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

Step 2: Add them.

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.