Geometrical Effects of Conjugating a Complex Number

What is the Conjugate of a Complex Number?

A conjugate of a complex number is formed by changing the sign of its imaginary part when the complex number is represented in the standard form a + bi.
The conjugate of a + bi is written as a - bi. This transformation is called conjugation.
The concept of conjugation is essential in calculations involving complex numbers, especially when dealing with division or rationalising the denominator.

Complex numbers can be visualised using the complex plane, also known as the Argand Diagram. A complex number a + bi is represented as a point (a, b) on the complex plane.
The x-axis represents the real part (a), and the y-axis represents the imaginary part (b).
When a complex number is reflected in the x-axis, the new complex number produced is the conjugate of the original number. This reflection represents the geometrical effect of conjugation.
Hence, conjugating a complex number is geometrically equivalent to reflecting the point about the x-axis in the complex plane.

Consider the complex number 3 + 4i. Its conjugate will be 3 - 4i.
In the Argand diagram, the point (3, 4) will represent the complex number 3 + 4i.
The point representing its conjugate, 3 - 4i, will be (3, -4). As we can see, these two points are mirror images of each other about the x-axis.
Notice that the conjugate is the same distance from the x-axis as the original number, but on the opposite side.

The modulus (or absolute value) of a complex number a + bi is the distance from the origin to the point (a, b) on the complex plane.
The modulus of a complex number and its conjugate are equal, i.e., a + bi = a - bi .
This property is vital in calculations and proofs involving complex numbers.

In disciplines like physics, engineering, and computer science, this geometrical interpretation of conjugating a complex number is hugely beneficial for mastering concepts involving complex quantities.
In addition, the ability to visually understand the process of conjugation assists in simplifying complex equations in pure mathematics.