Re: [中學] 循環小數的循環節

[JohnMash (Paul)]

※ 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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推 ATOU0526    :多謝不過有人可以幫忙翻譯一下嗎?原文已經離我太遙遠
→ BARGARYARLOO:寫這樣已經很清楚了吧  自己查字典比較快
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◆ From: 27.147.57.77
※ 編輯: JohnMash        來自: 27.147.57.77         (01/08 11:19)
→ lbBNbLX     :原來早就出現過XD
推 coolbetter33:也證明了循環節數不大於b