What is the name for a fraction whose expansion extends forever?

Question

Irrespective of whether it is a rational number like $\frac{1}{3}$, an irrational number, like $2^{1/2}$ or a transcendental number, like $\pi$, is there a word for all decimal fractions which continue without end?

Answer 1

Every real number can be written as a nonterminating decimal expansion using trickery like $.999999... = 1$, so the question should really be, which numbers are forced to be written as a nonterminating expansion.

First consider a slightly easier question, which numbers are forced to be written as a nonrepeating expansion? The answer to this question is independent of the base (so long as it is integral >= 2) chosen for the expansion, and it is the irrational numbers. The proof of this goes by long division and noting that division by an integer gives only finitely many possible choices for remainder, so repetition will eventually occur. Thus, in any base any irrational number must be represented in a nonterminating expansion.

Then the question remains, of the rational numbers, which amongst them can be written in a nonterminating expansion. E.g. $1/10 = .1$ but $1/3=.33333333...$ must repeat base 10. This question's answer obviously does depend on the base, so lets fix a base $b$. To say that $x$ has a terminating expansion in base $b$ means that $x = \sum_{i=-k}^na_i b^{-i}$ for a finite $n,k$ and some $a_{-k}, \ldots, a_0, a_1, \ldots, a_n \in \{0,1,\ldots, b-1\}$. This is possible iff multiplying by a large power of $b$ gives an integer, so we conclude that $x$ has a finite representation in base $b$ if and only if it is a rational number which can be written as $a/b^n$ for $a, n $ integers with $n \geq 0$.

Such numbers are called $b$-adic numbers, most commonly considered only for primes $p$, and referred to as $p$-adic numbers. https://en.wikipedia.org/wiki/Dyadic_rational

Thus the answer to your question is that $x$ must have a nonterminating expansion in base $b$ iff it $x$ is not $b$-adic (this includes all irrationals).