The multiplication of complex numbers is generally done using the transformation i2 = -1. Let’s immediately demonstrate this with an example. Let’s assume that z1 and z2 are two complex numbers, and they are defined as follows:

z1 = 2 + 3i and z2 = 4 - 2i

a. Addition
For the addition operation, we simply add the real and imaginary parts of the two complex numbers separately:

z1 + z2

= (2 + 4) + (3i - 2i)

= 6 + i

b. Subtraction
For the subtraction operation, we similarly subtract the real parts and the imaginary parts:

z1 - z2

= (2 - 4) - (3i + 2i)

= -2 + 5i

c. Multiplication
The multiplication operation is a bit more complex. We multiply the real and imaginary parts of both numbers and then add them appropriately:

z1 × z2

= (2 × 4) + (2 × -2i) + (3i × 4) + (3i × -2i)

= 14 + 8i

3.3.1 Conjugate of Complex Numbers

As we have mentioned several times throughout this work, complex numbers are expressed as a real part (a) and an imaginary part (b). Here, i is the imaginary unit, and its square is equal to -1. Taking the conjugate of a complex number means changing the sign of its imaginary part. This operation is particularly useful when converting the number to polar coordinates or solving equations involving complex numbers.

Let the complex number be z = 3 + 4i. The conjugate of this complex number is obtained by changing the sign of the imaginary part:

z* = 3 - 4i

The complex conjugate is commonly used when calculating the magnitude or absolute value of a complex number. The magnitude of a number |z| is calculated as the square root of the product of the number and its conjugate. If we want to calculate magnitude for the example above, we can say |z| is 5.