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Given a basis, like 10, what numbers can be exactly represented in a number base 10, without repeating decimals (is it called n-mals for a general base $n$)?

I assumed it is any number which could be written as a fraction of: $$ \frac{X}{2^p \times 5^q} $$ being 2 and 5 the prime factors of 10 and $n$ and $m$ and $X$ are integers.

Rule that applied to base 2 would be: $$ \frac{X}{2^p} $$

And to base 60 would be: $$ \frac{X}{2^p \times 3^q \times 5^r} $$

Is that correct? If not, what is the correct rule?

lvella
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  • I believe that is correct. Said another way, $x$ can be represented in base $b$ with a finite expansion if and only if $b^nx$ is an integer for some integer $n$. – MPW Nov 03 '16 at 14:31

1 Answers1

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Yes that is correct. If a number has a finite number of digits (the $i^{th}$ such digit being the value $a_i$) in base $b$ after the point then it can be represented by the sum:

$$\sum_{i=1}^n\frac{a_i}{b^i}=\sum_{i=1}^n\frac{a_i\cdot b^{n-i}}{b^n}=\frac{\sum_{i=1}^na_i\cdot b^{n-i}}{b^n}$$

If $b$ is a composite number then the denominator of each of these fractions can be split up and any common factors in the numerator can be cancelled out giving an expression like you listed.

Ian Miller
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