Prove that the decimal expression of any rational number is periodic. For this, let a/b be your rational number, with (a, b) = 1 and b belongs to N.
(a) Show that there are two different non-negative integers k > l >= 0 such that (10^k − 10^l) is a multiple of b. Hint: if (b, 10) = 1 then apply Euler’s theorem. If not, write b = b1 x b2 with (10, b1) = 1 and b2 = 2r1 x 5r2 . Apply Fermat’s little theorem to b1 and figure out what to do with b2.
(b) Say why the previous part implies periodicity of the decimal expression. What can you say about the period?
(c) The decimal expression is said to be pure periodic if the periodic part starts “right after the dot”. For example, 43/6 = 7.16, 1/56 = 017857142 and 2/15 = 0.06 are not pure, while 10/11 = 0.90 and 1/37 = 0.027 are pure. Show that the decimal expression of a/b is pure if and only if (10, b) = 1
2007-11-29
21:17:07
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2 answers
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azn137
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