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One can easily find numbers with finite decimal representation with infinite binary representation. (Like $0.3$ and $0.01010101..$)

I assume there is an opposite case, meaning a number with finite binary representation but infinite decimal representation, does any of you know such number? if the existence is impossible then why?

3 Answers 3

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It’s impossible.

A number has a finite binary representation if and only if it can be written as a fraction whose denominator is a power of $2$: $\dfrac{k}{2^n}$ for some integer $k$ and some non-negative integer $n$.

A number has a finite decimal representation if and only if it can be written in the form $\dfrac{k}{2^m5^n}$ for some integer $k$ and non-negative integers $m$ and $n$.

Clearly the first is a special case of the second.

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    Clear, to the point, and correct. +12012-12-19
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Well, one can be a wiseguy:

$0.1_2=0.5_{10}=0.49999....$

Any finite decimal expansion can be made infinite...

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    I like that. +12013-04-21
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If a number, $q$, has a finite binary expansion, that means that $ q=\frac p{2^n} $ for some integers $p$ and $n$. Since $\frac12=\frac5{10}$, we have that $q=\dfrac p{2^n}=\dfrac{5^np}{10^n}$ also has a finite decimal expansion.