# Correct term for the set of numbers which cannot be expressed in a particular base

For every base there's a set of fractions that can't be expressed with the exponent in that base, for instance 1/3 cannot be represented in decimal and 1/10 can't be represented in binary (particularly relevant to floating-point calculations in computers).

I've heard these described as irrational, but I think that's incorrect (although the true irrational set will be contained in this set for any base).

What is the correct term (if there is one)?

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What you're asking about is rational numbers with a finite expansion in some base $b$. I'm a bit tired so I don't have time to work through all the details. But I think it's something along the lines of $1/n$ has a finite decimal representation in base $b$ if and only if for ever prime $p$ dividing $n$ then $p$ also divides $b$. – JSchlather Jan 3 '13 at 8:46
@JacobSchlather - cheers, I'd kind of just assumed that bit (that every base has this set), but what I'd like to know is whether there's a term for it. – Keith Jan 3 '13 at 8:51
It's called finite decimal (or base b) expansion. Like 1/3 is a rational, but doesn't have a finite decimal expansion. You can show that regardless of the base, a number is rational IFF it can be expressed as a finite or repeating decimal. – Calvin Lin Jan 3 '13 at 8:57
I don't know of any special term. Also here Arturo Magidin proves the characterization I mentioned. – JSchlather Jan 3 '13 at 8:58
@Keith, more like "numbers without a finite binary expansion." – Eric Stucky Jan 3 '13 at 9:30