Definition:Wholly Real/Abbreviated Notation

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
• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory