# Definition:Complex Conjugate

## Definition

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

Then the **(complex) conjugate** of $z$ is denoted $\overline z$ and is defined as:

- $\overline z := a - i b$

That is, you get the complex conjugate of a complex number by negating its imaginary part.

### Complex Conjugation

The operation of **complex conjugation** is the mapping:

- $\overline \cdot: \C \to \C: z \mapsto \overline z$.

where $\overline z$ is the complex conjugate of $z$.

That is, it maps a complex number to its complex conjugate.

## Also known as

The complex conjugate of a complex number is usually just called its **conjugate** when (as is usual in the context) there is no danger of confusion with other usages of the word **conjugate**.

The notation $z^*$ is a frequently encountered alternative to $\overline z$.

The notation $\hat z$ is also occasionally seen.

## Also see

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 1.2$. The Algebraic Theory - 1964: Murray R. Spiegel:
*Theory and Problems of Complex Variables*... (previous) ... (next): $1$: Complex Numbers: The Complex Number System - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{II}$: Subgroups - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $6$: Complex Numbers - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms