# Definition:Wholly Real/Abbreviated Notation

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## Definition

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.

## 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*