{"id":1153,"date":"2021-12-02T19:36:45","date_gmt":"2021-12-03T00:36:45","guid":{"rendered":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/chapter\/chapter-1-1-integers\/"},"modified":"2023-08-30T12:07:39","modified_gmt":"2023-08-30T16:07:39","slug":"integers","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/chapter\/integers\/","title":{"raw":"1.1 Integers","rendered":"1.1 Integers"},"content":{"raw":"The ability to work comfortably with negative numbers is essential to success in algebra. For this reason, a quick review of adding, subtracting, multiplying, and dividing integers is necessary.[footnote]Read about\u00a0The History of Integers<\/a>.[\/footnote] Integers[footnote]The word \"integer\" is derived from the Latin word integer,<\/em> which means \"whole.\" Integers are written without using a fractional component. Examples are 2, 3, 1042, 28, 0, \u221242, \u22122. Numbers that are fractional\u2014such as \u00bc, 0.33, and 1.42\u2014are not integers.[\/footnote] are all the positive whole numbers, all the negative whole numbers, and zero. As this is intended to be a review of integers, descriptions and examples will not be as detailed as in a normal lesson.\r\n\r\nWhen adding integers, there are two cases to consider. The first is when the signs match\u2014that is, the two integers are both positive or both negative.\r\n\r\nIf the signs match, add the numbers together and retain the sign.\r\n
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Example 1.1.1<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nAdd [latex]-5 + (-3).[\/latex]\r\n\r\n[latex]\\begin{array}{rl}\r\n-5+(-3)&\\text{Same sign. Add }5+3\\text{. Keep the negative sign.} \\\\\r\n-8&\\text{Solution}\r\n\\end{array}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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Example 1.1.2<\/p>\r\n\r\n<\/header>\r\n

Add [latex]-7+(-5).[\/latex]<\/div>\r\n
[latex]\\begin{array}{rl}\r\n-7+(-5)&\\text{Same sign. Add }7+5\\text{. Keep the negative sign.} \\\\\r\n-12&\\text{Solution}\r\n\\end{array}[\/latex]<\/div>\r\n<\/div>\r\nThe second case is when the signs don't match, and there is one positive and one negative number. Subtract the numbers (as if they were all positive), then use the sign from the number with the greatest absolute value. This means that, if the number with the greater absolute value is positive, the answer is positive. If it is negative, the answer is negative.\r\n
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Example 1.1.3<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nAdd [latex]-7+2.[\/latex]\r\n

[latex]\\begin{array}{rl}\r\n-7+2&\\text{Different signs. Subtract }7-2\\text{. Negative number has greater absolute value.} \\\\\r\n-5&\\text{Solution}\r\n\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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Example 1.1.4<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nAdd [latex]-4+6.[\/latex]\r\n

[latex]\\begin{array}{rl}\r\n-4+6 & \\text{Different signs. Subtract }6-4\\text{. Positive number has greater absolute value.} \\\\\r\n2&\\text{Solution}\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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Example 1.1.5<\/p>\r\n\r\n<\/header>\r\n

Add [latex]4+(-3).[\/latex]<\/div>\r\n
[latex]\\begin{array}{rl}\r\n4+(-3)&\\text{Different signs. Subtract }4-3\\text{. Positive number has greater absolute value.} \\\\\r\n1&\\text{Solution}\\end{array}[\/latex]<\/div>\r\n<\/div>\r\n
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Example 1.1.6<\/p>\r\n\r\n<\/header>\r\n

Add [latex]7+(-10).[\/latex]<\/div>\r\n
[latex]\\begin{array}{rl}\r\n7+(-10)&\\text{Different signs. Subtract }10-7.\\text{ Negative number has greater absolute value.} \\\\\r\n-3&\\text{Solution}\r\n\\end{array}[\/latex]<\/div>\r\n<\/div>\r\nFor subtraction of negatives, change the problem to an addition problem, which is then solved using the above methods. The way to change a subtraction problem to an addition problem is by adding the opposite of the number after the subtraction sign to the number before the subtraction sign. Often, this method is referred to as \"adding the opposite.\"\r\n
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Example 1.1.7<\/p>\r\n\r\n<\/header>\r\n

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\r\n\r\nSubtract [latex]8-3.[\/latex]\r\n

[latex]\\begin{array}{rl}\r\n8-3&\\text{Add the opposite of 3 to 8.} \\\\\r\n8+(-3)&\\text{Different signs. Subtract }8-3.\\text{ Positive number has greater absolute value.} \\\\\r\n5&\\text{Solution}\r\n\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Example 1.1.8<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nSubtract [latex]-4-6.[\/latex]\r\n

[latex]\\begin{array}{rl}\r\n-4-6&\\text{Add the opposite of 6 to }-4. \\\\\r\n-4+(-6)&\\text{Same sign. Add }4+6.\\text{ Keep the negative sign.} \\\\\r\n-10&\\text{Solution}\r\n\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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Example 1.1.9<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nSubtract [latex]9-(-4).[\/latex]\r\n

[latex]\\begin{array}{rl}\r\n9-(-4)&\\text{Add the opposite of }-4\\text{ to 9.} \\\\\r\n9+4&\\text{Same sign. Add }9+4. \\text{ Keep the positive sign.} \\\\\r\n13&\\text{Solution}\r\n\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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Example 1.1.10<\/p>\r\n\r\n<\/header>\r\n

Subtract [latex]-6-(-2).[\/latex]<\/div>\r\n
[latex]\\begin{array}{rl}\r\n-6-(-2)&\\text{Add the opposite of }-2\\text{ to }-6. \\\\\r\n-6+2&\\text{Different signs. Subtract }6-2.\\text{ Negative number has greater absolute value.} \\\\\r\n-4&\\text{Solution}\r\n\\end{array}[\/latex]<\/div>\r\n<\/div>\r\nMultiplication and division of integers both work in a very similar pattern. The short description of the process is to multiply and divide like normal. If the signs match (numbers are both positive or both negative), the answer is positive. If the signs don't match (one positive and one negative), then the answer is negative.\r\n\r\n<\/div>\r\n
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Example 1.1.11<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nMultiply [latex](4)(-6).[\/latex]\r\n

[latex]\\begin{array}{rl}\r\n(4)(-6)&\\text{Signs do not match, so the answer is negative.} \\\\\r\n-24&\\text{Solution}\r\n\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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Example 1.1.12<\/p>\r\n\r\n<\/header>\r\n

Divide [latex]-36 \\div -9.[\/latex]<\/div>\r\n
[latex]\\begin{array}{rl}\r\n-36 \\div -9 & \\text{Signs match, so the answer is positive.} \\\\\r\n4&\\text{Solution}\r\n\\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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Example 1.1.13<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nMultiply [latex]-2(-6).[\/latex]\r\n

[latex]\\begin{array}{rl}\r\n-2(-6)&\\text{Signs match, so the answer is positive.} \\\\\r\n12&\\text{Solution}\r\n\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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Example 1.1.14<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nDivide [latex]15 \\div -3.[\/latex]\r\n

[latex]\\begin{array}{rl}\r\n15\\div -3 &\\text{Signs do not match, so the answer is negative.} \\\\\r\n-5&\\text{Solution}\r\n\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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Key Takeaways:\u00a0A few things to be careful of when working with integers.<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nBe sure not to confuse a problem like\u00a0\u22123\u00a0\u2212 8 with \u22123(\u22128).\r\n