# What are complex numbers and what are the rules to solve them? In mathematics complex numbers are used widely. It is mostly used to find the square root of negative numbers. We can find the square root of positive numbers, to find the square root of negative numbers, we use a complex numbers system. Complex numbers are also used in engineering and physics.

What are complex numbers?

A number that comes with the sum of a real and imaginary number is said to be a complex number. Real numbers are those numbers that can be expressed as infinite decimal places between two numbers. For example, there are infinite decimals between 1 and 2. On the other hand, imaginary numbers are those numbers whose square is negative and is denoted by an iota. Complex numbers are generally denoted by z and the equation of z is,
z = a + ib
if the real number in complex numbers is zero then we said it is a pure imaginary number and if the imaginary number is zero, we said that number is purely real, or every real number is a complex number with i0 as the imaginary number.

What is iota?

Iota is representing the complex number. It is denoted by i. We can find the square root of negative numbers with the help of iota. For example, √-4 = √4 i = 2i as i = √-1.
In order to solve the problems of complex numbers we must have sound knowledge about the powers of iota.
⦁ i = √-1
⦁ i2 = -1
⦁ i3 = -i
⦁ i4 = (i2)2 = (-1)2 = 1
and so on.

Rules of complex numbers
Some basic rules of solving the complex number are mentioned below. For the calculation of the addition, subtraction, multiplication, and division of complex numbers complex number calculator are very useful for accurate results.

Sum of Complex numbers

The Sum of the complex numbers is very simple. In this case, like terms are added, such as real number must be added in real number and the imaginary number is added in imaginary numbers. If we have two complex numbers such as z1 and z2 then their sum is,
Z1 = a + ib
Z2 = c + id
Z1 + Z2 = a + ib + c + id
= (a + c) + i (b + d)
Example
Find the sum of 3 + 2i and 4 + 5i?
Solution
Step 1: Give names z1 and z2 to the given numbers.
Z1 = 3 + 2i
Z2 = 4 + 5i
Z1 + Z2 = 3 + 2i + 4 + 5i
= (3 + 4) + i (2 + 5)
= 7 + 7i

Difference of Complex numbers

The difference of complex numbers is very simple. In this case, like terms are subtracted, such as real number must be subtracted from the real number and the imaginary number is subtracted from imaginary numbers. If we have two complex numbers such as z1 and z2 then their difference is,
Z1 = a + ib
Z2 = c + id
Z1 + Z2 = a + ib – (c + id)
= a + ib – c – id
= (a-c) + i (b – d)
Example
Find the difference of 13 + 12i and 4 + 5i?
Solution
Step 1: Give names z1 and z2 to the given numbers.
Z1 = 13 + 12i
Z2 = 4 + 5i
Step 2: Subtract them.
Z1 – Z2 = 13 + 12i – (4 + 5i)
= 13 + 12i – 4 – 5i
= (13 – 4) + i (12 – 5)
= 9 + 7i
Product of Complex numbers
Product of complex numbers are calculated by multiplying both the complex numbers equations and use i2 = -1, such as
Z1 = a + ib
Z2 = c + id
Z1 x Z2 = (a + ib) x (c + id)
= ac + iad + ibc + i2bd
= ac + iad + ibc – bd
= (ac – bd) + i (ad + bc)
Example
Find the product of 3 + i and 4 + 2i?
Solution
Step 1: Give names z1 and z2 to the given numbers.
Z1 = 3 + i
Z2 = 4 + 2i
Step 2: Multiply them.
Z1 x Z2 = (3 + i) x (4 + 2i)
= 12 + 6i + 4i + 2i2
= 12 + 6i + 4i – 2
= (12 – 2) + i (6 + 4)
= 10 + 10i
Quotient of Complex numbers
The quotient of two numbers can be calculated by rationalization. Such as,
Z1 = a + ib
Z2 = c + id
Z1 / Z2 = (a + ib) / (c + id)
Multiply and divide by c – id.
Z1 / Z2 = (a + ib) (c – id) / (c + id) (c – id)
Z1 / Z2 = (a + ib) (c – id) / (c2 – i2d2)
Z1 / Z2 = (a + ib) (c – id) / (c2 – (-1) d2)
Z1 / Z2 = (a + ib) (c – id) / (c2 + d2)
Z1 / Z2 = (ac + bd) + i (-ad + bc)/ (c2 + d2)
Example
Find the quotient of 3 + i and 4 + 2i?
Solution
Step 1: Give names z1 and z2 to the given numbers.
Z1 = 3 + i
Z2 = 4 + 2i
Step 2: Divide them.
Z1 / Z2 = (3 + i) / (4 + 2i)
Step 2: Multiply and divide by 4 – 2i.
Z1 / Z2 = (3 + i) (4 – 2i) / (4 + 2i) (4 – 2i)
Z1 / Z2 = (3 + i) (4 – 2i) / (42 – i222)
Z1 / Z2 = (3 + i) (4 – 2i) / (16 + 4)
Z1 / Z2 = (12 – 6i + 4i – 2i2) / (20)
Z1 / Z2 = (12 – 6i + 4i + 2) / (20)
Z1 / Z2 = (12 – 2) + i (-6 + 4) / (20)
Z1 / Z2 = (10 – 2i) / (20)
Z1 / Z2 = 10/20 – 2i/20
Z1 / Z2 = 1/2 – i/10

Summary

Now you can easily solve any athematic operation on complex numbers. A number that comes with the sum of a real and imaginary number is said to be a complex number. We can perform addition, subtraction, multiplication, and division on two complex numbers.