Complex numbers are the numbers which are expressed in the form of a+ib where i is an imaginary number calledÂ iotaÂ and has the value of (âˆš1). 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.Â 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. The key concepts are highlighted in this article will include the following:
Table of Contents:
 Introduction
 Algebraic Operations
 Formulas
 Power of Iota
 Identities
 Modulus and Conjugate
 Argand Plane and Polar Representation
 Examples
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,Â âˆš5 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:Â âˆš2,Â âˆš7,Â âˆš11 are all imaginary numbers.
In the 16th century, the complex numbers were introduced which made it possible to solve the equation x^{2}Â +1 = 0. The roots of the equation are of form x = Â±âˆš1Â and no real roots exist. Thus, with the introduction of complex numbers, we have Imaginary roots.
We denote âˆš1 with the symbol ‘i’, where iÂ 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.
See the table below to differentiate between a real number and an imaginary number.
Complex Number  Real Number  Imaginary Number 
1+2i  1  2i 
79i  7  9i 
6i  0  6i(Purely Imaginary) 
6  6  0i(Purely Real) 
Algebraic Operation on Complex numbers
There can be four types ofÂ algebraic operation on complex numbersÂ 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

Roots of Complex Numbers
When we solve a quadratic equation of the form ax^{2}Â +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
Â Â Â Â Â (a + ib) + (c + id) = (a + c) +Â 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)/ (c^{2} + d^{2}) + i(bc – ad)Â / (c^{2} + d^{2})
Power of Iota (i)
Depending upon the power of “i”, it can take the following values;
i^{4k+1} = i .Â i^{4k+2Â }= 1Â i^{4k+3} =Â iÂ .Â i^{4k} = 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 / iÂ Â Ã—Â i/iÂ Â Ã— i^{1}Â = i / i^{2} = i / 1 = iÂ / 1 = i
Note: âˆš1 Ã— âˆš1 = âˆš(1Â Ã— 1) = âˆš1 = 1 contradicts to the fact that i^{2}Â = 1.
Therefore, for an imaginary number,Â âˆšaÂ Ã— âˆšb is not equal toÂ âˆšab.
Complex Numbers Identities
Let us see some of the identities.


 (z_{1Â }+ z_{2})^{2} = (z_{1})^{2Â }+Â (z_{2})^{2}Â + 2 z_{1}Â Ã—Â z_{2}
 (z_{1 }–Â z_{2})^{2} = (z_{1})^{2Â }+Â (z_{2})^{2}Â – 2 z_{1}Â Ã—Â z_{2}
 (z_{1})^{2 }–Â (z_{2})^{2} =Â (z_{1Â }+ z_{2})(z_{1 }–Â z_{2})
 (z_{1Â }+ z_{2})^{3}Â = (z_{1})^{3}^{Â }+ 3(z_{1})^{2}Â z_{2Â }Â +3(z_{2})^{2}Â z_{1}_{Â }+ (z_{2}Â )^{3}
 (z_{1Â }– z_{2})^{3}Â = (z_{1})^{3}^{Â }– 3(z_{1})^{2}Â z_{2Â }Â +3(z_{2})^{2}Â z_{1}_{Â }– (z_{2}Â )^{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 Î¸, where r is the length of the vector and Î¸ is the angle made with the real axis. Check out the detailedÂ argand plane and polar representation of complex numbersÂ in this article and understand this concept in a detailed way along with solved examples.
Complex Numbers Examples
Example 1:
Simplify
a) 16i + 10i(3i)
b) (7i)(5i)
c) 11i + 13i – 2i
Solution:Â
a) 16i + 10i(3i)
= 16i + 10i(3) + 10i (i)
= 16i +30i – 10 i2
= 46 i – 10 (1)
= 46i + 10
b) (7i)(5i) = 35Â i2 = 35 (1) = 35
c) 11i + 13i – 2i = 22i
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