Re: [中學] 循環小數和分數的互換規則？
※ 引述《JohnMash》之銘言：
請問證明如何寫會比較清楚明瞭?
作者: JohnMash (Paul) 看板: Math
※ 引述《gary85238 (小黃)》之銘言：
: Let a,b 屬於 N , a<b and a is relative prime to b.
: Use the Pigeonhole Principle to show that the decimal representation
: of the fraction a/b either terminate or repeats.
if a/b terminates in its representation, done.
if a/b does not terminate
consider a, 10a, 100a, ....,10^b a,....
a=a_0 (mod b)
10a=a_1 (mod b)
100a=a_2 (mod b)
......
because a_0,a_1,a_2,...,a_b are all less than b
then there exist at least two a_i, a_j equal
that is 10^i a= kb+10^j a
10^i a/b=k+10^j a/b
that is 10^i a/b has the same decimal part as 10^j a/b
it means a/b repeats its decimal representation
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