By W. W. Rouse Ball

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Verify that Rl to Rll are satisfied a 18. Verify that Rl to = Rll are 2 satisfied a=J. 19. 20. > when: = J- c = -3 when: 6= -2 c = 4 that the natural mimbers are special cases of the rationals. that the integers are special cases of the rationals. • The Number System Sec. 9] 39 any pair of rational numbers o and b, there exists a rational nvimx = b. Moreover, any two rationals Xi and Xa with this ber X such that a property are equal. 22. Prove: For any three rational numbers a, b, and c, where a 9^ 0, there exists a rational number x such that ax -\- b = c.

Fundamentals of Freshman Mathematics 34 In Probs. 19 to 22 prove the given statement > ~i9. 2" 1 21. Let ai ^22. 6. + forn > 1. = 2 and a„ = 2a„_i. = 3 and a„ = 3a„_i. 20. 3" n Then Then = — a„ a„ [Chap. 2 by mathematical induction. > 1 + 2" for n 3" for n for n 2ii > > > 1. 1. 1. Special Properties of Zero The behavior of zero is one of the more troublesome parts of the study of real numbers. ^ by hypotheses, is it equivalent to = = 6 X c. Since Hence 0. The equation a/0 = c However, X c is equal to for all and hence cannot equal a, which is not zero.

There is no automatic way of doing this; you must develop skill through experience and the use of your originality. Mathematics and Science Sec. 7. (a) 15 Other Methods of Proof you have If Indirect Proof. difficulty in constructing a direct you can sometimes make progress by using other tactics. The method of "indirect proof" rehes on the fact that, if not-p is false, then we attempt to show that p is true. Hence, to prove that p is true, this is to show that not-p accomplish The best way to not-p is false.