A **complex conjugate** z, has one real part and one imaginary part; the parts have the same magnitude but different signs. In other words, the conjugate of a complex number is the same number but a reversed sign for the imaginary part.

Generally, speaking, the complex conjugate of *a* + *bi* is *a* – *bi* (where *a* and *b* are two real numbers).

A few examples:

- Conjugate of z = 5 + 3i is z = 5 – 3i
- Conjugate of z = -6 – i is z = -6 + i
- Conjugate of z = 9i is z = -9i

A complex conjugate negates the imaginary part, so geometrically it is a transformation of the complex plane where all points are reflected over the real axis. All points above and below the axis are exchanged; in other words, you can find the complex conjugate of any complex number geometrically by reflecting *z* across the real axis.

## Complex Conjugate Properties and Rules

The following properties and rules apply to complex conjugates:

## Usefulness of the Complex Conjugate

The complex conjugate is very useful because if you multiply any complex number by its conjugate, you end up with a real number [1]:

(a + jb)(a – jb) = a^{2} – j^{2}b^{2} = a^{2} + b^{2}.

It also gives us another way to interpret **reciprocals**. A complex number multiplied by its conjugate is the square of its absolute value (or complex modulus):

z · z = |z|^{2}.

Therefore:

Geometrically, 1/|z| and z are on the same ray from the origin, but 1/|z| is a quarter of the length [2], as the following image shows:

## References

Complex conjugate image: Oleg Alexandrov,

[1] Multiplication and Division. Retrieved November 9, 2021 from: http://hyperphysics.phy-astr.gsu.edu/hbase/cmplx2.html

[2] Joyce, D. Dave’s Short Course on Complex Numbers. Reciprocals, conjugates, and division. Retrieved November 9, 2021 from: https://www2.clarku.edu/faculty/djoyce/complex/div.html