2.7: Infinite decimal expansions

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The standard written algorithms for calculating with integers extend naturally to terminating decimals. But how is one supposed to calculate exactly with decimals that go on for ever?

Problem 64 The decimals listed here all continue forever, recurring in the expected way. Calculate:

(a) $0.55555 \dots + 0.66666 \dots =$

(b) $0.99999 \dots + 0.11111 \dots =$

(d) $0.33333 \dots \times 0.66666 \dots =$

(e) $1.22222 \dots \times 0.818181 \dots =$

Problem 65

(a) Show that any decimal b n b n - 1 ⋯ b 0 . b -1 b -2 ⋯ b - k that terminates can be written as a fraction with denominator a power of 10.

(b) Show that any fraction that is equivalent to a fraction with denominator a power of 10 has a decimal that terminates.

(c) Conclude that a fraction $\frac{p}{q}$ , for which HCF(p,q) = 1, has a decimal that terminates precisely when q divides some power of 10 (that is, when $q = 2^{a} \times 5^{b}$ for some non-negative integers a, b).

(d) Prove that any fraction $\frac{p}{q}$ , for which HCF(p,q) = 1, and where q is not of the form $q = 2^{a} \times 5^{b}$ , has a decimal which recurs, with a recurring block of length at most q – 1.

(e) Prove that any decimal which recurs is the decimal of some fraction.

Problem 66

(a) Find the fraction equivalent to each of these recurring decimals:

(i) 0.037037037···

(ii) 0.370370370 ···

(iii) 0.703703703 ···

(b) Let a, b, c be digits ( $0 \leq a, b, c \leq 9$ ).

(i) Write the recurring decimal “0.aaaaa · · · “ as a fraction.

(ii) Write the recurring decimal “0.ababababab · · · “ as a fraction.

(iii) Write the recurring decimal “0.abcabcabcabcabc · · · “ as a fraction.

Problem 67 Find the lengths of the recurring blocks for:

(a) $\frac{1}{6}, \frac{5}{6}$

(b) $\frac{1}{7}, \frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7}$

(c) $\frac{1}{11}, \frac{2}{11}, \frac{3}{11}, \frac{4}{11}, \frac{5}{11}, \frac{6}{11}, \frac{7}{11}, \frac{8}{11}, \frac{9}{11}, \frac{10}{11}$

(d) $\frac{1}{13}, \frac{2}{13}, \frac{3}{13}, \frac{4}{13}, \frac{5}{13}, \frac{6}{13}, \frac{7}{13}, \frac{8}{13}, \frac{9}{13}, \frac{10}{13}, \frac{11}{13}, \frac{12}{13}$

Problem 68 Decide whether each of these numbers has a decimal that recurs. Prove each claim.

(a) $0.12345678910111213141516171819202122232425262728293031 \dots$

(b) $0.100100010000100000100000010000000100000000100000000010 \dots$

Problem 69 For which real numbers x is the decimal representation of x unique?

Problem 68 raises the question as to whether one person, who has total control, can specify the digits of a decimal so as to be sure that it neither terminates nor recurs: that is, so that it represents an irrational number. The next problem asks whether one person can achieve the same outcome with less control over the choice of digits.

Problem 70 Players A and B specify a real number between 0 and 1. The first player A tries to make sure that the resulting number is rational; the second player B tries to make sure that the resulting number is irrational. In each of the following scenarios, decide whether either player has a strategy that guarantees success.

(a) Can either player guarantee a “win” if the two players take turns to specify successive digits: first A chooses the entry in the first decimal place, then B chooses the entry in the second decimal place, then A chooses the entry in the third decimal place, and so on?

(b) Can either player guarantee a win if A chooses the digits to go in the odd-numbered places, and (entirely separately) B chooses the digits to go in the even-numbered places?

(c) What if A chooses the digits that go in almost all the places, but allows B to choose the digits that are to go in a sparse infinite collection of decimal places (e.g. the prime-numbered positions; or the positions numbered by the powers of 2; or ...)?

(d) What if A controls the choice of all but a finite number of decimal digits?