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 :
(a + jb)(a – jb) = a2 – j2b2 = a2 + b2.
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.
Geometrically, 1/|z| and z are on the same ray from the origin, but 1/|z| is a quarter of the length , as the following image shows:
Complex conjugate image: Oleg Alexandrov,
 Multiplication and Division. Retrieved November 9, 2021 from: http://hyperphysics.phy-astr.gsu.edu/hbase/cmplx2.html
 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