Answer by Alon Amit:

When someone says that 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- expansion is but one special class, of which the common base-10 or “decimal” expansion is but one instance.

For example, the number can be represented in base 10 like this:

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

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

(base 2, terminating)

(base 2, non-terminating)

(base 3, non-terminating).

Generally speaking a number *can* be represented by a finite expansion in base 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 . 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 allows a terminating decimal expansion but 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 is non-terminating”.

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

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