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Identity, Grouping, and Signs
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Let s review the mechanics of addition and subtraction, and make sure you understand how the signs work Then we ll proceed to two important rules, called laws, that govern addition and subtraction These laws apply not only to the integers, but to all quantities, expressions, and numbers you ll encounter in algebra
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The identity element There is one special integer that you can add to, or subtract from, any integer, and it will never change the value That integer is 0 Because adding 0 to a number gives you the same number again, 0 is called the additive identity element You can also call it the subtractive identity element, although that term is informal The word identity means that you always get an output value identical to the input value For any integer a, the following two statements are always true:
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a+0=0 and a 0=0
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Grouping with parentheses A couple of minutes ago, you saw some expressions containing parentheses around a negative number or quantity Here s one of those expressions, with the right-hand side changed to a variable:
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3 ( 5) = a These parentheses indicate that you should consider 5 as negative 5, not the subtraction of 5 If you don t write the parentheses, the above expression is 3 5 = a This is an example of improper mathematical grammar It s a problem because it s ambiguous People can t be sure what it means Should you subtract 5 from 3 That would give you 8, not 2 Should you subtract 5 from 3 twice That would give you 13 Does the double minus sign mean something different from plain old subtraction or addition Pay attention to parentheses when you see them When you write an expression, be sure to include parentheses when you need them You can add extra parentheses to an expression as long as you don t change the meaning Always be sure the number of opening parentheses in an expression is the same as the number of closing parentheses
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Signs in addition and subtraction Here s a summary of how addition and subtraction work for negative integers as well as for positive ones:
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When you add a positive, the result grows larger When you subtract a positive, the result grows smaller When you add a negative, the result grows smaller; it s like subtracting a positive When you subtract a negative, the result grows larger; it s like adding a positive
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You might want to memorize two general facts For any two integers a and b, a + ( b) = a b and a ( b) = a + b
Are you confused
Here s a real-life situation where the idea of subtracting a negative number makes pretty good sense Suppose the people in your town are trying to reduce their driving because of high gasoline prices They ve started a support group People go to meetings once a month to share information about how much less they ve driven in the past month compared with the month before that One after another, the members of the group tell their stories William says, I drove 45 fewer miles Anna says, I cut back by 65 miles Maria says, I drove 200 miles less last month! Then it comes to you You have
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not done so well The group leader asks, How much did you cut back Go on, don t be afraid to tell us You blush, clear your throat, hold your head up, and declare, I reduced my driving last month by negative 80 miles
Here s a challenge!
Start with the integer 5, add 3 to it, then subtract 6 from that, then subtract 10 from that, then add 14 to that, and finally subtract 21 from that What s the result
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