What is the complex conjugate

This is not a math tutorial. See Why am I writing about math?

The complex conjugate of a complex number is the number with an equal real part but with an imaginary part that has an equal magnitude, but the opposite sign.

If $a$ and $b$ are real numbers, then the complex conjugate of $a + bi$ is $a - bi$.

Notation

The complex conjugate of $z$ is written as $\bar{z}$ (LaTex: $\bar{z}$, for multiple characters or longer expressions $\overline{ab}).

References

Wikipedia contributors. “Complex conjugate.” Accessed on: January 20, 2026. https://en.wikipedia.org/wiki/Complex_conjugate .