# Complex conjugate

The complex conjugate of a complex number is formed by changing the sign between the real and imaginary components of the complex number. Given a complex number of the form,

z = a + bi

where a is the real component and bi is the imaginary component, the complex conjugate, z*, of z is:

z* = a - bi

The complex conjugate can also be denoted using z. Note that a + bi is also the complex conjugate of a - bi.

The complex conjugate is particularly useful for simplifying the division of complex numbers. This is because any complex number multiplied by its conjugate results in a real number:

(a + bi)(a - bi) = a^{2} + b^{2}

Thus, a division problem involving complex numbers can be multiplied by the conjugate of the denominator to simplify the problem.

Example

Simplify

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## Properties of complex conjugates

Below are some properties of complex conjugates given two complex numbers, z and w. Conjugation is distributive for the operations of addition, subtraction, multiplication, and division.

If a complex number only has a real component:

The complex conjugate of the complex conjugate of a complex number is the complex number:

Below are a few other properties.

## Graph of the complex conjugate

Below is a geometric representation of a complex number and its conjugate in the complex plane.

As can be seen in the figure above, the complex conjugate of a complex number is the reflection of the complex number across the real axis.