62 Addition and Subtraction

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We can use the associative law for addition to get a + [( b) + ( c)] We can simplify this to a + ( b c) Now we know that when both operations are subtraction, (a b) c = a + ( b c)

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We haven t started to dissect the anatomy of multiplication yet That will come in the next chapter But you ve had basic multiplication in your arithmetic classes, so let s cheat for a moment and take advantage of that What are you actually doing when you change c to c Here s an alternative to the number reflector idea When you want to find the negative (also called its additive inverse) of any integer, multiply by 1 It works like this: c ( 1) = c and c ( 1) = c

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Here s a challenge!

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Based on the commutative law for the sum of two integers and the associative law for the sum of three integers, show that for any three integers a, b, and c a+b+c=c+b+a

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Solution

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If you can manipulate the left-hand side of this equation to get the expression on the right-hand side, that s good enough Because these statements are not very complicated and the proof is not too hard, you can write it as a table with statements on the left and reasons on the right Table 4-1 shows how it s done This is a simple statements/reasons (S/R) proof

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Practice Exercises

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Table 4-1 Here is a proof that shows how you can reverse the order in which three integers a, b, and c are added, and get the same sum As you read down the left-hand column, each statement is equal to all the statements above it

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Statements a+b+c a + (b + c) a + (c + b) (c + b) + a c+b+a QED Reasons Begin here Group the second two integers Commutative law for the sum of b and c Commutative law for the sum of a and (c + b) Ungroup the first two integers Latin Quod erat demonstradum, translated into English as Which was to be proved

Practice Exercises

This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App A The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 Evaluate and compare these two sums: a = | 3 + 4 + ( 5) + 6| and b = | 3| + |4| + | 5| + |6| What general fact can you deduce from the results 2 To illustrate the importance of the placement of parentheses in a mixed sum and difference, evaluate the following two expressions In long strings of sums and differences, you should first perform the operations inside the parentheses from left to right, and then perform the operations outside the parentheses from left to right Here are the expressions: (3 + 5) (7 + 9) (11 + 13) 15 and 3 + (5 7) + (9 11) + (13 15) 3 Using the rules explained in the previous exercise, how should you evaluate the string of sums and differences if there are no parentheses at all Here it is: 3 + 5 7 + 9 11 + 13 15 4 Suppose someone tells you that there was a significant trend in the mid-winter average temperatures in the town of Hoodopolis during the period 1998 through 2005 You want to find out if this is true You come across some old heating bills from the utility company that show how much warmer or cooler a given month was, on the average,