What are complex conjugate numbers

In this publication, we will consider what complex conjugate numbers are, and also list their main properties. The theoretical information presented is accompanied by practical examples for better understanding.

Definition of complex conjugate numbers

Given a complex number z = a + bi. Its complex conjugate is the number z = a – bi (a dash above is used for designation).

Thus, for complex conjugate numbers, the real parts are the same, and the imaginary parts differ in sign.

Example:

For a number z=3+2i complex conjugate is z = 3 – 2i.

Geometric interpretation

If we transfer complex conjugate numbers to the complex plane, then they will be a mirror image of each other about the real axis (RE).

Properties of complex conjugate numbers

1. If z = z, means the number z is valid.

Example:

z = 2, so z ∈ R, Consequently z = 2Ie z = z.

2. complex conjugate numbers are equal, i.e. | from | = |z|. And since such numbers are mirrored on the complex plane, their arguments differ in sign.

3. The sum of complex conjugate numbers is a real number: with + z = 2 RE z.

Example:

z=5+2i

z = 5 – 2i

with + z = 5 + 2i + 5 – 2i = 5 + 5 = 10, 10 ∈ R.

4. The product of complex conjugate numbers is equal to the square of their modulus and is a real number: with ⋅ z = | z |² ∈ R.

Example:

z = 6 – 4i

z = 6 + 4i

with ⋅ z = (6 – 4i)(6 + 4i) = 36 + 24i – 24i – 16i² = 36 – 16 ⋅ (-1) = 52, 52 ∈ R.

The module is considered as follows:

5 For z = a + bi и z = a – bi rightly:

6. For arbitrary complex numbers z₁ и z₂: