What does it mean when a number is non-terminating?

Answer by Alon Amit:
When someone says that \pi is non-terminating” it almost invariably means that they are rather confused.
A number cannot be terminating or non-terminating. The representation of a real number in the very special form of an expansion in some base may be terminating or not. Numbers can be represented in many ways, of which the base-b expansion is but one special class, of which the common base-10 or “decimal” expansion is but one instance.

For example, the number \frac{1}{4} can be represented in base 10 like this:

\frac{1}{4} = 0.25

which is a terminating expansion (only finitely many digits are needed). But the same number in the same base can also be represented as the non-terminating decimal

\frac{1}{4} = 0.249999\ldots

and the same number can be represented in other bases like this:

\frac{1}{4} = 0.01_2     (base 2, terminating)

\frac{1}{4} = 0.00111111\ldots_2     (base 2, non-terminating)

\frac{1}{4} = 0.0202020\ldots_3     (base 3, non-terminating).

Generally speaking a number can be represented by a finite expansion in base b if (and only if) it is rational and the denominator of its reduced form has no prime factor which isn’t also a factor of b. So in base 10, which is what we normally use, a number has a non-terminating expansion if (and only if) it is a rational number whose denominator is the product of any number (possibly none) of 2’s and 5’s (but nothing else). So \frac{23}{20} allows a terminating decimal expansion but \frac{20}{23} does not.

This is a rather useless and ad-hoc property of a number. When people say “a number is non-terminating” they almost always intend to say that it is irrational, but they’re using the wrong terminology.

Specifically, nothing good ever comes out of a text that starts with \pi is non-terminating”.

What does it mean when a number is non-terminating?

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