Definition:Complex Conjugate

Let $$z = a + \imath b$$ be a complex number.

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

$$\overline z \ \stackrel {\mathbf {def}} {=\!=} \ a - \imath b$$

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 $$\overline z$$.