- \r\n \t
- in 4.306, the 0 holds the hundredths place<\/li>\r\n \t
- in 17.0002, the 0's hold the tenth, hundredth and thousandth places.<\/li>\r\n<\/ul>\r\nA zero is usually placed to the left of the decimal point if there is no whole number.\r\n
0.5\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.937<\/p>\r\n\r\n
Which Zeros are not Needed?<\/h2>\r\n
- \r\n \t
- Zeros at the beginning, or far left, of whole numbers are not needed.\r\n
- \r\n \t
- 00004 = 4<\/li>\r\n \t
- 00100 = 100<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
- Zeros at the end, or far right, of decimal are not needed.\r\n
- \r\n \t
- 3.210 = 3.21<\/li>\r\n \t
- 34.062000 = 34.062<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n\r\n\r\nZeros on the outside edges of [pb_glossary id=\"669\"]mixed decimals[\/pb_glossary] do not change the value of the number and are not necessary.\r\n
00<\/strong>28.9710 <\/strong>= 28.971<\/p>\r\n
00<\/strong>100.003000 <\/strong>= 100.003<\/p>\r\n
890.407 = 00<\/strong>890.4070000000<\/strong><\/p>\r\n\r\n<\/div>\r\n
\r\n Exercise One<\/p>\r\n\r\n<\/header>\r\n
\r\n\r\nCross out the zeros that are not needed.\r\n- \r\n \t
- 70.0390<\/li>\r\n \t
- 00.2906<\/li>\r\n \t
- 06.30<\/li>\r\n \t
- 087.50<\/li>\r\n \t
- 60.0205<\/li>\r\n \t
- 4 020.0020<\/li>\r\n \t
- 400.080<\/li>\r\n \t
- 340.600<\/li>\r\n \t
- 03.5830<\/li>\r\n \t
- 9.0<\/li>\r\n \t
- 0.002030<\/li>\r\n \t
- 50.300<\/li>\r\n<\/ol>\r\nAnswers to Exercise One<\/strong>\r\n
- \r\n \t
- 709.039<\/li>\r\n \t
- 0.2906<\/li>\r\n \t
- 6.3<\/li>\r\n \t
- 87.5<\/li>\r\n \t
- 60.0205<\/li>\r\n \t
- 4 020.002<\/li>\r\n \t
- 400.08<\/li>\r\n \t
- 340.6<\/li>\r\n \t
- 3.583<\/li>\r\n \t
- 9.0<\/li>\r\n \t
- 0.00203<\/li>\r\n \t
- 50.3<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nZeros at the end of a decimal do not change the value.\r\n
6. = 6.0 = 6.00<\/p>\r\nAnd zeros at the beginning of a whole number do not change the value.\r\n
8 = 08 = 00008<\/p>\r\nBut zeros between a decimal point and a digit do change the value.\r\n
\r\n Example A<\/p>\r\n\r\n<\/header>\r\n
\r\n\r\n405 is very different than 45. And 0.05 (five hundredths) is very different than 0.5 (five tenths).\r\n\r\n<\/div>\r\n<\/div>\r\nComparing Decimals<\/h1>\r\nYou have probably heard the old saying: \"You cannot compare apples to oranges!\" And it's true, it is tough to compare things that do not have much in common. So before we compare decimals, we give the decimals something in common\u2014the same number of decimal places which gives them a common understood [pb_glossary id=\"625\"]denominator[\/pb_glossary].\r\n
\r\n\r\nBefore comparing decimals, put zeros at the end or cross out any unnecessary zeros so the decimals have a common (same) number of decimal places. If you write the decimals that you are comparing right underneath each other, your eye will often tell you which is the larger amount or if the amounts are equal.\r\n\r\n<\/div>\r\n\r\n Example B<\/p>\r\n\r\n<\/header>\r\n
\r\n\r\nCompare 0.43 and 0.4. Which is larger?\r\n\r\n0.43 has two decimal places; it is forty-three hundredths\u00a0 = [latex]\\tfrac{43}{100}[\/latex]. 0.4 has one decimal place; it is four tenths = [latex]\\tfrac{4}{10}[\/latex].\r\n\r\nGive them a common number of decimal places.\r\n<\/strong>\r\n\r\nAdd a zero to 0.4 to make it 0.40; now we read it as forty hundredths = [latex]\\tfrac{40}{100}[\/latex].\r\n\r\nNow, which is larger? [latex]\\tfrac{40}{100} \\text{ or }\\tfrac{43}{100}[\/latex].\r\n\r\nYou can easily see that 0.43 is the larger amount.\r\n\r\n<\/div>\r\n<\/div>\r\nReview of Greater Than > and Less Than <<\/h2>\r\nAn easy way to remember these signs is to think that the big (wide) end of the sign is closer to the bigger (greater) number, and the small end of the sign is closer to the smaller number.\r\n
0.43 is larger than 0.40, 0.43 ><\/strong> 0.40<\/p>\r\n
0.52 is smaller than 0.60, 0.52 <<\/strong> 0.60<\/p>\r\n\r\n
\r\n Exercise Two<\/p>\r\n\r\n<\/header>\r\n
\r\n\r\nWhich is greater? Draw a box around the bigger decimal fraction in each pair and write a greater than > or a less than < sign to make a true statement.\r\n- \r\n \t
- 0.6 \u00a0 <\u00a0\u00a0<\/span> \u00a00.65<\/li>\r\n \t
- 14\u00a0 \u00a0>\u00a0 \u00a0<\/span> 1.4<\/li>\r\n \t
- 0.17\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 0.02<\/li>\r\n \t
- 0.009 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 0.09<\/li>\r\n \t
- 0.23 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 0.215<\/li>\r\n \t
- 0.51 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 0.159<\/li>\r\n \t
- 0.03 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 0.0352<\/li>\r\n \t
- 0.07 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 0.063<\/li>\r\n \t
- 630.3 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 630.03<\/li>\r\n \t
- 0.80 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 0.6989<\/li>\r\n<\/ol>\r\nAnswers to Exercise Two<\/strong>\r\n
- \r\n \t
- <<\/li>\r\n \t
- ><\/li>\r\n \t
- ><\/li>\r\n \t
- <<\/li>\r\n \t
- ><\/li>\r\n \t
- ><\/li>\r\n \t
- <<\/li>\r\n \t
- ><\/li>\r\n \t
- ><\/li>\r\n \t
- ><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
Comparing Decimals Using a Number Line<\/h2>\r\nYou can compare decimals using a number line. A number line organizes what you are thinking about on paper \u2013 or on a ruler. You can plot your decimals on the number line and then be able to see which number is larger. Take a look:\r\n
\r\n Example C<\/p>\r\n\r\n<\/header>\r\n
\r\n\r\nFirst try to put the following numbers in order without looking at the number line below:\r\n2.347\u00a0\u00a0\u00a0\u00a0 2.3\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 2.37\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 2.33\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 2.39\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 2.341\u00a0 \u00a0 \u00a0 \u00a02.41<\/p>\r\nThen, look at the number line and see if you ordered your numbers correctly.\u00a0 The number line has a jagged edge which means it does not start at zero.\u00a0 It starts in the middle of a ruler. Using a number line can help you see your work and think about it at the same time.
![\"A](\"https:\/\/opentextbc.ca\/alfm4\/wp-content\/uploads\/sites\/421\/2022\/12\/number-line.png\")
\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n Exercises<\/p>\r\n\r\n<\/header>\r\n
\r\n\r\nTry plotting the following decimals on the number line below:\r\n4.59\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 4.32\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 4.7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 5.23\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 4.47\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 4.3\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 4.17<\/p>\r\n \r\n\r\n
![\"a](\"https:\/\/opentextbc.ca\/alfm4\/wp-content\/uploads\/sites\/421\/2022\/12\/image44.png\")
\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n Exercises<\/p>\r\n\r\n<\/header>\r\n
\r\n\r\nAnd now, plot these numbers on an empty number line. (You need to fill in the numbers yourself. \u00a0Plot 7.3 on the first large vertical line on the left)\r\n7.35\u00a0 \u00a07.3\u00a0 \u00a0 \u00a0 8.2\u00a0\u00a0\u00a0\u00a0 7.53\u00a0\u00a0\u00a0\u00a0\u00a0 7.98\u00a0\u00a0\u00a0\u00a0 8.34\u00a0\u00a0\u00a0 7.9\u00a0 \u00a07.5<\/p>\r\n
![\"An](\"https:\/\/opentextbc.ca\/alfm4\/wp-content\/uploads\/sites\/421\/2022\/12\/image45.png\")
\r\n\r\n<\/div>\r\n<\/div>\r\nThis is one way to organize and order decimals, please use it if the system is helpful for you.\r\n\r\n Exercise Three<\/p>\r\n\r\n<\/header>\r\n
\r\n\r\nDraw a circle around the smallest decimal fraction in each group.\r\n- \r\n \t
- 0.3,\u00a0 0.03 ,\u00a0 0.23<\/li>\r\n \t
- 0.04\u00a0 ,\u00a0 0.14,\u00a0 0.41<\/li>\r\n \t
- 0.016,\u00a0 0.2,\u00a0 0.216<\/li>\r\n \t
- 0.62,\u00a0 0.6103,\u00a0 0.5<\/li>\r\n \t
- 0.041,\u00a0 0.04,\u00a0 0.4<\/li>\r\n \t
- 100.1,\u00a0 100.13,\u00a0 100.01<\/li>\r\n<\/ol>\r\nAnswers to Exercise Three<\/strong>\r\n
- \r\n \t
- 0.03<\/li>\r\n \t
- 0.04<\/li>\r\n \t
- 0.016<\/li>\r\n \t
- 0.5<\/li>\r\n \t
- 0.04<\/li>\r\n \t
- 100.01<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
\r\n Exercise Four<\/p>\r\n\r\n<\/header>\r\n
\r\n\r\nIdentify whether the pair of numbers is equal (=) or not equal (\u2260).\r\n- \r\n \t
- 0.8 \u00a0 \u00a0=\u00a0 \u00a0<\/span> 0.80<\/li>\r\n \t
- 0.25\u00a0 \u00a0 \u2260 \u00a0 \u00a0<\/span> 0.205<\/li>\r\n \t
- 5.503\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 5.330<\/li>\r\n \t
- 9.3200\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 9.32<\/li>\r\n \t
- 0.75\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 0.750<\/li>\r\n \t
- 23.3\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 2.33<\/li>\r\n \t
- 0.040\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 0.40<\/li>\r\n \t
- 0.8010\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 0.801<\/li>\r\n<\/ol>\r\nAnswers to Exercise Four<\/strong>\r\n
- \r\n \t
- =<\/li>\r\n \t
- \u2260<\/li>\r\n \t
- \u2260<\/li>\r\n \t
- =<\/li>\r\n \t
- =<\/li>\r\n \t
- \u2260<\/li>\r\n \t
- \u2260<\/li>\r\n \t
- =<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
Topic C: Self-Test<\/h1>\r\nMark \/14\u00a0 \u00a0 \u00a0 Aim 12\/14<\/strong>\r\n
- \r\n \t
- Cross out the unnecessary zeros. (4 marks)\r\n
- \r\n \t
- 0401.02<\/li>\r\n \t
- 1 000.001<\/li>\r\n \t
- 68.3020<\/li>\r\n \t
- 050.200<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
- Draw a box around the larger amount in each pair. (4 marks)\r\n
- \r\n \t
- 0.8 or\u00a0 0.08<\/li>\r\n \t
- 0.004 or\u00a0 0.04<\/li>\r\n \t
- 0.006 or\u00a0 0.6<\/li>\r\n \t
- 33.006 or\u00a0 33.06<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
- Put the correct sign between the decimals.\u00a0 Choose\u00a0 =\u00a0 or >\u00a0 or <. (6 marks)\r\n
- \r\n \t
- 2.3\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 23<\/li>\r\n \t
- 0.2\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 0.003<\/li>\r\n \t
- 4.7\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 4.700<\/li>\r\n \t
- 0.25\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 0.250<\/li>\r\n \t
- 042.9002\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 042.9<\/li>\r\n \t
- 0.4092\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 0.411<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n
Answers to Topic C Self-Test<\/h2>\r\n
- \r\n \t
- Cross out the unnecessary zeros.\r\n
- \r\n \t
- 401.02<\/li>\r\n \t
- 1 000.001<\/li>\r\n \t
- 68.302<\/li>\r\n \t
- 50.2<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
- Draw a box around the larger amount in each pair.\r\n
- \r\n \t
- 0.8<\/li>\r\n \t
- 0.04<\/li>\r\n \t
- \u00a00.6<\/li>\r\n \t
- 33.06<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
- Put the correct sign between the decimals.\u00a0 Choose\u00a0 =\u00a0 or >\u00a0 or <.\r\n
- \r\n \t
- <<\/li>\r\n \t
- ><\/li>\r\n \t
- =<\/li>\r\n \t
- =<\/li>\r\n \t
- ><\/li>\r\n \t
- <<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n
- \r\n \t
- <\/li>\r\n<\/ol>","rendered":"
More about Zeros…<\/h1>\n
A whole number can have a decimal point and many zeros after it without changing its value.<\/p>\n
47 = 47.0 = 47.000 = 47.0000000000000000<\/p>\n
Zeros are used to hold a place when we write whole numbers.<\/p>\n
- \n
- in 100, the 0s hold the tens place and the ones place<\/li>\n
- in 1206<\/span>, the 0 holds the tens place<\/li>\n<\/ul>\n
In decimals, any zero to the right of the decimal point and to the left of another digit<\/a> is important because the zero is holding a place and giving the decimal the correct value.<\/p>\n
- \n
- in 4.306, the 0 holds the hundredths place<\/li>\n
- in 17.0002, the 0’s hold the tenth, hundredth and thousandth places.<\/li>\n<\/ul>\n
A zero is usually placed to the left of the decimal point if there is no whole number.<\/p>\n
0.5\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.937<\/p>\n
Which Zeros are not Needed?<\/h2>\n
- \n
- Zeros at the beginning, or far left, of whole numbers are not needed.\n
- \n
- 00004 = 4<\/li>\n
- 00100 = 100<\/li>\n<\/ul>\n<\/li>\n
- Zeros at the end, or far right, of decimal are not needed.\n
- Zeros at the beginning, or far left, of whole numbers are not needed.\n
- <\/li>\r\n<\/ol>","rendered":"
- 0.2\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 0.003<\/li>\r\n \t
- 2.3\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> 23<\/li>\r\n \t
- Cross out the unnecessary zeros. (4 marks)\r\n
- 0.25\u00a0 \u00a0 \u2260 \u00a0 \u00a0<\/span> 0.205<\/li>\r\n \t
- 0.8 \u00a0 \u00a0=\u00a0 \u00a0<\/span> 0.80<\/li>\r\n \t
- 14\u00a0 \u00a0>\u00a0 \u00a0<\/span> 1.4<\/li>\r\n \t
- 0.6 \u00a0 <\u00a0\u00a0<\/span> \u00a00.65<\/li>\r\n \t
- Zeros at the beginning, or far left, of whole numbers are not needed.\r\n