Definition:Complex Number/Wholly Real

Definition

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

• Definition:Wholly Imaginary

Sources

• 1957: E.G. Phillips: Functions of a Complex Variable (8th ed.) ... (previous) ... (next): Chapter $\text I$: Functions of a Complex Variable: $\S 1$. Complex Numbers: The abbreviated notation
• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: The Complex Number System