Complex Numbers

Complex numbers are the numbers which are expressed in the form of a+ib where i is an imaginary number called iota and has the value of (√-1). For example, 2+3i is a complex number, where 2 is a real number and 3i is an imaginary number. Therefore, the combination of both numbers is a complex one.

See the table below to differentiate between a real number and an imaginary number.

Complex Number Real Number Imaginary Number
-1+2i -1 2i
7-9i 7 -9i
-6i 0 -6i(Purely Imaginary)
6 6 0i(Purely Real)

The main application of these numbers is to represent periodic motions such as water waves, alternating current, light waves, etc., which relies on sine or cosine waves etc. There are certain formulas which are used to solve the problems based on complex numbers. Also, the mathematical operations such as addition, subtraction and multiplication are performed on these numbers.

Complex Numbers Definition

The complex number is basically the combination of a real number and an imaginary number. The real numbers are the numbers which we usually work on to do the mathematical calculations. But the imaginary numbers are not generally used for calculations but only in the case of imaginary numbers. let us check the definitions for both the numbers.

What are Real Numbers?

Any number which is present in a number system such as positive, negative, zero, integer, rational, irrational, fractions, etc. are real numbers. It is represented as Re(). For example: 12, -45, 0, 1/7, 2.8, √5 are all real numbers.

What are Imaginary Numbers?

The numbers which are not real are imaginary numbers. When we square an imaginary number, it gives a negative result. It is represented as Im(). Example: √-2, √-7, √-11 are all imaginary numbers.

In the 16th century, the complex numbers were introduced which made it possible to solve the equation x2  +1 = 0. The roots of the equation are of form x = ±√-1 and no real roots exist. Thus, with the introduction of complex numbers, we have Imaginary roots.

We denote √-1 with the symbol ‘i’, where i denotes Iota (Imaginary number).

An equation of the form z= a+ib, where a and b are real numbers, is defined to be a complex number. The real part is denoted by Re z = a and the imaginary part is denoted by Im z = b.

Algebraic Operation on Complex numbers

There can be four types of algebraic operation on complex numbers which are mentioned below. Visit the linked article to know more about these algebraic operations along with solved examples. The four operations on the complex numbers include:

  • Addition
  • Subtraction
  • Multiplication
  • Division

Quadratic Equations-Complex Numbers

When we solve a quadratic equation of the form ax2 +bx+c = 0, the roots of the equations can be determined in three forms;

  • Two Distinct Real Roots
  • Similar Root
  • No Real roots (Complex Roots)

Complex Number Formulas

  • Addition

          (a + ib) + (c + id) = (a + c) + i(b + d)

  • Subtraction

(a + ib) – (c + id) = (a – c) + i(b – d)

  • Multiplication

(a + ib) – (c + id) = (ac – bd) + i(ad + bc)

  • Division

(a + ib) / (c + id) = (ac+bd)/ (c2 + d2) + i(bc – ad) / (c2 + d2)

Power of Iota (i)

Depending upon the power of “i”, it can take the following values;

i4k+1 = i . i4k+2 = -1 i4k+3 =  -i . i4k = 1

where k can have an integral value (positive or negative).

Similarly, we can find for the negative power of i, which are as follows;

i-1 = 1 / i

Multiplying and dividing the above term with i, we have;

i-1 = 1 / i  ×  i/i  × i-1  = i / i2 = i / -1 = -i / -1 = -i

Note: √-1 × √-1 = √(-1  × -1) = √1 = 1 contradicts to the fact that i2 = -1.

Therefore, for an imaginary number, √a × √b is not equal to √ab.


Let us see some of the identities.

  1. (z+ z2)2 = (z1)+ (z2)2 + 2 z1 × z2
  2. (z1 – z2)2 = (z1)+ (z2)2 – 2 z1 × z2
  3. (z1)2 – (z2)2 =  (z+ z2)(z1 – z2)
  4. (z+ z2)3 = (z1)3 + 3(z1)2 z +3(z2)2 z1 + (z2 )3
  5. (z– z2)3 = (z1)3 – 3(z1)2 z +3(z2)2 z1 – (z2 )3

Modulus and Conjugate

Let z=a+ib be a complex number.

The Modulus of z is represented by |z|.

Mathematically, \(\left | z \right |= \sqrt{a^{2}+b^{2}}\)

The conjugate of “z” is denoted by \(\bar{z}\).

Mathematically, \(\bar{z}\)= a – ib

Argand Plane and Polar Representation

Similar to the XY plane, the Argand(or complex) plane is a system of rectangular coordinates in which the complex number a+ib is represented by the point whose coordinates are a and b.

We find the real and complex components in terms of r and θ, where r is the length of the vector and θ is the angle made with the real axis. Check out the detailed argand plane and polar representation of complex numbers in this article and understand this concept in a detailed way along with solved examples.

Problems and Solutions

Example 1: Simplify 

  1. a) 16i + 10i(3-i)
  2. b) (7i)(5i)
  3. c) 11i + 13i – 2i


  1. a) 16i + 10i(3-i)

= 16i + 10i(3) + 10i (-i)

= 16i +30i – 10 i2

= 46 i – 10 (-1)

= 46i + 10

  1. b) (7i)(5i) = 35  i2 = 35 (-1) = -35
  2. c) 11i + 13i – 2i = 22i