td>\\(\\begin{array}{c}0.64\\\\ 0.6\\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td>Add a zero to 0.6 to make it a decimal with 2 decimal places.\nNow they are both hundredths.<\/td>\n<td>\\(\\begin{array}{c}0.64\\\\ 0.60\\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td>64 is greater than 60.<\/td>\n<td>\\(64\\) &gt; \\(60\\)<\/td>\n<\/tr>\n<tr>\n<td>64 hundredths is greater than 60 hundredths.<\/td>\n<td>\\(0.64\\) &gt; \\(0.60\\)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>\\(0.64\\) &gt; \\(0.6\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 10<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655162906\" data-type=\"problem\">\n<p id=\"fs-id1170655162908\">Order each of the following pairs of numbers, using \\(&lt;\\phantom{\\rule{0.2em}{0ex}}\\text{or}\\phantom{\\rule{0.2em}{0ex}}\\) &gt; \\(\\phantom{\\rule{0.2em}{0ex}}\\text{:}\\phantom{\\rule{0.2em}{0ex}}0.42\\rule{2em}{0.4pt}0.4\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1170654903658\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1170654903660\">&gt;<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 11<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654903625\" data-type=\"problem\">\n<p id=\"fs-id1170654903627\">Order \\(0.83\\rule{2em}{0.4pt}0.803\\) using &lt; or &gt;.<\/p>\n\n<\/div>\n<div id=\"fs-id1170655059396\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-28\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>\\(0.83\\rule{2em}{0.4pt}0.803\\)<\/td>\n<\/tr>\n<tr>\n<td>Write the numbers one under the other, lining up the decimals.<\/td>\n<td>\\(\\begin{array}{c}0.83\\\\ 0.803\\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td>They do not have the same number of digits.\nWrite one zero at the end of 0.83.<\/td>\n<td>\\(\\begin{array}{c}0.830\\\\ 0.803\\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td>Since \\(830\\) &gt; \\(803\\), 830 thousandths is greater than 803 thousandths.<\/td>\n<td>\\(0.830\\) &gt; \\(0.803\\)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>\\(0.83\\) &gt; \\(0.803\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 11<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654944217\" data-type=\"problem\">\n<p id=\"fs-id1170654944219\">Order the following pair of numbers, using \\(&lt;\\phantom{\\rule{0.2em}{0ex}}\\text{or}\\phantom{\\rule{0.2em}{0ex}}\\) &gt; \\(\\phantom{\\rule{0.2em}{0ex}}\\text{:}\\phantom{\\rule{0.2em}{0ex}}0.76\\rule{2em}{0.4pt}0.706\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1170655090488\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1170655090490\">&gt;<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170654943214\">When we order negative decimals, it is important to remember how to order negative integers. Recall that larger numbers are to the right on the number line. For example, because \\(-2\\) lies to the right of \\(-3\\) on the number line, we know that \\(-2\\) &gt; \\(-3\\). Similarly, smaller numbers lie to the left on the number line. For example, because \\(-9\\) lies to the left of \\(-6\\) on the number line, we know that \\(-9&lt;-6\\). See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_015\">Figure 7<\/a>.<\/p>\n\n<div id=\"CNX_ElemAlg_Figure_01_08_015\" class=\"bc-figure figure\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"656\"]<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_015_new.jpg\" alt=\"There is a number line shown that runs from negative 10 to 0. There are not points given and the hashmarks exist at every integer between negative 10 and 0.\" width=\"656\" height=\"40\" data-media-type=\"image\/jpeg\"> Figure 7[\/caption]\n\n<\/div>\n<p id=\"fs-id1170654941319\">If we zoomed in on the interval between 0 and \\(-1\\), as shown in <a class=\"autogenerated-content\" href=\"#fs-id1170655098079\">Example 10<\/a>, we would see in the same way that \\(-0.2\\) &gt; \\(-0.3\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}-0.9&lt;-0.6\\).<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 12<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655098079\" data-type=\"problem\">\n\nUse &lt; or &gt; to order \\(-0.1\\rule{2em}{0.4pt}-0.8\\).\n\n<\/div>\n<div id=\"fs-id1170655060964\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-770\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>\\(-0.1\\rule{2em}{0.4pt}-0.8\\)<\/td>\n<\/tr>\n<tr>\n<td>Write the numbers one under the other, lining up the decimal points.\nThey have the same number of digits.<\/td>\n<td>\\(\\begin{array}{c}-0.1\\\\ -0.8\\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td>Since \\(-1\\) &gt; \\(-8\\), \u22121 tenth is greater than \u22128 tenths.<\/td>\n<td>\\(-0.1\\) &gt; \\(-0.8\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 12<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655065251\" data-type=\"problem\">\n<p id=\"fs-id1170655099866\">Order the following pair of numbers, using &lt; or &gt;: \\(-0.3\\rule{2em}{0.4pt}-0.5\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1170655059478\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1170655059480\">&gt;<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<h1>Key Concepts<\/h1>\n<ul id=\"fs-id1170654944002\" data-bullet-style=\"bullet\">\n \t<li><strong data-effect=\"bold\">Order Decimals<\/strong>\n<ol id=\"fs-id1170654942859\" class=\"stepwise\" type=\"1\">\n \t<li>Write the numbers one under the other, lining up the decimal points.<\/li>\n \t<li>Check to see if both numbers have the same number of digits. If not, write zeros at the end of the one with fewer digits to make them match.<\/li>\n \t<li>Compare the numbers as if they were whole numbers.<\/li>\n \t<li>Order the numbers using the appropriate inequality sign.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<h1>Glossary<\/h1>\n<div class=\"textbox shaded\">\n<dl id=\"fs-id1166425286362\">\n \t<dt>equivalent decimals<\/dt>\n \t<dd id=\"fs-id1166425286367\">Two decimals are equivalent if they convert to equivalent fractions.<\/dd>\n<\/dl>\n<dl id=\"fs-id1166425286372\">\n \t<dt>irrational number<\/dt>\n \t<dd id=\"fs-id1166425286377\">An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.<\/dd>\n<\/dl>\n<dl id=\"fs-id1166424794533\">\n \t<dt>rational number<\/dt>\n \t<dd id=\"fs-id1166424794538\">A rational number is a number of the form \\(\\frac{p}{q}\\), where <em data-effect=\"italics\">p<\/em> and <em data-effect=\"italics\">q<\/em> are integers and \\(q\\ne 0\\). A rational number can be written as the ratio of two integers. Its decimal form stops or repeats.<\/dd>\n<\/dl>\n<dl id=\"fs-id1166424761145\">\n \t<dt>real number<\/dt>\n \t<dd id=\"fs-id1166424794538\">A real number is a number that is either rational or irrational<\/dd>\n<\/dl>\n<\/div>\n<h1 id=\"fs-id1170655029650\">1.7 Exercise Set<\/h1>\n<p id=\"fs-id1166422646844\">In the following exercises, write as the ratio of two integers.<\/p>\n\n<ol class=\"twocolumn\">\n \t<li>\n<ol type=\"a\">\n \t<li>5<\/li>\n \t<li>3.19<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(-12\\phantom{\\rule{0.2em}{0ex}}\\)<\/li>\n \t<li>9.279<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<div id=\"fs-id1170655029657\" data-type=\"exercise\">\n<div id=\"fs-id1170655073127\" data-type=\"problem\">In the following exercises, list the a) rational numbers, b) irrational numbers<\/div>\n<ol class=\"twocolumn\" start=\"3\">\n \t<li data-type=\"problem\">\\(0.75,0.22\\stackrel{-}{3},1.39174\\)<\/li>\n \t<li data-type=\"problem\">\\(0.4\\stackrel{-}{5},1.919293\\text{\u2026},3.59\\)<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1170655037954\" data-type=\"exercise\">\n<div data-type=\"problem\">\n<div id=\"fs-id1169144768977\" data-type=\"exercise\">\n<div id=\"fs-id1169144768979\" data-type=\"problem\"><span style=\"font-size: 14pt; text-align: initial;\">In the following exercises, list the a) whole numbers, b) integers, c) rational numbers, d) irrational numbers, e) real numbers for each set of numbers.<\/span><\/div>\n<ol class=\"twocolumn\" start=\"5\">\n \t<li data-type=\"problem\">\\(-8,0,\\sqrt[5]{-32} , 1.95286\\text{\u2026},\\frac{12}{5},\\sqrt[2]{-9},\\sqrt[3]{9}\\)<\/li>\n \t<li data-type=\"problem\">\\(-7,\\sqrt[3]{512} ,-\\phantom{\\rule{0.2em}{0ex}}\\frac{8}{3},-1, \\sqrt[4]{-75}, 0.77,3\\frac{1}{4}\\)<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170655029153\" data-type=\"exercise\">\n<div id=\"fs-id1170655062368\" data-type=\"problem\">\n<p id=\"fs-id1166425252416\">In the following exercises, locate the numbers on a number line.<\/p>\n\n<ol class=\"twocolumn\" start=\"7\">\n \t<li>\\(\\frac{3}{4},\\frac{8}{5},\\frac{10}{3}\\)<\/li>\n \t<li>\\(\\frac{3}{10},\\frac{7}{2},\\frac{11}{6},4\\)<\/li>\n \t<li>\\(\\frac{2}{5},-\\phantom{\\rule{0.2em}{0ex}}\\frac{2}{5}\\)<\/li>\n \t<li>\\(\\frac{3}{4},-\\phantom{\\rule{0.2em}{0ex}}\\frac{3}{4},1\\frac{2}{3},-1\\frac{2}{3},\\frac{5}{2},-\\phantom{\\rule{0.2em}{0ex}}\\frac{5}{2}\\)<\/li>\n<\/ol>\n<div id=\"fs-id1170655088969\" data-type=\"exercise\">\n<div id=\"fs-id1170655084984\" data-type=\"problem\">In the following exercises, order each of the pairs of numbers, using &lt; or &gt;.<\/div>\n<ol class=\"twocolumn\" start=\"11\">\n \t<li data-type=\"problem\">\\(-1\\rule{2em}{0.4pt}-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{4}\\)<\/li>\n \t<li data-type=\"problem\">\\(-2\\frac{1}{2}\\rule{2em}{0.4pt}-3\\)<\/li>\n \t<li data-type=\"problem\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{5}{12}\\rule{2em}{0.4pt}-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{12}\\)<\/li>\n \t<li data-type=\"problem\">\\(-3\\rule{2em}{0.4pt}-\\phantom{\\rule{0.2em}{0ex}}\\frac{13}{5}\\)<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1170653193072\" data-type=\"exercise\">\n<div id=\"fs-id1170653193074\" data-type=\"problem\">Locate Decimals on the Number Line In the following exercises, locate the number on the number line.<\/div>\n<ol class=\"twocolumn\" start=\"15\">\n \t<li data-type=\"problem\">0.8<\/li>\n \t<li data-type=\"problem\">\\(-1.6\\)<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1170655096647\" data-type=\"exercise\">\n<div id=\"fs-id1170655096649\" data-type=\"problem\">In the following exercises, order each pair of numbers, using &lt; or &gt;.<\/div>\n<ol class=\"twocolumn\" start=\"17\">\n \t<li data-type=\"problem\">\\(0.37\\rule{2em}{0.4pt}0.63\\)<\/li>\n \t<li data-type=\"problem\">\\(0.91\\rule{2em}{0.4pt}0.901\\)<\/li>\n \t<li data-type=\"problem\">\\(-0.5\\rule{2em}{0.4pt}-0.3\\)<\/li>\n \t<li data-type=\"problem\">\\(-0.62\\rule{2em}{0.4pt}-0.619\\)<\/li>\n<\/ol>\n<ol start=\"21\">\n \t<li><strong>Child care.<\/strong> Serena wants to open a licensed child care center. Her state requires there be no more than 12 children for each teacher. She would like her child care centre to serve 40 children.\n<ol type=\"a\">\n \t<li style=\"list-style-type: none;\">\n<ol type=\"a\">\n \t<li>How many teachers will be needed?<\/li>\n \t<li>Why must the answer be a whole number?<\/li>\n \t<li>Why shouldn\u2019t you round the answer the usual way, by choosing the whole number closest to the exact answer?<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2 data-type=\"title\"><span style=\"font-size: 1.2em;\">Answers:<\/span><\/h2>\n<ol>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(\\frac{5}{1}\\)<\/li>\n \t<li>\\(\\frac{319}{100}\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(\\frac{-12}{1}\\)<\/li>\n \t<li>\\(\\frac{9297}{1000}\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(0.75,0.22\\stackrel{-}{3}\\)<\/li>\n \t<li>\\(1.39174\\text{\u2026}\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(0.4\\stackrel{-}{5},3.59\\)<\/li>\n \t<li>\\(1.919293\\text{\u2026}\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(0\\)<\/li>\n \t<li>\\(-8,0, \\sqrt[5]{-32}\\)<\/li>\n \t<li>\\(-8,0,\\sqrt[5]{-32}, \\frac{12}{5}\\)<\/li>\n \t<li>\\(1.95286\\text{\u2026}, \\sqrt[3]{9}\\)<\/li>\n \t<li>\\(-8,0, \\sqrt[5]{-32}, 1.95286\\text{\u2026},\\frac{12}{5},\\sqrt[3]{9} \\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(\\sqrt[3]{512}\\)<\/li>\n \t<li>\\(-7,-1, \\(\\sqrt[3]{512}\\)<\/li>\n \t<li>-7, \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{8}{3},-1,0.77,3\\frac{1}{4}, \\(\\sqrt[3]{512} \\)<\/li>\n \t<li>none<\/li>\n \t<li>\\(-7,-\\phantom{\\rule{0.2em}{0ex}}\\frac{8}{3}, -1,0.77,3\\frac{1}{4}, \\(\\sqrt[3]{512} \\)<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><span id=\"fs-id1170655080413\" data-type=\"media\" data-alt=\"There is a number line shown that runs from 0 to 6. From left to right the points read 3\/4, 8\/5, and 10\/3. The point for 3\/4 is between 0 and 1. The point for 8\/5 is between 1 and 2. The point for 10\/3 is between 3 and 4.\">7.<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_202_img_new.jpg\" alt=\"There is a number line shown that runs from 0 to 6. From left to right the points read 3\/4, 8\/5, and 10\/3. The point for 3\/4 is between 0 and 1. The point for 8\/5 is between 1 and 2. The point for 10\/3 is between 3 and 4.\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<td style=\"width: 50%;\">\u00a08. <span id=\"fs-id1170653193526\" data-type=\"media\" data-alt=\"There is a number line shown that runs from 0 to 6. From left to right the points read 3\/10, 11\/6, 7\/2, and 4. The point for 3\/10 is between 0 and 1. The point for 11\/6 is between 1 and 2. The point for 7\/2 is between 3 and 4.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_203_img_new.jpg\" alt=\"There is a number line shown that runs from 0 to 6. From left to right the points read 3\/10, 11\/6, 7\/2, and 4. The point for 3\/10 is between 0 and 1. The point for 11\/6 is between 1 and 2. The point for 7\/2 is between 3 and 4.\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">9. <span id=\"fs-id1170655353874\" data-type=\"media\" data-alt=\"There is a number line shown that runs from negative 1 to 1. From left to right the points read negative 2\/5 and 2\/5. The point for negative 2\/5 is between negative 1 and 0. The point for 2\/5 is between 0 and 1.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_205_img_new.jpg\" alt=\"There is a number line shown that runs from negative 1 to 1. From left to right the points read negative 2\/5 and 2\/5. The point for negative 2\/5 is between negative 1 and 0. The point for 2\/5 is between 0 and 1.\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<td style=\"width: 50%;\">10. <span id=\"fs-id1170655124914\" data-type=\"media\" data-alt=\"There is a number line shown that runs from negative 4 to 4. From left to right the points read negative 5\/2, negative 1 and 2\/3, negative 3\/4, \u00be, 1 and 2\/3, and 5\/2. The point for negative 5\/2 is between negative 3 and negative 2. The point for negative 1 and 2\/3 is between negative 2 and negative 1. The point for negative 3\/4 is between negative 1 and 0. The point for 3\/4 is between 0 and 1. The point for 1 and 2\/3 is between 1 and 2. The point for 5\/2 is between 2 and 3.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_207_img_new.jpg\" alt=\"There is a number line shown that runs from negative 4 to 4. From left to right the points read negative 5\/2, negative 1 and 2\/3, negative 3\/4, \u00be, 1 and 2\/3, and 5\/2. The point for negative 5\/2 is between negative 3 and negative 2. The point for negative 1 and 2\/3 is between negative 2 and negative 1. The point for negative 3\/4 is between negative 1 and 0. The point for 3\/4 is between 0 and 1. The point for 1 and 2\/3 is between 1 and 2. The point for 5\/2 is between 2 and 3.\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<dl id=\"fs-id1166424761164\"><\/dl>\n<ol start=\"11\">\n \t<li>&lt;<\/li>\n \t<li>&gt;<\/li>\n \t<li>&gt;<\/li>\n \t<li>&lt;<\/li>\n<\/ol>\n<table style=\"border-collapse: collapse; width: 100%; height: 120px;\" border=\"0\">\n<tbody>\n<tr style=\"height: 17px;\">\n<td style=\"width: 33.2078%; height: 17px;\">15. <span id=\"fs-id1170655095903\" data-type=\"media\" data-alt=\"There is a number line shown that runs from negative 4 to 4. The point 0.8 is between 0 and 1.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_209_img_new.jpg\" alt=\"There is a number line shown that runs from negative 4 to 4. The point 0.8 is between 0 and 1.\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<td style=\"width: 33.4588%; height: 17px;\">16. <span id=\"fs-id1170654941068\" data-type=\"media\" data-alt=\"There is a number line shown that runs from negative 4 to 4. The point negative 1.6 is between negative 2 and negative 1.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_211_img_new.jpg\" alt=\"There is a number line shown that runs from negative 4 to 4. The point negative 1.6 is between negative 2 and negative 1.\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol start=\"17\">\n \t<li style=\"list-style-type: none;\">\n<ol start=\"17\">\n \t<li>&lt;<\/li>\n \t<li>&gt;<\/li>\n \t<li>&lt;<\/li>\n \t<li>&lt;<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>4 buses<\/li>\n \t<li>answers may vary<\/li>\n \t<li>answers may vary<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<h1>Attributions<\/h1>\nThis chapter has been adapted from \u201cThe Real Numbers\u201d in <a href=\"https:\/\/openstax.org\/details\/books\/elementary-algebra\"><em>Elementary Algebra<\/em><\/a> (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a <a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY 4.0 Licence<\/a>. Adapted by Izabela Mazur. See the Copyright page for more information.","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>By the end of this section it is expected that you will be able to:<\/p>\n<ul>\n<li>Identify integers, rational numbers, irrational numbers, and real numbers<\/li>\n<li>Locate fractions on the number line<\/li>\n<li>Locate decimals on the number line<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h1>Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers<\/h1>\n<p id=\"fs-id1170654983255\">We have already described numbers as <span class=\"no-emphasis\" data-type=\"term\"><em data-effect=\"italics\">counting number<\/em><em data-effect=\"italics\">s<\/em><\/span>, <span class=\"no-emphasis\" data-type=\"term\"><em data-effect=\"italics\">whole number<\/em><em data-effect=\"italics\">s<\/em><\/span>, and <span class=\"no-emphasis\" data-type=\"term\"><em data-effect=\"italics\">integers<\/em><\/span>. What is the difference between these types of numbers?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bf2d69fb260e39309d9776fd7f44ed0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#111;&#117;&#110;&#116;&#105;&#110;&#103;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#49;&#44;&#50;&#44;&#51;&#44;&#52;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#104;&#111;&#108;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#48;&#44;&#49;&#44;&#50;&#44;&#51;&#44;&#52;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#110;&#116;&#101;&#103;&#101;&#114;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#45;&#51;&#44;&#45;&#50;&#44;&#45;&#49;&#44;&#48;&#44;&#49;&#44;&#50;&#44;&#51;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"61\" width=\"387\" style=\"vertical-align: -26px;\" \/><\/p>\n<p id=\"fs-id1170655000002\">What type of numbers would we get if we started with all the integers and then included all the fractions? The numbers we would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.<\/p>\n<div id=\"fs-id1170655026966\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Rational Number<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A <strong data-effect=\"bold\">rational number<\/strong> is a number of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b9606a6a8bc8c28efca6d4be67f31e5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#125;&#123;&#113;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"8\" style=\"vertical-align: -9px;\" \/>, where <em data-effect=\"italics\">p<\/em> and <em data-effect=\"italics\">q<\/em> are integers and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2e3f318f8865aafe5e97953eb6c5e2ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"41\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p>A rational number can be written as the ratio of two integers.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>All signed fractions, such as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1f59080f1163547c886c23ae0cb49bc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#44;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#56;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#51;&#125;&#123;&#52;&#125;&#44;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#48;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"111\" style=\"vertical-align: -6px;\" \/> are rational numbers. Each numerator and each denominator is an integer.<\/p>\n<p id=\"fs-id1170654944763\">Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. Each integer can be written as a ratio of integers in many ways. For example, 3 is equivalent to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ff176d646cff725cd406a98246e110af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#125;&#123;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"95\" style=\"vertical-align: -7px;\" \/><\/p>\n<p id=\"fs-id1170654984452\">An easy way to write an integer as a ratio of integers is to write it as a fraction with denominator one.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d97e5959058f499c41339514e0093e95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#56;&#61;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#48;&#125;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"256\" style=\"vertical-align: -7px;\" \/><\/p>\n<p id=\"fs-id1170655195958\">Since any integer can be written as the ratio of two integers, <em data-effect=\"italics\">all integers are rational numbers<\/em>! Remember that the counting numbers and the whole numbers are also integers, and so they, too, are rational.<\/p>\n<p id=\"fs-id1170655059465\">What about decimals? Are they rational? Let\u2019s look at a few to see if we can write each of them as the ratio of two integers.<\/p>\n<p id=\"fs-id1170654928833\">We\u2019ve already seen that integers are rational numbers. The integer <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ca32393b1b5af7c55a95d89cf9d610f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> could be written as the decimal <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c96697c04ae3cbfe7de993ec4cc7faea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#46;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"36\" style=\"vertical-align: 0px;\" \/>. So, clearly, some decimals are rational.<\/p>\n<p id=\"fs-id1170655083456\">Think about the decimal 7.3. Can we write it as a ratio of two integers? Because 7.3 means <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6b215732248857a9b3a88182283bc202_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/>, we can write it as an improper fraction, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-40f4f6193e9a9069d9112e26dd128e5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#51;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/>. So 7.3 is the ratio of the integers 73 and 10. It is a rational number.<\/p>\n<p id=\"fs-id1170655026632\">In general, any decimal that ends after a number of digits (such as 7.3 or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-cfc628917a3e1eb86891becfb753a5ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#46;&#50;&#54;&#56;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"63\" style=\"vertical-align: 0px;\" \/> is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654943006\" data-type=\"problem\">\n<p id=\"fs-id1170655111565\">Write as the ratio of two integers: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-95f4287cdd4dbed37de8e3c463100f37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/> b) 7.31<\/p>\n<\/div>\n<div id=\"fs-id1170655163665\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<table id=\"eip-639\" summary=\"-\">\n<tbody>\n<tr>\n<td>a)<br \/>\nWrite it as a fraction with denominator 1.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-942404a5107cc5134553a1e70c2dc42e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#50;&#55;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#55;&#125;&#123;&#49;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"31\" style=\"vertical-align: -18px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>b)<br \/>\nWrite it as a mixed number. Remember, 7 is the whole number and the decimal part, 0.31, indicates hundredths.<br \/>\nConvert to an improper fraction.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b1393f503fcfbe22832bb1cf60cd4e89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#55;&#46;&#51;&#49;&#92;&#92;&#32;&#55;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#49;&#125;&#123;&#49;&#48;&#48;&#125;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#51;&#49;&#125;&#123;&#49;&#48;&#48;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"32\" style=\"vertical-align: -29px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170655128317\">So we see that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-95f4287cdd4dbed37de8e3c463100f37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/> and 7.31 are both rational numbers, since they can be written as the ratio of two integers.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id1170654982461\">Write as the ratio of two integers: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d15aa1d60ab00024197b84d5e3e3d75e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> b) 3.57<\/p>\n<\/div>\n<div id=\"fs-id1170655098810\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170655083207\">a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-43129ba242c83b1617c4e2d884f863b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#52;&#125;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"25\" style=\"vertical-align: -7px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-57f156260dd7efb57ba7ac7807e3e4e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#53;&#55;&#125;&#123;&#49;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"21\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170654988762\">Let\u2019s look at the decimal form of the numbers we know are rational.<\/p>\n<p id=\"fs-id1170654905314\">We have seen that <em data-effect=\"italics\">every<\/em> <em data-effect=\"italics\">integer is a rational number<\/em>, since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6ec739fb75662dd29f7282286aad4a84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"43\" style=\"vertical-align: -7px;\" \/> for any integer, <em data-effect=\"italics\">a<\/em>. We can also change any integer to a decimal by adding a decimal point and a zero.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 14.2857%;\" scope=\"row\">Integer<\/th>\n<td style=\"width: 14.2857%;\">-2<\/td>\n<td style=\"width: 14.2857%;\">-1<\/td>\n<td style=\"width: 14.2857%;\">0<\/td>\n<td style=\"width: 14.2857%;\">1<\/td>\n<td style=\"width: 14.2857%;\">2<\/td>\n<td style=\"width: 14.2857%;\">3<\/td>\n<\/tr>\n<tr>\n<th style=\"width: 14.2857%;\" scope=\"row\">Decimal form<\/th>\n<td style=\"width: 14.2857%;\">-2.0<\/td>\n<td style=\"width: 14.2857%;\">-1.0<\/td>\n<td style=\"width: 14.2857%;\">0.0<\/td>\n<td style=\"width: 14.2857%;\">1.0<\/td>\n<td style=\"width: 14.2857%;\">2.0<\/td>\n<td style=\"width: 14.2857%;\">3.0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>These decimal numbers stop.<\/p>\n<p id=\"fs-id1170655108063\">We have also seen that <em data-effect=\"italics\">every<\/em> <em data-effect=\"italics\">fraction is a rational number<\/em>. Look at the decimal form of the fractions we considered above.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 128px;\">\n<tbody>\n<tr style=\"height: 55px;\">\n<th style=\"width: 20%; height: 55px;\" scope=\"row\">Ratio of integers<\/th>\n<td style=\"width: 20%; height: 55px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c995a9c919f066bf6863b7f22d9cc88f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 20%; height: 55px;\">&#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-27a06edc5fe1932bd0c3274d9af93920_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 20%; height: 55px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-603f954d2dfd7453214afc11a07fab94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 20%; height: 55px;\">&#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-eda29b4ce6e30ac27eb5c46dc5eba6ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#48;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 55px;\">\n<th style=\"width: 20%; height: 55px;\" scope=\"row\">The decimal form<\/th>\n<td style=\"width: 20%; height: 55px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1e4419032e9b2a7c997243dc95aced4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"23\" style=\"vertical-align: 0px;\" \/><\/td>\n<td style=\"width: 20%; height: 55px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d079f44bac785047ece87b816dc283c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#56;&#55;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"53\" style=\"vertical-align: 0px;\" \/><\/td>\n<td style=\"width: 20%; height: 55px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a005928d9ea75d0170b694f5cffc8c65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/td>\n<td style=\"width: 20%; height: 55px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c400b813f7ab8fe85b10531ef3e64127_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#46;&#54;&#54;&#54;&#46;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"63\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>These decimals either stop or repeat.<\/p>\n<p id=\"fs-id1170654986576\">What do these examples tell us?<\/p>\n<p id=\"fs-id1170655196916\"><em data-effect=\"italics\">Every rational number can be written both as a ratio of integers<\/em>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b9606a6a8bc8c28efca6d4be67f31e5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#125;&#123;&#113;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"8\" style=\"vertical-align: -9px;\" \/>,<em data-effect=\"italics\">where p and q are integers and <\/em><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2e3f318f8865aafe5e97953eb6c5e2ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"41\" style=\"vertical-align: -4px;\" \/>,<em data-effect=\"italics\">and as a decimal that either stops or repeats.<\/em><\/p>\n<p id=\"fs-id1170654942363\">Here are the numbers we looked at above expressed as a ratio of integers and as a decimal:<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<caption>Fractions<\/caption>\n<tbody>\n<tr valign=\"top\">\n<th style=\"width: 5.77386%;\" scope=\"row\" data-valign=\"top\" data-align=\"left\">Number<\/th>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c995a9c919f066bf6863b7f22d9cc88f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9dbe942ff30fccbe3cafef9973811e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 6.89655%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-603f954d2dfd7453214afc11a07fab94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-999fd045616dd916304b645dcc9eb04f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#48;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<th style=\"width: 5.77386%;\" scope=\"row\" data-valign=\"top\" data-align=\"left\">Ratio of Integers<\/th>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c995a9c919f066bf6863b7f22d9cc88f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9dbe942ff30fccbe3cafef9973811e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 6.89655%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-603f954d2dfd7453214afc11a07fab94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-999fd045616dd916304b645dcc9eb04f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#48;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<th style=\"width: 5.77386%;\" scope=\"row\" data-valign=\"top\" data-align=\"left\">Decimal Form<\/th>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1e4419032e9b2a7c997243dc95aced4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"23\" style=\"vertical-align: 0px;\" \/><\/td>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d079f44bac785047ece87b816dc283c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#56;&#55;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"53\" style=\"vertical-align: 0px;\" \/><\/td>\n<td style=\"width: 6.89655%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a005928d9ea75d0170b694f5cffc8c65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/td>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3ab6c617e59f40e57616f410b08f8445_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#46;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<caption>Integers<\/caption>\n<tbody>\n<tr valign=\"top\">\n<th style=\"width: 5.77386%;\" scope=\"row\" data-valign=\"top\" data-align=\"left\">Number<\/th>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/td>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/><\/td>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a5e437be25f29374d30f66cd46adf81c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/td>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4868771cbc422b5818f85500909ce433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"7\" style=\"vertical-align: -1px;\" \/><\/td>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/td>\n<td style=\"width: 0.801925%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4a1d3ea4963f568cabd97329456036b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<th style=\"width: 5.77386%;\" scope=\"row\" data-valign=\"top\" data-align=\"left\">Ratio of Integers<\/th>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-85cb188f2ff69584857b4fa308ec8cca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"25\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b813c0b1b59eca458f88e38451483d86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"25\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ee1cce477463dd0bed4f613b38e20264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#48;&#125;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"7\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-68d581ab5be39d19aaf7b9cb740d5662_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"7\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-52d566d0ef86628a48b1bd293562662b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"7\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 0.801925%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b24e9c560dcf409991d36b54933232dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"7\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<th style=\"width: 5.77386%;\" scope=\"row\" data-valign=\"top\" data-align=\"left\">Decimal Form<\/th>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-369cac3502395cc4fac70051300db53f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#46;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"36\" style=\"vertical-align: 0px;\" \/><\/td>\n<td style=\"width: 14.1139%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ee8fa04fc2bf1dbbff787fc33f469ffe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#46;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"36\" style=\"vertical-align: -1px;\" \/><\/td>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0b0fea539d13a19676a3acbb7aec5534_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"23\" style=\"vertical-align: 0px;\" \/><\/td>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2b23fc34a5447966d71dba505f0d8e9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/><\/td>\n<td style=\"width: 6.17482%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3d3004dd9e4a8cd76474f9f0ac507f2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"23\" style=\"vertical-align: 0px;\" \/><\/td>\n<td style=\"width: 0.801925%;\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c38bed5d290bc84ef07f7bd5619e31e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"23\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1170655007226\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Rational Number<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A <strong data-effect=\"bold\">rational number<\/strong> is a number of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b9606a6a8bc8c28efca6d4be67f31e5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#125;&#123;&#113;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"8\" style=\"vertical-align: -9px;\" \/>, where <em data-effect=\"italics\">p<\/em> and <em data-effect=\"italics\">q<\/em> are integers and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2e3f318f8865aafe5e97953eb6c5e2ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"41\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p>Its decimal form stops or repeats.<\/p>\n<\/div>\n<\/div>\n<p>Are there any decimals that do not stop or repeat? Yes!<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170655027582\">The number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-26d6788550ffd50fe94542bb3e8ee615_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> (the Greek letter <em data-effect=\"italics\">pi<\/em>, pronounced \u201cpie\u201d), which is very important in describing circles, has a decimal form that does not stop or repeat.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-db7b7c0dd4c368acbdf78b452ba5f5d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#105;&#32;&#61;&#51;&#46;&#49;&#52;&#49;&#53;&#57;&#50;&#54;&#53;&#52;&#46;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"138\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<p id=\"fs-id1170655162537\">We can even create a decimal pattern that does not stop or repeat, such as<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-da93dd94b6087c3b6cfc1515cfa1d5fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#48;&#49;&#48;&#48;&#49;&#48;&#48;&#48;&#49;&#48;&#48;&#48;&#48;&#49;&#92;&#100;&#111;&#116;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"161\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"fs-id1170655097911\">Numbers whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call these numbers irrational. More on irrational numbers later on is this course.<\/p>\n<div id=\"fs-id1170655121425\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Irrational Number<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>An irrational number is a number that cannot be written as the ratio of two integers.<\/p>\n<p>Its decimal form does not stop and does not repeat.<\/p>\n<\/div>\n<\/div>\n<p>Let\u2019s summarize a method we can use to determine whether a number is rational or irrational.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170655353858\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Rational or Irrational?<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655353858\" data-type=\"note\">\n<p id=\"fs-id1170655353986\">If the decimal form of a number<\/p>\n<ul id=\"fs-id1170655107719\" data-bullet-style=\"bullet\">\n<li><em data-effect=\"italics\">repeats or stops<\/em>, the number is <strong data-effect=\"bold\">rational<\/strong>.<\/li>\n<li><em data-effect=\"italics\">does not repeat and does not stop<\/em>, the number is irrational<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170655128145\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1170655161217\" data-type=\"exercise\">\n<div id=\"fs-id1170655161219\" data-type=\"problem\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654967985\" data-type=\"problem\">\n<p id=\"fs-id1170654967987\">Given the numbers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d871fb9d10c19946bf29675d84038749_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#53;&#56;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#125;&#123;&#51;&#125;&#44;&#48;&#46;&#52;&#55;&#44;&#51;&#46;&#54;&#48;&#53;&#53;&#53;&#49;&#50;&#55;&#53;&#46;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"196\" style=\"vertical-align: -4px;\" \/>. list the a) rational numbers b) irrational numbers.<\/p>\n<\/div>\n<div id=\"fs-id1170655025175\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-798\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td>a)<br \/>\nLook for decimals that repeat or stop.<\/td>\n<td>The 3 repeats in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-75ca9b826c23f3d85618d1367696806f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#53;&#56;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -1px;\" \/>.<br \/>\nThe decimal 0.47 stops after the 7.<br \/>\nSo <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-31a9a8759b159028785bc2ceb74b16aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#53;&#56;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -1px;\" \/> and 0.47 are rational.<\/td>\n<\/tr>\n<tr>\n<td>b)<br \/>\nLook for decimals that neither stop nor repeat.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c98d2260e880ab3f85268383f4a3421a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#54;&#48;&#53;&#53;&#53;&#49;&#50;&#55;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"94\" style=\"vertical-align: -1px;\" \/> has no repeating block of digits and it does not stop.<br \/>\nSo <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c98d2260e880ab3f85268383f4a3421a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#54;&#48;&#53;&#53;&#53;&#49;&#50;&#55;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"94\" style=\"vertical-align: -1px;\" \/> is irrational.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655161219\" data-type=\"problem\">\n<p id=\"fs-id1170654943758\">For the given numbers list the a) rational numbers b) irrational numbers: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0bf1f73616a1030844256d73f9cee007_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#50;&#57;&#44;&#48;&#46;&#56;&#49;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#125;&#123;&#54;&#125;&#44;&#50;&#46;&#53;&#49;&#53;&#49;&#49;&#53;&#49;&#49;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"206\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654963540\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170655117899\">a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-82aa690d4f889245dbcf2f9bcc7a34b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#50;&#57;&#44;&#48;&#46;&#56;&#49;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"85\" style=\"vertical-align: -4px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-65bdfdd6261141355e511ed8c5c989a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#53;&#49;&#53;&#49;&#49;&#53;&#49;&#49;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"94\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170655128145\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1170655161217\" data-type=\"exercise\">\n<div id=\"fs-id1170655161219\" data-type=\"problem\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655218302\" data-type=\"problem\">\n<p id=\"fs-id1170655083134\">For each number given, identify whether it is rational or irrational: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-98077b985c4d3e263c7a4b085c65329f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-87505831bed95ee2989b5d4ae4a3f8e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1170655154779\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p style=\"padding-left: 40px;\">a) Recognize that 36 is a perfect square, since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-27ea3efffc0047b7ccf8ae62ae82b944_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#54;&#125;&#94;&#123;&#50;&#125;&#61;&#51;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"62\" style=\"vertical-align: 0px;\" \/> So <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bfe334e48313d3d2cce1b69f1ccaf61d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#54;&#125;&#61;&#54;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"68\" style=\"vertical-align: -4px;\" \/> therefore <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-98077b985c4d3e263c7a4b085c65329f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> is rational.<\/p>\n<p style=\"padding-left: 40px;\">b) Remember that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2dd37d6dd2f77686116d9bc09019fa9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#54;&#125;&#94;&#123;&#50;&#125;&#61;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"58\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3140118e82fdadeaeebd01eca45f4121_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#55;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#57;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"62\" style=\"vertical-align: -4px;\" \/> so 44 is not a perfect square. Therefore, the decimal form of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a13bd8e5e731a414b9dd4445c5b0260d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> will never repeat and never stop, so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a13bd8e5e731a414b9dd4445c5b0260d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> is irrational.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655083580\" data-type=\"problem\">\n<p id=\"fs-id1170655202060\">For each number given, identify whether it is rational or irrational: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3d78fc7bc47f4b31f9b5c66f4c786945_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-edb2860779e381702328f2852a201359_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#55;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170655216149\"><span class=\"token\">a)<\/span> rational b) irrational<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. The irrational numbers are numbers whose decimal form does not stop and does not repeat. When we put together the rational numbers and the irrational numbers, we get the set of <span data-type=\"term\">real number<\/span><strong data-effect=\"bold\">s<\/strong>.<\/p>\n<div id=\"fs-id1170655121069\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Real Number<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A <strong data-effect=\"bold\">real number<\/strong> is a number that is either rational or irrational.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"CNX_ElemAlg_Figure_01_08_001\" class=\"bc-figure figure\">\n<div>\n<p id=\"fs-id1170655000486\">All the numbers we use in algebra are real numbers. <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_001\">\u00a0Figure 1<\/a> illustrates how the number sets we\u2019ve discussed in this section fit together.<\/p>\n<div id=\"CNX_ElemAlg_Figure_01_08_001\" class=\"bc-figure figure\">\n<figure style=\"width: 654px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_01_08_001_new.jpg\" alt=\"This figure consists of a Venn diagram. To start there is a large rectangle marked Real Numbers. The right half of the rectangle consists of Irrational Numbers. The left half consists of Rational Numbers. Within the Rational Numbers rectangle, there are Integers \u2026, negative 2, negative 1, 0, 1, 2, \u2026. Within the Integers rectangle, there are Whole Numbers 0, 1, 2, 3, \u2026 Within the Whole Numbers rectangle, there are Counting Numbers 1, 2, 3, \u2026\" width=\"654\" height=\"396\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 1 This chart shows the number sets that make up the set of real numbers. Does the term \u201creal numbers\u201d seem strange to you? Are there any numbers that are not \u201creal,\u201d and, if so, what could they be?<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<p id=\"fs-id1170655082021\">Do you remember that the square root of a negative number was not a real number?<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654925513\" data-type=\"problem\">\n<p id=\"fs-id1170654925515\">For each number given, identify whether it is a real number or not a real number: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8c3b9ea68a1a2eb8f9f1cb89bf27632d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#54;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -3px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-be30997fbfbecdd027524fddb71dcf65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1170654944094\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<p>&nbsp;<\/p>\n<p><span class=\"token\">a)<\/span> There is no real number whose square is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-84c73b332360577423989d5ed3b23169_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#54;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: -1px;\" \/> Therefore, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8c3b9ea68a1a2eb8f9f1cb89bf27632d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#54;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -3px;\" \/> is not a real number.<\/p>\n<p>b) Since the negative is in front of the radical, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3b6616169b7490db4d681c4f04663aa8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -2px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-95b507f07085049276cbc062895c8807_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"26\" style=\"vertical-align: -4px;\" \/> Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ca32393b1b5af7c55a95d89cf9d610f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> is a real number, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3b6616169b7490db4d681c4f04663aa8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -2px;\" \/> is a real number.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655062079\" data-type=\"problem\">\n<p id=\"fs-id1170655062081\">For each number given, identify whether it is a real number or not a real number: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1154b34d2f8ec68cebe7a9283e374a45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#57;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -3px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b8ac7f0c2949653b991cd7c9e84ebaab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1170655164781\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170655164783\"><span class=\"token\">a)<\/span> not a real number b) real number<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170655062462\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1170655062465\" data-type=\"exercise\">\n<div id=\"fs-id1170655083274\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655162698\" data-type=\"problem\">\n<p id=\"fs-id1170655162700\">Given the numbers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5c5d5669448b554ca86dd345007e7c74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#125;&#123;&#53;&#125;&#44;&#56;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#44;&#53;&#46;&#57;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"176\" style=\"vertical-align: -6px;\" \/> list the a) whole numbers b) integers c) rational numbers d) irrational numbers e) real numbers.<\/p>\n<\/div>\n<div id=\"fs-id1170654943557\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1167835374083\"><span class=\"token\">a)<\/span> Remember, the whole numbers are 0, 1, 2, 3, \u2026 and 8 is the only whole number given.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<p><span class=\"token\">b)<\/span> The integers are the whole numbers, their opposites, and 0. So the whole number 8 is an integer, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-00c6d30c5f7439a21caf981437f64be1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: 0px;\" \/> is the opposite of a whole number so it is an integer, too. Also, notice that 64 is the square of 8 so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a8d9ea4d09d13d80d26750b73a030c5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#125;&#61;&#45;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -2px;\" \/> So the integers are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bdeec592c83f04413cec9b69c2f702f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#44;&#56;&#44;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"97\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<p><span class=\"token\">c)<\/span> Since all integers are rational, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d9e4261bab6d4ba37a837226be0427bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#44;&#56;&#44;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"93\" style=\"vertical-align: -4px;\" \/> are rational. Rational numbers also include fractions and decimals that repeat or stop, so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-81669546781ec8602982a147240e1e41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#125;&#123;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#46;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"74\" style=\"vertical-align: -6px;\" \/> are rational. So the list of rational numbers is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c8a13c0b9c56c12d425ee564cb34ae23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#125;&#123;&#53;&#125;&#44;&#56;&#44;&#53;&#46;&#57;&#44;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"153\" style=\"vertical-align: -6px;\" \/><span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<p>d) Remember that 5 is not a perfect square, so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7ba329d3f2f86c0fb47d848202e0ee7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\" \/> is irrational.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<p>e) All the numbers listed are real numbers.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655166461\" data-type=\"problem\">\n<p id=\"fs-id1170655166463\">For the given numbers, list the a) whole numbers b) integers c) rational numbers d) irrational numbers e) real numbers: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e9c816f38c47e0d2e39d68c2f2652f57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#44;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#44;&#48;&#46;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#53;&#125;&#44;&#52;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#57;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"182\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1170655098112\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170655106839\"><span class=\"token\">a) <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-acbce00f4d6bdd2ab7e2f9f29f41aa81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"50\" style=\"vertical-align: -4px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-78feb015bf7d488e9d493ee526306cd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#44;&#52;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"79\" style=\"vertical-align: -4px;\" \/> c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-27f904b95cf990567469dcd26bb4ce54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#44;&#48;&#46;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#53;&#125;&#44;&#52;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"133\" style=\"vertical-align: -6px;\" \/> d) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b8e59a7113f28c9f6309179eac0b8693_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"30\" style=\"vertical-align: -2px;\" \/> e) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e5d79ce1b72f2c386c82a24399cd72bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#44;&#48;&#46;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#53;&#125;&#44;&#52;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"171\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"CNX_ElemAlg_Figure_01_08_001\" class=\"bc-figure figure\"><\/div>\n<h1>Locate Fractions on the Number Line<\/h1>\n<p>The last time we looked at the <span class=\"no-emphasis\" data-type=\"term\">number line<\/span>, it only had positive and negative integers on it. We now want to include <span class=\"no-emphasis\" data-type=\"term\">fraction<\/span>s and decimals on it.<\/p>\n<p id=\"fs-id1170655091189\">Let\u2019s start with fractions and locate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-98c27d0083866e5b2a96c95a08cd3f44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#44;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#44;&#51;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#52;&#125;&#44;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#125;&#44;&#45;&#53;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"199\" style=\"vertical-align: -6px;\" \/> on the number line.<\/p>\n<p id=\"fs-id1170655203491\">We\u2019ll start with the whole numbers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4a1d3ea4963f568cabd97329456036b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b5b9d9f382b11767d19f257afca0019_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: 0px;\" \/>. because they are the easiest to plot. See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_002\">Figure 2<\/a>.<\/p>\n<p id=\"fs-id1170655353090\">The <span class=\"no-emphasis\" data-type=\"term\">proper fractions<\/span> listed are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-30e6a846480f6b8d0bfd23e9cdc650d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"79\" style=\"vertical-align: -6px;\" \/>. We know the proper fraction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e174a73678f6bd6f70d7aaec8f911349_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> has value less than one and so would be located between <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7e43f6f1ae753f402a682e1f0502d84d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#97;&#110;&#100;&#32;&#49;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"62\" style=\"vertical-align: -1px;\" \/> The denominator is 5, so we divide the unit from 0 to 1 into 5 equal parts <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e5576e47a03b6e188fce571a56918139_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"63\" style=\"vertical-align: -6px;\" \/>. We plot <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e174a73678f6bd6f70d7aaec8f911349_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/>. See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_002\">Figure 2<\/a>.<\/p>\n<p id=\"fs-id1170655188953\">Similarly, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ab05a56cbb64f6adca854aeeacc1b321_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/> is between 0 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/>. After dividing the unit into 5 equal parts we plot <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ab05a56cbb64f6adca854aeeacc1b321_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/>. See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_002\">Figure 2<\/a>.<\/p>\n<p id=\"fs-id1170655215831\">Finally, look at the <span class=\"no-emphasis\" data-type=\"term\">improper fractions<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9a4e82ac034b2748d4c4785db09d5ed4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#52;&#125;&#44;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"62\" style=\"vertical-align: -6px;\" \/>. These are fractions in which the numerator is greater than the denominator. Locating these points may be easier if you change each of them to a mixed number. See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_002\">Figure 2<\/a>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b94640586d0ea03a61508adebeebc82e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#52;&#125;&#61;&#49;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#125;&#61;&#45;&#52;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;&#61;&#50;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"287\" style=\"vertical-align: -6px;\" \/><\/p>\n<p id=\"fs-id1170654944810\"><a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_002\">Figure 2<\/a> shows the number line with all the points plotted.<\/p>\n<div id=\"CNX_ElemAlg_Figure_01_08_002\" class=\"bc-figure figure\">\n<figure style=\"width: 656px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_002_new.jpg\" alt=\"There is a number line shown that runs from negative 6 to positive 6. From left to right, the numbers marked are negative 5, negative 9\/2, negative 4\/5, 1\/5, 4\/5, 8\/3, and 3. The number negative 9\/2 is halfway between negative 5 and negative 4. The number negative 4\/5 is slightly to the right of negative 1. The number 1\/5 is slightly to the right of 0. The number 4\/5 is slightly to the left of 1. The number 8\/3 is between 2 and 3, but a little closer to 3.\" width=\"656\" height=\"75\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 2<\/figcaption><\/figure>\n<\/div>\n<div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655166545\" data-type=\"problem\">\n<p id=\"fs-id1170655166548\">Locate and label the following on a number line: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-48e3ce958bebf428e559f7674f37f1d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#44;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#44;&#45;&#51;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#53;&#125;&#44;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"201\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654905871\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1170654905873\">Locate and plot the integers, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ff61b3d56ac562f8eb2db2013b99a050_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#44;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"40\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p id=\"fs-id1170655098863\">Locate the proper fraction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-469181e98c8ec4b1bce8f57ea4ba4e31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> first. The fraction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-469181e98c8ec4b1bce8f57ea4ba4e31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> is between 0 and 1. Divide the distance between 0 and 1 into four equal parts then, we plot <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-469181e98c8ec4b1bce8f57ea4ba4e31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/>. Similarly plot <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3579c51ce3351cefb94bd858e9dc5596_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<p id=\"fs-id1170654941093\">Now locate the improper fractions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-17a6356131b1c2c8d511ea587e449553_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#53;&#125;&#44;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"62\" style=\"vertical-align: -6px;\" \/>. It is easier to plot them if we convert them to mixed numbers and then plot them as described above: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e0ef252356fff91c751a9747a0de1ff6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#53;&#125;&#61;&#49;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#44;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#61;&#45;&#50;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#51;&#125;&#61;&#50;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"205\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<p><span id=\"fs-id1170655098989\" data-type=\"media\" data-alt=\"There is a number line shown that runs from negative 6 to positive 6. From left to right, the numbers marked are negative 3, negative 5\/2, negative 1\/4, 3\/4, 6\/5, 7\/3, and 4. The number negative 5\/2 is halfway between negative 3 and negative 2. The number negative 1\/4 is slightly to the left of 0. The number 3\/4 is slightly to the left of 1. The number 6\/5 is slightly to the right of 1. The number 7\/3 is between 2 and 3, but a little closer to 2.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_003_img_new.jpg\" alt=\"There is a number line shown that runs from negative 6 to positive 6. From left to right, the numbers marked are negative 3, negative 5\/2, negative 1\/4, 3\/4, 6\/5, 7\/3, and 4. The number negative 5\/2 is halfway between negative 3 and negative 2. The number negative 1\/4 is slightly to the left of 0. The number 3\/4 is slightly to the left of 1. The number 6\/5 is slightly to the right of 1. The number 7\/3 is between 2 and 3, but a little closer to 2.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id1170655133229\">Locate and label the following on a number line: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5d26045e2beca6f577b4760237223e1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#53;&#125;&#44;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#125;&#44;&#53;&#44;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"164\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<details>\n<summary>Show answer<\/summary>\n<div id=\"fs-id1170654914689\" data-type=\"solution\"><span data-type=\"media\" data-alt=\"There is a number line shown that runs from negative 4 to positive 5. From left to right, the numbers marked are negative 8\/3, negative 7\/4, negative 1, 1\/3, 6\/5, 9\/2, and 5. The number negative 8\/3 is between negative 3 and negative 2 but slightly closer to negative 3. The number negative 7\/4 is slightly to the right of negative 2. The number 1\/3 is slightly to the right of 0. The number 6\/5 is slightly to the right of 1. The number 9\/2 is halfway between 4 and 5.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_004_img_new.jpg\" alt=\"There is a number line shown that runs from negative 4 to positive 5. From left to right, the numbers marked are negative 8\/3, negative 7\/4, negative 1, 1\/3, 6\/5, 9\/2, and 5. The number negative 8\/3 is between negative 3 and negative 2 but slightly closer to negative 3. The number negative 7\/4 is slightly to the right of negative 2. The number 1\/3 is slightly to the right of 0. The number 6\/5 is slightly to the right of 1. The number 9\/2 is halfway between 4 and 5.\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170655353172\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1170655353176\" data-type=\"exercise\">\n<div data-type=\"problem\">\n<p>In <a href=\"#fs-id1170655150929\">Example 5<\/a>, we\u2019ll use the inequality symbols to order fractions. In previous chapters we used the number line to order numbers.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<ul id=\"fs-id1166425210298\" data-bullet-style=\"bullet\">\n<li><em data-effect=\"italics\">a &lt; b<\/em> \u201c<em data-effect=\"italics\">a<\/em> is less than <em data-effect=\"italics\">b<\/em>\u201d when <em data-effect=\"italics\">a<\/em> is to the left of <em data-effect=\"italics\">b<\/em> on the number line<\/li>\n<li><em data-effect=\"italics\">a &gt; b<\/em> \u201c<em data-effect=\"italics\">a<\/em> is greater than <em data-effect=\"italics\">b<\/em>\u201d when <em data-effect=\"italics\">a<\/em> is to the right of <em data-effect=\"italics\">b<\/em> on the number line<\/li>\n<\/ul>\n<p id=\"fs-id1170655121376\">As we move from left to right on a number line, the values increase.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id1170655150929\">Order each of the following pairs of numbers, using &lt; or &gt;. It may be helpful to refer <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_006\">Figure 3<\/a>.<\/p>\n<p>a) &#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6e89c4f69688b0dd6ab75a55841059de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/>____<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/> b) -3<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b8fa03e1b526c6d07ec843385490ca4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/>____<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-469181e98c8ec4b1bce8f57ea4ba4e31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/>____ &#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-83dc4a903812856a369855508d0b2637_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> d) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/>____ &#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-44def33f7275fe3dbc5caccd0de620c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<div id=\"CNX_ElemAlg_Figure_01_08_006\" class=\"bc-figure figure\">\n<figure style=\"width: 656px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_006_img_new.jpg\" alt=\"There is a number line shown that runs from negative 4 to positive 4. From left to right, the numbers marked are negative 3 and 1\/2, negative 3, negative 8\/3, negative 2, negative 1, negative 3\/4, negative 2\/3, and negative 1\/4. The number negative 3 and 1\/2 is between negative 4 and negative 3 The number negative 8\/3 is between negative 3 and negative 2, but closer to negative 3. The numbers negative 3\/4, negative 2\/3, and negative 1\/4 are all between negative 1 and 0.\" width=\"656\" height=\"75\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 3<\/figcaption><\/figure>\n<\/div>\n<div><\/div>\n<\/div>\n<div><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-232f2faf108f1a943a949da9e8aec8fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#51;&#125;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#32;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#32;&#45;&#51;&#32;&#92;&#92;&#32;&#45;&#51;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#32;&#60;&#32;&#45;&#51;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"92\" style=\"vertical-align: -17px;\" \/><\/div>\n<div>\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<table id=\"eip-313\" summary=\"\/\">\n<tbody>\n<tr>\n<td style=\"width: 513.406px;\">a)<br \/>\n&#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6e89c4f69688b0dd6ab75a55841059de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> is to the right of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/> on the number line.<\/td>\n<td style=\"width: 834.406px;\">&#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6e89c4f69688b0dd6ab75a55841059de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/>___-1<\/p>\n<p>&#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6e89c4f69688b0dd6ab75a55841059de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> &gt; -1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 513.406px;\">b)<br \/>\n&#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4260474bd3509e25524bef571fc3c20a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"18\" style=\"vertical-align: -6px;\" \/> is to the right of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> on the number line.<\/td>\n<td style=\"width: 834.406px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8049cf0225a15314a43cdab885ccbba8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#51;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#45;&#51;&#32;&#92;&#92;&#32;&#45;&#51;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#60;&#45;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"99\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 513.406px;\">c)<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e8ab9ebbace69ceb967d69cd3d13f20c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/> is to the right of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3579c51ce3351cefb94bd858e9dc5596_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/> on the number line.<\/td>\n<td style=\"width: 834.406px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-47e3ce5a8fbbee820a209a0949b12bde_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#92;&#92;&#32;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#60;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"97\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 513.406px;\">d)<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> is to the right of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4273cdb3286c50165fe922304b2fa543_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/> on the number line.<\/td>\n<td style=\"width: 834.406px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-13457\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/Screenshot-2021-02-22-at-9.35.54-AM-e1614015412311.png\" alt=\"negative 2 is greater than negative 8 divided by 3.\" width=\"117\" height=\"59\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654941144\" data-type=\"problem\">\n<p id=\"fs-id1170654941146\">Order each of the following pairs of numbers, using &lt; or &gt;:<\/p>\n<p id=\"fs-id1166422659176\">a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3720a33382ac59189b503d6f70c1c6b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"92\" style=\"vertical-align: -6px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2eaf8509496d2a1ff826a83bce339a7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"98\" style=\"vertical-align: -6px;\" \/> c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-cbb12dc1f9175e4cba726967b56dd866_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -6px;\" \/> d) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e3fa5bf74022b5a25d53cbea109031f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"91\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654940483\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654940485\">a) &gt; b) &gt; c) &lt; d) &lt;<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<h1>Locate Decimals on the Number Line<\/h1>\n<p id=\"fs-id1170655098623\">Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655029934\" data-type=\"problem\">\n<p id=\"fs-id1170655029936\">Locate 0.4 on the number line.<\/p>\n<\/div>\n<div id=\"fs-id1170655029941\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1170655081279\">A proper fraction has value less than one. The decimal number 0.4 is equivalent to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ddd0eb29ad8084ce92e3d980a048b90c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/>, a proper fraction, so 0.4 is located between 0 and 1. On a number line, divide the interval between 0 and 1 into 10 equal parts. Now label the parts 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. We write 0 as 0.0 and 1 and 1.0, so that the numbers are consistently in tenths. Finally, mark 0.4 on the number line. See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_007\">Figure 4<\/a>.<\/p>\n<div id=\"CNX_ElemAlg_Figure_01_08_007\" class=\"bc-figure figure\">\n<figure style=\"width: 667px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_007_new.jpg\" alt=\"There is a number line shown that runs from 0.0 to 1. The only point given is 0.4, which is between 0.3 and 0.5.\" width=\"667\" height=\"42\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 4<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654962328\" data-type=\"problem\">\n<p id=\"fs-id1170654962330\">Locate on the number line: 0.6<\/p>\n<\/div>\n<details>\n<summary>Show answer<\/summary>\n<div id=\"fs-id1170654968605\" data-type=\"solution\"><span id=\"fs-id1170654968607\" data-type=\"media\" data-alt=\"There is a number line shown that runs from 0.0 to 1. The only point given is 0.6, which is between 0.5 and 0.7.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_008_img_new.jpg\" alt=\"There is a number line shown that runs from 0.0 to 1. The only point given is 0.6, which is between 0.5 and 0.7.\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"try\" data-type=\"note\">\n<div id=\"fs-id1170654962326\" data-type=\"exercise\">\n<div id=\"fs-id1170654962328\" data-type=\"problem\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655195763\" data-type=\"problem\">\n<p id=\"fs-id1170655195765\">Locate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9d30ea149e810e5e4675dea74e975f29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#55;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -1px;\" \/> on the number line.<\/p>\n<\/div>\n<div id=\"fs-id1170655090205\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1170655090207\">The decimal <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9d30ea149e810e5e4675dea74e975f29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#55;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -1px;\" \/> is equivalent to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-90e905a6fd04a544f4e0a4094af9d7f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#52;&#125;&#123;&#49;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" style=\"vertical-align: -6px;\" \/>, so it is located between 0 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/>. On a number line, mark off and label the hundredths in the interval between 0 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/>. See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_010\">Figure 5<\/a>.<\/p>\n<div id=\"CNX_ElemAlg_Figure_01_08_010\" class=\"bc-figure figure\">\n<figure style=\"width: 676px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_010_new.jpg\" alt=\"There is a number line shown that runs from negative 1.00 to 0.00. The only point given is negative 0.74, which is between negative 0.8 and negative 0.7.\" width=\"676\" height=\"61\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 5<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655057804\" data-type=\"problem\">\n<p id=\"fs-id1170655057806\">Locate on the number line: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8bc8f85cb5db1c89a8372b13802bb793_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"36\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<\/div>\n<details>\n<summary>Show answer<\/summary>\n<div id=\"fs-id1170655003781\" data-type=\"solution\"><span id=\"fs-id1170655003783\" data-type=\"media\" data-alt=\"There is a number line shown that runs from negative 1.00 to 0.00. The only point given is negative 0.6, which is between negative 0.8 and negative 0.4.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_011_img_new.jpg\" alt=\"There is a number line shown that runs from negative 1.00 to 0.00. The only point given is negative 0.6, which is between negative 0.8 and negative 0.4.\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170655082976\">Which is larger, 0.04 or 0.40? If you think of this as money, you know that ?0.40 (forty cents) is greater than ?0.04 (four cents). So,<\/p>\n<p id=\"fs-id1170654940800\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b4dc20a251fc8e8d2578f848411e92a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#52;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: -1px;\" \/> &gt; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8d9a18298d71046d4938c101e1367c19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#48;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"fs-id1170655061503\">Again, we can use the number line to order numbers.<\/p>\n<ul id=\"fs-id1166422832290\" data-bullet-style=\"bullet\">\n<li><em data-effect=\"italics\">a &lt; b<\/em> \u201c<em data-effect=\"italics\">a<\/em> is less than <em data-effect=\"italics\">b<\/em>\u201d when <em data-effect=\"italics\">a<\/em> is to the left of <em data-effect=\"italics\">b<\/em> on the number line<\/li>\n<li><em data-effect=\"italics\">a &gt; b<\/em> \u201c<em data-effect=\"italics\">a<\/em> is greater than <em data-effect=\"italics\">b<\/em>\u201d when <em data-effect=\"italics\">a<\/em> is to the right of <em data-effect=\"italics\">b<\/em> on the number line<\/li>\n<\/ul>\n<p id=\"fs-id1170655060667\">Where are 0.04 and 0.40 located on the number line? See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_013\">Figure 6<\/a>.<\/p>\n<div id=\"CNX_ElemAlg_Figure_01_08_013\" class=\"bc-figure figure\">\n<figure style=\"width: 676px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_013_new.jpg\" alt=\"There is a number line shown that runs from negative 0.0 to 1.0. From left to right, there are points 0.04 and 0.4 marked. The point 0.04 is between 0.0 and 0.1. The point 0.4 is between 0.3 and 0.5.\" width=\"676\" height=\"58\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 6<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1170654934978\">We see that 0.40 is to the right of 0.04 on the number line. This is another way to demonstrate that 0.40 &gt; 0.04<\/p>\n<p id=\"fs-id1170654934983\">How does 0.31 compare to 0.308? This doesn\u2019t translate into money to make it easy to compare. But if we convert 0.31 and 0.308 into fractions, we can tell which is larger.<\/p>\n<table id=\"eip-id1169746116939\" style=\"width: 100%;\" summary=\"Two numbers are given: 0.31 and 0.308. There is a table with instructions on the left and mathematical steps on the right. In the first row we have \u201cConvert to fractions.\u201d To the right of this we have 31\/100 and 308\/1000. In the following row, it says \u201cWe need a common denominator to compare them. To the right of this we have the quantity (31 times 10) over the quantity (100 times 10). The fraction 308\/1000 remains the same. Below this, we have the fractions 310\/1000 and 308 over 1000.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>0.31<\/td>\n<td>0.308<\/td>\n<\/tr>\n<tr>\n<td>Convert to fractions.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-24d15db5e2105ea4e499bd001aadf727_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#49;&#125;&#123;&#49;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"21\" style=\"vertical-align: -7px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-dfa5b364c04734a1fa2b5a97fc2ef93a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#48;&#56;&#125;&#123;&#49;&#48;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"28\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>We need a common denominator to compare them.<\/td>\n<td><span id=\"eip-id1169746116992\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_014a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<td><span id=\"eip-id1169746117002\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_014b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-249f22e14451eff91766760af76f8365_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#49;&#48;&#125;&#123;&#49;&#48;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"28\" style=\"vertical-align: -7px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-dfa5b364c04734a1fa2b5a97fc2ef93a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#48;&#56;&#125;&#123;&#49;&#48;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"28\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170654943714\">Because 310 &gt; 308, we know that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-249f22e14451eff91766760af76f8365_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#49;&#48;&#125;&#123;&#49;&#48;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"28\" style=\"vertical-align: -7px;\" \/> &gt; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-dfa5b364c04734a1fa2b5a97fc2ef93a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#48;&#56;&#125;&#123;&#49;&#48;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"28\" style=\"vertical-align: -7px;\" \/>. Therefore, 0.31 &gt; 0.308<\/p>\n<p id=\"fs-id1170654935899\">Notice what we did in converting 0.31 to a fraction\u2014we started with the fraction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-24d15db5e2105ea4e499bd001aadf727_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#49;&#125;&#123;&#49;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"21\" style=\"vertical-align: -7px;\" \/> and ended with the equivalent fraction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-249f22e14451eff91766760af76f8365_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#49;&#48;&#125;&#123;&#49;&#48;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"28\" style=\"vertical-align: -7px;\" \/>. Converting <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-249f22e14451eff91766760af76f8365_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#49;&#48;&#125;&#123;&#49;&#48;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"28\" style=\"vertical-align: -7px;\" \/> back to a decimal gives 0.310. So 0.31 is equivalent to 0.310. Writing zeros at the end of a decimal does not change its value!<\/p>\n<div id=\"fs-id1166420392202\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2fd302d58779091348abd0c0f9664101_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#49;&#125;&#123;&#49;&#48;&#48;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#49;&#48;&#125;&#123;&#49;&#48;&#48;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#46;&#51;&#49;&#61;&#48;&#46;&#51;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"238\" style=\"vertical-align: -7px;\" \/><\/div>\n<p>We say 0.31 and 0.310 are equivalent decimals.<\/p>\n<div data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Equivalent Decimals<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Two decimals are equivalent if they convert to equivalent fractions.<\/p>\n<\/div>\n<\/div>\n<p>We use equivalent decimals when we order decimals.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170654944395\">The steps we take to order decimals are summarized here.<\/p>\n<div id=\"fs-id1170654944399\" class=\"howto\" data-type=\"note\">\n<div data-type=\"title\">\n<div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">HOW TO: Order Decimals.<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol id=\"fs-id1166422524060\" class=\"stepwise\" type=\"1\">\n<li>Write the numbers one under the other, lining up the decimal points.<\/li>\n<li>Check to see if both numbers have the same number of digits. If not, write zeros at the end of the one with fewer digits to make them match.<\/li>\n<li>Compare the numbers as if they were whole numbers.<\/li>\n<li>Order the numbers using the appropriate inequality sign.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id1170655083061\">Order <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d3a2db723058176a6ad8afa312602369_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#54;&#52;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#48;&#46;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"90\" style=\"vertical-align: 0px;\" \/> using &lt; or &gt;.<\/p>\n<\/div>\n<div id=\"fs-id1170654943856\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<table id=\"eip-61\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td>Write the numbers one under the other, lining up the decimal points.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e34cd0b080365cded7f3b49020436183_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#48;&#46;&#54;&#52;&#92;&#92;&#32;&#48;&#46;&#54;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"32\" style=\"vertical-align: -11px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add a zero to 0.6 to make it a decimal with 2 decimal places.<br \/>\nNow they are both hundredths.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-075482abf8e030dbe2bbf3b7c1b7da9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#48;&#46;&#54;&#52;&#92;&#92;&#32;&#48;&#46;&#54;&#48;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"32\" style=\"vertical-align: -11px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>64 is greater than 60.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-de25eb06760758d2c59eb63fca270413_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: -1px;\" \/> &gt; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7029ea134aa43ac5ffd9f780e196307d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>64 hundredths is greater than 60 hundredths.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b786125fbd4c36104af7fa3eef1dfd11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> &gt; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8774eaae0b69c9dff8ac00c632570137_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#54;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b786125fbd4c36104af7fa3eef1dfd11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> &gt; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6467b5d4c60ec605876c9aa484e54ddd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"23\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655162906\" data-type=\"problem\">\n<p id=\"fs-id1170655162908\">Order each of the following pairs of numbers, using <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-755484e03a32c417de8f2c06c64fbcd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#60;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"37\" style=\"vertical-align: -2px;\" \/> &gt; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e264f26dc408fd1b0c2bf6db6bcd1277_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#58;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#46;&#52;&#50;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#48;&#46;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"97\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654903658\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654903660\">&gt;<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 11<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654903625\" data-type=\"problem\">\n<p id=\"fs-id1170654903627\">Order <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8ea76d27e5559b93058daeee71ad1eba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#56;&#51;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#48;&#46;&#56;&#48;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"108\" style=\"vertical-align: 0px;\" \/> using &lt; or &gt;.<\/p>\n<\/div>\n<div id=\"fs-id1170655059396\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-28\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8ea76d27e5559b93058daeee71ad1eba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#56;&#51;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#48;&#46;&#56;&#48;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"108\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Write the numbers one under the other, lining up the decimals.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a25ef24ed31f0657f29b208b11a2d8a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#48;&#46;&#56;&#51;&#92;&#92;&#32;&#48;&#46;&#56;&#48;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"41\" style=\"vertical-align: -11px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>They do not have the same number of digits.<br \/>\nWrite one zero at the end of 0.83.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d80444b256414b62f5906f389ffd8976_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#48;&#46;&#56;&#51;&#48;&#92;&#92;&#32;&#48;&#46;&#56;&#48;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"41\" style=\"vertical-align: -11px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d2d890496efe83b04abd4ec3c58ec7bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> &gt; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-abebf72de39f4e2597a5f008ecd5ef32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#48;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/>, 830 thousandths is greater than 803 thousandths.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-75453d4e993e41a6a00fe5c09d971885_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#56;&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/> &gt; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f049b775e25f14487dedeb8c4b81c6b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#56;&#48;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-79198520370cb4b8021e33e351e1bffc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#56;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> &gt; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f049b775e25f14487dedeb8c4b81c6b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#56;&#48;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 11<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654944217\" data-type=\"problem\">\n<p id=\"fs-id1170654944219\">Order the following pair of numbers, using <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-755484e03a32c417de8f2c06c64fbcd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#60;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"37\" style=\"vertical-align: -2px;\" \/> &gt; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1ec48943b9f83309165ad1d682d9469c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#58;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#46;&#55;&#54;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#48;&#46;&#55;&#48;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"115\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655090488\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170655090490\">&gt;<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170654943214\">When we order negative decimals, it is important to remember how to order negative integers. Recall that larger numbers are to the right on the number line. For example, because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> lies to the right of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> on the number line, we know that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> &gt; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/>. Similarly, smaller numbers lie to the left on the number line. For example, because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9672181aec15f8334b80ada7de4e4fc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> lies to the left of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4797c874a138ca175d7c2cd8b3ed9a98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> on the number line, we know that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-60d3cd2430f14efd3470472a0d563128_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;&#60;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"68\" style=\"vertical-align: -2px;\" \/>. See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_08_015\">Figure 7<\/a>.<\/p>\n<div id=\"CNX_ElemAlg_Figure_01_08_015\" class=\"bc-figure figure\">\n<figure style=\"width: 656px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_015_new.jpg\" alt=\"There is a number line shown that runs from negative 10 to 0. There are not points given and the hashmarks exist at every integer between negative 10 and 0.\" width=\"656\" height=\"40\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 7<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1170654941319\">If we zoomed in on the interval between 0 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/>, as shown in <a class=\"autogenerated-content\" href=\"#fs-id1170655098079\">Example 10<\/a>, we would see in the same way that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b71e6473c1bd63144fee0c0d03721d39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"35\" style=\"vertical-align: 0px;\" \/> &gt; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-dc1d15a02f557f8825ca225e58bb6adc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#48;&#46;&#57;&#60;&#45;&#48;&#46;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"176\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 12<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655098079\" data-type=\"problem\">\n<p>Use &lt; or &gt; to order <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b842abcbaee13424ae38730b292dca2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#49;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#45;&#48;&#46;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"116\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655060964\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-770\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b842abcbaee13424ae38730b292dca2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#49;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#45;&#48;&#46;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"116\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Write the numbers one under the other, lining up the decimal points.<br \/>\nThey have the same number of digits.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5270bd022c2cfb67d3794746cadd4a8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#48;&#46;&#49;&#92;&#92;&#32;&#45;&#48;&#46;&#56;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"36\" style=\"vertical-align: -11px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/> &gt; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ca32393b1b5af7c55a95d89cf9d610f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/>, \u22121 tenth is greater than \u22128 tenths.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-65193683874a338a58c1e613cb1beafb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"35\" style=\"vertical-align: -1px;\" \/> &gt; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-df61d07f5a3371fbbdf489a534427bc1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"36\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 12<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655065251\" data-type=\"problem\">\n<p id=\"fs-id1170655099866\">Order the following pair of numbers, using &lt; or &gt;: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-94931016a478e06d4ad8ed9e01c64f5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#51;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#45;&#48;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"115\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655059478\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170655059480\">&gt;<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<h1>Key Concepts<\/h1>\n<ul id=\"fs-id1170654944002\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Order Decimals<\/strong>\n<ol id=\"fs-id1170654942859\" class=\"stepwise\" type=\"1\">\n<li>Write the numbers one under the other, lining up the decimal points.<\/li>\n<li>Check to see if both numbers have the same number of digits. If not, write zeros at the end of the one with fewer digits to make them match.<\/li>\n<li>Compare the numbers as if they were whole numbers.<\/li>\n<li>Order the numbers using the appropriate inequality sign.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<h1>Glossary<\/h1>\n<div class=\"textbox shaded\">\n<dl id=\"fs-id1166425286362\">\n<dt>equivalent decimals<\/dt>\n<dd id=\"fs-id1166425286367\">Two decimals are equivalent if they convert to equivalent fractions.<\/dd>\n<\/dl>\n<dl id=\"fs-id1166425286372\">\n<dt>irrational number<\/dt>\n<dd id=\"fs-id1166425286377\">An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.<\/dd>\n<\/dl>\n<dl id=\"fs-id1166424794533\">\n<dt>rational number<\/dt>\n<dd id=\"fs-id1166424794538\">A rational number is a number of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b9606a6a8bc8c28efca6d4be67f31e5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#125;&#123;&#113;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"8\" style=\"vertical-align: -9px;\" \/>, where <em data-effect=\"italics\">p<\/em> and <em data-effect=\"italics\">q<\/em> are integers and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2e3f318f8865aafe5e97953eb6c5e2ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"41\" style=\"vertical-align: -4px;\" \/>. A rational number can be written as the ratio of two integers. Its decimal form stops or repeats.<\/dd>\n<\/dl>\n<dl id=\"fs-id1166424761145\">\n<dt>real number<\/dt>\n<dd id=\"fs-id1166424794538\">A real number is a number that is either rational or irrational<\/dd>\n<\/dl>\n<\/div>\n<h1 id=\"fs-id1170655029650\">1.7 Exercise Set<\/h1>\n<p id=\"fs-id1166422646844\">In the following exercises, write as the ratio of two integers.<\/p>\n<ol class=\"twocolumn\">\n<li>\n<ol type=\"a\">\n<li>5<\/li>\n<li>3.19<\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7370d314c10f88ec8b86d7749a473921_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"30\" style=\"vertical-align: 0px;\" \/><\/li>\n<li>9.279<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<div id=\"fs-id1170655029657\" data-type=\"exercise\">\n<div id=\"fs-id1170655073127\" data-type=\"problem\">In the following exercises, list the a) rational numbers, b) irrational numbers<\/div>\n<ol class=\"twocolumn\" start=\"3\">\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-331edbeff35ec7874e4f89142d620f86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#55;&#53;&#44;&#48;&#46;&#50;&#50;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#51;&#125;&#44;&#49;&#46;&#51;&#57;&#49;&#55;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/li>\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-780c4b2e43e4cda1ad91f499fc16022c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#52;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#53;&#125;&#44;&#49;&#46;&#57;&#49;&#57;&#50;&#57;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#44;&#51;&#46;&#53;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"177\" style=\"vertical-align: -4px;\" \/><\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1170655037954\" data-type=\"exercise\">\n<div data-type=\"problem\">\n<div id=\"fs-id1169144768977\" data-type=\"exercise\">\n<div id=\"fs-id1169144768979\" data-type=\"problem\"><span style=\"font-size: 14pt; text-align: initial;\">In the following exercises, list the a) whole numbers, b) integers, c) rational numbers, d) irrational numbers, e) real numbers for each set of numbers.<\/span><\/div>\n<ol class=\"twocolumn\" start=\"5\">\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1d85f484469818ec318ae0b46a0d8057_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#44;&#48;&#44;&#92;&#115;&#113;&#114;&#116;&#091;&#53;&#093;&#123;&#45;&#51;&#50;&#125;&#32;&#44;&#32;&#49;&#46;&#57;&#53;&#50;&#56;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#53;&#125;&#44;&#92;&#115;&#113;&#114;&#116;&#091;&#50;&#093;&#123;&#45;&#57;&#125;&#44;&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"288\" style=\"vertical-align: -6px;\" \/><\/li>\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-30d30b15249cb7338af8575d45732729_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#44;&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#53;&#49;&#50;&#125;&#32;&#44;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;&#44;&#45;&#49;&#44;&#32;&#92;&#115;&#113;&#114;&#116;&#091;&#52;&#093;&#123;&#45;&#55;&#53;&#125;&#44;&#32;&#48;&#46;&#55;&#55;&#44;&#51;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"259\" style=\"vertical-align: -6px;\" \/><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170655029153\" data-type=\"exercise\">\n<div id=\"fs-id1170655062368\" data-type=\"problem\">\n<p id=\"fs-id1166425252416\">In the following exercises, locate the numbers on a number line.<\/p>\n<ol class=\"twocolumn\" start=\"7\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-68c7af8aa98ed5081fc86293834ed8ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#53;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"51\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-33b5ff7f9a409125784cafca354daa19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#48;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#54;&#125;&#44;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"77\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e5b63323677059742e739ee00d501c67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#44;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d87ea92d0b65e7b84b3f13805656723c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#44;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#44;&#49;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#44;&#45;&#49;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#44;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"166\" style=\"vertical-align: -6px;\" \/><\/li>\n<\/ol>\n<div id=\"fs-id1170655088969\" data-type=\"exercise\">\n<div id=\"fs-id1170655084984\" data-type=\"problem\">In the following exercises, order each of the pairs of numbers, using &lt; or &gt;.<\/div>\n<ol class=\"twocolumn\" start=\"11\">\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-00ae0375759cdfbf690c7f901076c649_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"91\" style=\"vertical-align: -6px;\" \/><\/li>\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-17a0bace0701ba4f06918f0173ee4848_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"99\" style=\"vertical-align: -6px;\" \/><\/li>\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-76fad47e83d07c7c043a59a803504743_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#50;&#125;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"111\" style=\"vertical-align: -6px;\" \/><\/li>\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-850f2cf6215ea048401acbb2c784d31d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#51;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"98\" style=\"vertical-align: -6px;\" \/><\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1170653193072\" data-type=\"exercise\">\n<div id=\"fs-id1170653193074\" data-type=\"problem\">Locate Decimals on the Number Line In the following exercises, locate the number on the number line.<\/div>\n<ol class=\"twocolumn\" start=\"15\">\n<li data-type=\"problem\">0.8<\/li>\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6dcf50e0b47145a50e261490f61cf521_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#46;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"36\" style=\"vertical-align: -1px;\" \/><\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1170655096647\" data-type=\"exercise\">\n<div id=\"fs-id1170655096649\" data-type=\"problem\">In the following exercises, order each pair of numbers, using &lt; or &gt;.<\/div>\n<ol class=\"twocolumn\" start=\"17\">\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-14849ceafd3b3f2a8cdbd68dad2ae1db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#51;&#55;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#48;&#46;&#54;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"99\" style=\"vertical-align: 0px;\" \/><\/li>\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e1600a57ee6f98186f211853f9e2029f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#57;&#49;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#48;&#46;&#57;&#48;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"107\" style=\"vertical-align: 0px;\" \/><\/li>\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-381dd5c2acb4795da1bf25e50b1d8846_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#53;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#45;&#48;&#46;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"116\" style=\"vertical-align: 0px;\" \/><\/li>\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0322ed19833bee00667219c9e9c9f828_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#54;&#50;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#46;&#52;&#112;&#116;&#125;&#45;&#48;&#46;&#54;&#49;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"142\" style=\"vertical-align: 0px;\" \/><\/li>\n<\/ol>\n<ol start=\"21\">\n<li><strong>Child care.<\/strong> Serena wants to open a licensed child care center. Her state requires there be no more than 12 children for each teacher. She would like her child care centre to serve 40 children.\n<ol type=\"a\">\n<li style=\"list-style-type: none;\">\n<ol type=\"a\">\n<li>How many teachers will be needed?<\/li>\n<li>Why must the answer be a whole number?<\/li>\n<li>Why shouldn\u2019t you round the answer the usual way, by choosing the whole number closest to the exact answer?<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2 data-type=\"title\"><span style=\"font-size: 1.2em;\">Answers:<\/span><\/h2>\n<ol>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-80a4ab2b90b70cc865af39f0bdbd7bc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"7\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-eba805dda9989172983cf56c42e7a901_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#49;&#57;&#125;&#123;&#49;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"21\" style=\"vertical-align: -7px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2546ade224b4da440b14e2c3c529fb91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#49;&#50;&#125;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"25\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b7be1834de2c89b306cd48dc766933d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#50;&#57;&#55;&#125;&#123;&#49;&#48;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"28\" style=\"vertical-align: -7px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-42a5803f515733816d93afd3ccada0c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#55;&#53;&#44;&#48;&#46;&#50;&#50;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"86\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-02a4735ca98c05f4c551a8713b22e140_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#51;&#57;&#49;&#55;&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"58\" style=\"vertical-align: -1px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c685bd923092268da3e27dfc7ddcc45f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#52;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#53;&#125;&#44;&#51;&#46;&#53;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-58dea56d36ce67cfc103b75a3a2d6653_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#57;&#49;&#57;&#50;&#57;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"67\" style=\"vertical-align: -1px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a5e437be25f29374d30f66cd46adf81c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-78b9f9eb6df48f7c94fe21e84c04f4b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#44;&#48;&#44;&#32;&#92;&#115;&#113;&#114;&#116;&#091;&#53;&#093;&#123;&#45;&#51;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"94\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0a58f966900a99c10f5ce303e2bcd561_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#44;&#48;&#44;&#92;&#115;&#113;&#114;&#116;&#091;&#53;&#093;&#123;&#45;&#51;&#50;&#125;&#44;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"117\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a7956177e941387825bfd06259bdf875_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#57;&#53;&#50;&#56;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#44;&#32;&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7ba7319f055030d2ca433993135f5a5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#44;&#48;&#44;&#32;&#92;&#115;&#113;&#114;&#116;&#091;&#53;&#093;&#123;&#45;&#51;&#50;&#125;&#44;&#32;&#49;&#46;&#57;&#53;&#50;&#56;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#53;&#125;&#44;&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"241\" style=\"vertical-align: -6px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-07742bd0b9c49b651cd1aa956d22b337_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#53;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f81d769284bc127cf1af90efdd017f3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#44;&#45;&#49;&#44;&#32;&#92;&#40;&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#53;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"103\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>-7, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9e51bea8cadd04b1072c1733e85eff55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;&#44;&#45;&#49;&#44;&#48;&#46;&#55;&#55;&#44;&#51;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#44;&#32;&#92;&#40;&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#53;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"175\" style=\"vertical-align: -6px;\" \/><\/li>\n<li>none<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-96c2da2e381d7646d419cdb199147763_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#44;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;&#44;&#32;&#45;&#49;&#44;&#48;&#46;&#55;&#55;&#44;&#51;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#44;&#32;&#92;&#40;&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#53;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"206\" style=\"vertical-align: -6px;\" \/><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><span id=\"fs-id1170655080413\" data-type=\"media\" data-alt=\"There is a number line shown that runs from 0 to 6. From left to right the points read 3\/4, 8\/5, and 10\/3. The point for 3\/4 is between 0 and 1. The point for 8\/5 is between 1 and 2. The point for 10\/3 is between 3 and 4.\">7.<img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_202_img_new.jpg\" alt=\"There is a number line shown that runs from 0 to 6. From left to right the points read 3\/4, 8\/5, and 10\/3. The point for 3\/4 is between 0 and 1. The point for 8\/5 is between 1 and 2. The point for 10\/3 is between 3 and 4.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td style=\"width: 50%;\">\u00a08. <span id=\"fs-id1170653193526\" data-type=\"media\" data-alt=\"There is a number line shown that runs from 0 to 6. From left to right the points read 3\/10, 11\/6, 7\/2, and 4. The point for 3\/10 is between 0 and 1. The point for 11\/6 is between 1 and 2. The point for 7\/2 is between 3 and 4.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_203_img_new.jpg\" alt=\"There is a number line shown that runs from 0 to 6. From left to right the points read 3\/10, 11\/6, 7\/2, and 4. The point for 3\/10 is between 0 and 1. The point for 11\/6 is between 1 and 2. The point for 7\/2 is between 3 and 4.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">9. <span id=\"fs-id1170655353874\" data-type=\"media\" data-alt=\"There is a number line shown that runs from negative 1 to 1. From left to right the points read negative 2\/5 and 2\/5. The point for negative 2\/5 is between negative 1 and 0. The point for 2\/5 is between 0 and 1.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_205_img_new.jpg\" alt=\"There is a number line shown that runs from negative 1 to 1. From left to right the points read negative 2\/5 and 2\/5. The point for negative 2\/5 is between negative 1 and 0. The point for 2\/5 is between 0 and 1.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td style=\"width: 50%;\">10. <span id=\"fs-id1170655124914\" data-type=\"media\" data-alt=\"There is a number line shown that runs from negative 4 to 4. From left to right the points read negative 5\/2, negative 1 and 2\/3, negative 3\/4, \u00be, 1 and 2\/3, and 5\/2. The point for negative 5\/2 is between negative 3 and negative 2. The point for negative 1 and 2\/3 is between negative 2 and negative 1. The point for negative 3\/4 is between negative 1 and 0. The point for 3\/4 is between 0 and 1. The point for 1 and 2\/3 is between 1 and 2. The point for 5\/2 is between 2 and 3.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_207_img_new.jpg\" alt=\"There is a number line shown that runs from negative 4 to 4. From left to right the points read negative 5\/2, negative 1 and 2\/3, negative 3\/4, \u00be, 1 and 2\/3, and 5\/2. The point for negative 5\/2 is between negative 3 and negative 2. The point for negative 1 and 2\/3 is between negative 2 and negative 1. The point for negative 3\/4 is between negative 1 and 0. The point for 3\/4 is between 0 and 1. The point for 1 and 2\/3 is between 1 and 2. The point for 5\/2 is between 2 and 3.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<dl id=\"fs-id1166424761164\"><\/dl>\n<ol start=\"11\">\n<li>&lt;<\/li>\n<li>&gt;<\/li>\n<li>&gt;<\/li>\n<li>&lt;<\/li>\n<\/ol>\n<table style=\"border-collapse: collapse; width: 100%; height: 120px;\">\n<tbody>\n<tr style=\"height: 17px;\">\n<td style=\"width: 33.2078%; height: 17px;\">15. <span id=\"fs-id1170655095903\" data-type=\"media\" data-alt=\"There is a number line shown that runs from negative 4 to 4. The point 0.8 is between 0 and 1.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_209_img_new.jpg\" alt=\"There is a number line shown that runs from negative 4 to 4. The point 0.8 is between 0 and 1.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td style=\"width: 33.4588%; height: 17px;\">16. <span id=\"fs-id1170654941068\" data-type=\"media\" data-alt=\"There is a number line shown that runs from negative 4 to 4. The point negative 1.6 is between negative 2 and negative 1.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_211_img_new.jpg\" alt=\"There is a number line shown that runs from negative 4 to 4. The point negative 1.6 is between negative 2 and negative 1.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol start=\"17\">\n<li style=\"list-style-type: none;\">\n<ol start=\"17\">\n<li>&lt;<\/li>\n<li>&gt;<\/li>\n<li>&lt;<\/li>\n<li>&lt;<\/li>\n<li>\n<ol type=\"a\">\n<li>4 buses<\/li>\n<li>answers may vary<\/li>\n<li>answers may vary<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<h1>Attributions<\/h1>\n<p>This chapter has been adapted from \u201cThe Real Numbers\u201d in <a href=\"https:\/\/openstax.org\/details\/books\/elementary-algebra\"><em>Elementary Algebra<\/em><\/a> (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a <a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY 4.0 Licence<\/a>. Adapted by Izabela Mazur. See the Copyright page for more information.<\/p>\n","protected":false},"author":125,"menu_order":7,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-269","chapter","type-chapter","status-publish","hentry"],"part":21,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/269","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/users\/125"}],"version-history":[{"count":1,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/269\/revisions"}],"predecessor-version":[{"id":270,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/269\/revisions\/270"}],"part":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/parts\/21"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/269\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/media?parent=269"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapter-type?post=269"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/contributor?post=269"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/license?post=269"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}