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We can break this down step-by-step, paying careful attention to signs and using parentheses when we need them Here we go: 5 + ( 3) = 5 3 = 2 2 ( 6) = 2 + 6 = 8 8 10 = 2 2 + 14 = 12 12 ( 21) = 12 + 21 = 33
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The Commutative Law for Addition
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In basic arithmetic, you learned that you can add a long string of numbers backward or forward, and it doesn t matter Good accountants take advantage of this when checking their work They ll add up a column of numbers from top to bottom, then again from bottom to top, just to be sure they have done the arithmetic correctly
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It works when you add The fact that you can add two integers in either order and get the same result is called the commutative law for addition It means you can commute (interchange) the two numbers you re adding, called the addends, and get the same sum either way In formal terms, a mathematician would say that for any two integers a and b,
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a+b=b+a This works whether the numbers are positive, negative, or 0 It also works if there are three, four, five, or more numbers in a sum, as long as the number of addends is not infinite
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It fails when you subtract In subtraction, the order does matter It s easy to find an example that shows why Consider this:
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3 5 = 2
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58 Addition and Subtraction
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but 5 3=2 In formal terms we would say that for any two integers a and b, it is not always true that a b=b a In fact, it is almost never true It only works if a and b happen to be the same
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Turning it inside-out We can use a trick that will make the commutative law sort of work with subtraction This trick is often used by accountants who must work with long columns of credits (money added) mixed with debits (money taken away) This trick involves taking every subtraction and turning it into the addition of a negative number Remember that adding a negative is the same as subtracting a positive That is, for any two integers a and b,
a b = a + ( b) Because the right-hand side of this equation is an addition problem, we can apply the commutative law and get a + ( b) = b + a Now we can combine the above two equations into a three-way equation: a b = a + ( b) = b + a Then we can get rid of the middle term and write a b = b + a Let s call this the inside-out commutative law for subtraction That s not a formal name, but you might find it useful as a memory aid
Are you confused
Imagine that you have a checking account and your balance on January 1 was exactly $700 Consider that as a starting deposit By the end of June, you ve made 15 deposits and written 20 checks You want to figure out your balance as of June 30 You convert all the checks to negative deposits For example, a check for $25 becomes a negative deposit of $25 Now you can add all the positive deposits and negative deposits in any order, and you ll always end up with the same final balance if you don t make any calculation errors!
The Associative Law for Addition
Here s a challenge!
Suppose you have bought a new car and you want to go for a test drive You live on a flat plain that seems to stretch forever in all directions You start driving on a straight highway that runs north and south for hundreds of miles on either side of your home town You drive 25 miles north, then turn around and drive 45 miles south Then you turn around again, driving 50 miles north Then you go 7 miles south, 12 miles north, 49 miles south, and finally 5 more miles south How far from your home town, and it what direction, will you finish Solve this problem in two different ways
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