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.
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.