Definition:Complex Number/Wholly Real

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A complex number $z = a + i b$ is wholly real if and only if $b = 0$.

Abbreviated Notation

Let $z = a + i b$ be a complex number such that $b = 0$.

That is, let $z$ be wholly real: $z = a + 0 i$, or $\tuple {a, 0}$

Despite the fact that $z$ is still a complex number, it is commonplace to use the same notation as if it were a real number, and hence say $z = a$.

While it is in theory important to distinguish between a real number and its corresponding wholly real complex number, in practice it makes little difference.

Also known as

Variants on wholly real are:

completely real
entirely real

and so on.

Some sources gloss over the distinction between a real number and a wholly real complex number and merely refer to a number of the form $z = a + 0 i$ as a real number.

Also see

  • Results about wholly real can be found here.