Complex Conjugates

Complex conjugates are pairs of complex numbers, where both have the same real part but opposite imaginary parts.
The complex conjugate of a + bi is identified as a - bi. Reverse is true too: the complex conjugate of a - bi is a + bi.

The symbol for complex conjugate is an overline, for example, if z = a + bi, then the complex conjugate is denoted by z̅ = a - bi.

Adding or subtracting complex conjugates always results in a real number. In other words, the sum or difference of a complex number and its conjugate has no imaginary part.
Multiplication or division of a complex number with its conjugate results in a real number as well. These operations often simplify complex number equations.
The product of a complex number and its conjugate equals the square of the modulus of the complex number. That is, **zz̅ = z ²**.

Complex conjugates are handy in simplifying fractions involving complex numbers. The process is called rationalising the denominator.
Complex conjugates also have notable applications in fields like physics and engineering, for instance, in signal processing and quantum physics.

On the complex plane, a complex number and its conjugate are reflections of each other over the x-axis. That is, they have the same ‘x’ value (real part) but mirrored ‘y’ values (imaginary parts).