Definition:Complex Conjugate
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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
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
- 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
