# Definition:Complex Number/Wholly Real

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## 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:

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**.

## 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