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?
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$\begingroup$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.
$\endgroup$ 1 $\begingroup$Well, one can be a wiseguy:
$$0.1_2=0.5_{10}=0.49999....$$
Any finite decimal expansion can be made infinite...
$\endgroup$ 1 $\begingroup$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.
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