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) = a2 + b2

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




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.