Complex Number Formula
A complex number is the sum of a real number and an imaginary number. So, a complex number is of the form x + iy. A complex number is nothing but a combination of a real number and an imaginary number. In this section, we will be discussing the complex number formulas to do addition, subtraction, multiplication, and division on the complex numbe$ Let us learn the complex number formula with a few solved examples.
What is the Complex Number Formula?
Operations on complex numbers are similar to polynomials. To add or subtract two complex numbers, use the following complex number formula.
(a + ib) + (c + id) = (a + c) + i(b + d)
(a + ib)  (c + id) = (a  c) + i(b  d)
To multiply two complex numbers, use the following complex number formula.
(a + ib) × (c + id) = (ac + bd) + i(bc + ad)
To divide two complex numbers, use the following complex number formula.
(a + ib) ÷ (c + id) = [(ac + bd)/(c^{2} + d^{2})] + i[(bc  ad)/(c^{2} + d^{2})]
To multiply complex conjugates, use the following complex number formula.
(a + ib) × (a  ib) = a^{2} + b^{2}
Let's take a quick look at a few solved examples to understand the complex number formulas better.
Solved Examples Using Complex Number Formulas

Example 1: Find the sum of 4  6i and 2 + 4i using the complex number formula.
Solution:
(4  6i) + (2 + 4i) = (4  2) + i(6 + 4) = 2  2i
Answer: The required sum is 2  2i.

Example 2: Find the product of 1+2i and its conjugate using the complex number formula.
Solution:
(1+2i) × (1  2i) = 1^{2} + 2^{2} = 1 + 4 = 5
Answer: The required product is 5