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.

It follows directly from this definition that $z$ is wholly real iff $z = \overline z$.

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.

Complex Conjugation
The operation of complex conjugation is the mapping $\overline \cdot: \C \to \C: z \mapsto \overline z$.

It maps a complex number its complex conjugate.