CH 24 IDENTITIES. [Each product is 35] Ch 24 Identities. Introduction

Transcription

1 139 CH 4 IDENTITIES Introduction First we need to recall that there are many ways to indicate multiplication; for eample the product of 5 and 7 can be written in a variety of ways: (7) (5)7 (5)(7) Repeated multiplication has its own terms and notation. To square a number means to multiply that number by itself. For eample the square of 5 is 5 5 = 5, and we can write 5 = 5. This is read either 5 squared or 5 to the second power or 5 to the second. In general, the square of n is written n. In the epression n, we call n the base and the eponent. We have a set of rules called the Order of Operations, and they tell us that multiplication is done before addition and subtraction. The rules also tell us that parentheses are always done first. Here are three eamples: = = 13 Multiply before adding [Each product is 35] In spreadsheets and computer languages, 5 times 7 is written 5 * 7 (3 + 10)(9 7) = (13)() = 6 Parentheses before anything else (8 + ) = 10 = 100 Parentheses before anything else Identities An identity in mathematics is a statement that two epressions are equal. If there are any letters (variables) in the epressions, then the two epressions must be equal no matter what numbers are assigned to the letters.

2 140 EXAMPLE 1: Give evidence that a + b = b + a is an identity. Solution: This statement is true because if a and b are replaced with any numbers whatsoever, the two epressions will be equal. For instance, if we let a = 10 and b = 5, then a + b b + a Since both a + b and b + a resulted in 15 for the chosen values of a and b, we have some good evidence that a + b = b + a, no matter what values of a and b we choose to test. (We could never really prove that it s an identity in this course because there are infinitely many numbers that a and b could be.) There s a fancy word we use to describe the fact that addition can be done in either order. What has a judge done when she commutes a death sentence and changes it to life in prison? She has reversed the original sentence. When you commute to wor, you go in the morning, wor, and then reverse direction to get home in the evening, The fact that a + b = b + a ending up where you started. for any numbers a and b is Thus, to commute means to called the commutative switch, or reverse. Since the property of addition. statement a + b = b + a is a switching of the order in which we add the a and the b, we say that addition is a commutative (accent on the mut ) operation.

3 141 Note #1: There s another commutative property; it s for multiplication. By using a pair of (different) numbers, convince yourself that y = y is a true statement for any values of and y. In short, multiplication is also a commutative operation. Note #: There would be no need for any mention of the commutative properties of addition and multiplication were it not for the fact that there are operations which are not commutative. For eample, consider the division problem 6, which equals. If 3 we commute the top and bottom, we get the division problem 3 6, which equals 1. Since switching, or reversing the two numbers in the division problem resulted in different answers, we conclude that division is not a commutative operation. Any ideas about whether subtraction is, or is not, a commutative operation? EXAMPLE : Give evidence that ( + 5)( 5) = 5 is an identity. Solution: Let s plug the value = 7 (just as an eample) into both sides of this statement: ( + 5)( 5) 5 (7 + 5)(7 5) 7 5 (1)() Both sides came out to the same number. Although we used only one value of to test the statement, it truly is an identity; any number you use for will result in equal numbers. [In an algebra course you ll see a way to prove conclusively that ( + 5)( 5) = 5 without using any number values of at all.] On the other hand, to prove that something is not an identity, we need only give one eample where the statement fails. Consider the net eample.

4 14 EXAMPLE 3: Prove that ( + 3)( + ) = + 6 is not an identity. Solution: Let = 10 (I just chose this number off the top of my head). Then, woring each side of the statement separately, we obtain: ( + 3)( + ) + 6 An eample which maes a statement fail is called a countereample. (10 + 3)(10 + ) (13)(1) We conclude that ( + 3)( + ) = + 6 is not an identity (even if there is some value of that would mae it true). In other words, if you ever wrote this statement in algebra, you d be mared wrong (very wrong!). EXAMPLE 4: Give evidence that ( + 3) = is an identity. Solution: 7 is a nice number. Let s use it to test the statement: ( + 3) (7 + 3) 7 + 6(7)

5 143 EXAMPLE 5: Determine whether the statement 3 = 6 is an identity. Solution: statement: We ll let = 3. We now evaluate each side of the 3 6 (3 )(3 3 ) 3 6 (9)(7) We conclude that 3 = 6 is not an identity. (Apparently, multiplying the eponents does not wor.) EXAMPLE 6: Determine whether = 5 3 is an identity. Solution: Let s test the number = 4: , The statement wored for = 4. Remember that for Pre-Algebra, this is enough evidence to conclude that it is an identity. An Algebra Proof: 5 = = = = 3

6 144 Homewor 1. By using numbers, show that the statement a(b + c) = ab + c is false (that is, it s not an identity). Note: When you calculate the left side, do the parentheses first by adding b and c, and then multiply by a. On the right side, multiply and then add.. By using numbers, give some evidence that the statement a(b + c) = ab + ac is true. 3. Consider the statement: ( + y) 3 = 3 + y 3. Decide if it s true or false, and give your reasoning. 4. Now do the same problem again, this time using = 10 and y = 0. What conclusion do you draw? How does this compare with the answer to the previous problem? What gives here? 5. Consider the statement =. Test its validity by using =. Now test it again using = 10. Suggestion: Never use the numbers 0, 1, or when testing whether a statement is an identity. They can be very misleading. Determine whether each statement is true or false; that is, whether it s an identity: 6. ( + 3)( 3) = 6 7. ( + 3)( 3) = 9 8. ( + 3)( 3) = 9 9. (a + b)(a b) = a b 10. (a + b)(a b) = a + b 11. (a + b)(a b) = (a b) 1. (a + b)(a b) = a b 13. (n + 3)(n + ) = n (n + 3)(n + ) = n (n + 3)(n + ) = n + 5n + 6

7 (a + 5) = a (a + 5) = a (a + 5) = a + 10a (a + 5) = a + 5a n(n 3 ) = n 4 1. a 3 a 3 = a = = 3 Solutions 1. Let a =, b = 3, and c = 4. Then a(b + c) = (3 + 4) = (7) = 14; but, ab + c = (3) + 4 = = 10.. Using the same numbers as above: a(b + c) = (3 + 4) = (7) = 14; and, ab + ac = (3) + (4) = = 14 Note: Just because these numbers made the formula wor doesn t necessarily mean that the formula wors all the time. All we ve done at this point is give some justification for the formula. 3. It s a false statement. For eample, if we let = and y = 3, we get ( + y) 3 = ( + 3) 3 = 5 3 = 15; but, 3 + y 3 = = = This problem shows one reason to avoid using 0 when testing an identity, as indicated in the bo net to the problem. It may lead you to the wrong conclusion.

8 When =, the statement appears to be true; but when = 10, it comes out false. Again, heed the suggestion in the bo. Since we found a value of ( = 10) that made the statement false, we have found a countereample, and that is all that is needed to declare the statement false. 6. Let = 5, for instance. Then ( + 3)( 3) = (5 + 3)(5 3) = (8)() = 16 6 = 5 6 = 5 6 = 19 The statement is therefore false. 7. False 8. Let = 10, for eample. Then ( + 3)( 3) = (10 + 3)(10 3) = (13)(7) = 91 9 = 10 9 = = 91 This situation (letting = 10) does not prove it conclusively; after all, maybe it wors for = 10 but not for some other value of. Indeed, we should try the epression for other values of -- the more s we use, the more sure we can be of our result. But for Pre-Algebra, one good eample is good enough. 9. False 10. False 11. False 1. True 13. False 14. False 15. True 16. False 17. False 18. True 19. False 0. True 1. False. False 3. True It is hard to fail, but it is worse never to have tried to succeed. Theodore Roosevelt