30 1. THE CONSTRUCTION OF REAL AND COMPLEX NUMBERS where aj S and 0 x Sn 1/10n. We conclude that Sn converges to x and we get (1.3) x = a1/10 + a2/102 + · · · + an/10n + · · · . Exercise 1.10.35. Let x be a irrational number between 0 and 1. Show that there is only one way to express x in the form (1.3). We now turn to rational numbers between 0 and 1. We can apply the above procedure to rational numbers with the possibility of equality in any of the inequalities above. Suppose that x has a terminating decimal expansion. That is, suppose there exists N so that an = 0 for all n N and aN = 0. Then we can write x = a1/10 + a2/102 + · · · + aN/10N. Exercise 1.10.36. (i) Show that if r is a rational number in (0, 1), then the decimal expansion of r terminates if and only if the denominator of r has the form 2a5b where a and b are nonnegative integers and are not both zero. (ii) With r as above, show that the last nonzero digit of r is in the m-th place where m = max(a, b). Note that rational numbers with terminating decimal expansions are the only real numbers between 0 and 1 for which equality can occur in the procedure above. Next consider a rational number r = p/q in (0, 1) for which q is relatively prime to 10. From Euler’s theorem (see the project in Section A.10.1), q divides 10φ(q) 1. Let n be the smallest natural number so that q divides 10n 1. Then (p/q)(10n 1) is an integer which we denote by m. That is, m = p q (10n 1) or p q = m 10n 1 . We can now write p q = m 10n 1 = m 10n (1 10−n)−1 = m 10n (1 + 10−n + 10−2n + · · · ) = m/10n + m/102n + · · · . As 0 p/q 1, we have m 10n. Thus the right-hand side of the equation above gives us a periodic decimal expansion of p/q whose period has length n. Exercise 1.10.37. Let p/q be a rational number between 0 and 1. If q and 10 are relatively prime, show that p/q has a unique periodic decimal expansion with the length of the period equal to the order of 10 mod q.
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