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.

Examples

Let $z_1 = 4 - 3 i$ and $z_2 = -1 + 2 i$.

Example: $\overline {z_1} - \overline {z_2}$

$\overline {z_1} - \overline {z_2} = 5 + 5 i$

Also see

• Complex Number equals Conjugate iff Wholly Real

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory
• 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
• 1974: Robert Gilmore: Lie Groups, Lie Algebras and Some of their Applications ... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $3$. FIELD
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: The Complex Number System
• 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