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

Conjugate of $2 + 3 i$

The complex conjugate of $2 + 3 i$ is given by:

$\overline {2 + 3 i} = 2 - 3 i$


Conjugate of $5 - 2 i$

The complex conjugate of $5 - 2 i$ is given by:

$\overline {5 - 2 i} = 5 + 2 i$


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

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

Then:

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


Also see

  • Results about complex conjugates can be found here.


Sources

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