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 the real number and imaginary number is a complex number.
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
 Properties
 Modulus and Conjugate
 Argand Plane and Polar Representation
 Examples
 FAQs
What are Complex Numbers?
The complex number is basically the combination of a real number and an imaginary number. The complex number is of the form a+ib. 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, etc., 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.
The complex numbers were introduced to solve the equation x^{2}+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 = ib.
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 Operations 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 in the form of 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
While performing the arithmetic operations of complex numbers such as addition and subtraction, combine similar terms. It means that combine the real number with the real number and imaginary number with the imaginary number.
Addition
(a + ib) + (c + id) = (a + c) +Â i(b + d)
Subtraction
(a + ib) – (c + id) = (a – c) + i(b – d)
Multiplication
When two complex numbers are multiplied by each other, the multiplication process should be similar to the multiplication of two binomials. It means that FOIL method (Distributive multiplication process) is used.
(a + ib). (c + id) = (ac – bd) + i(ad + bc)
Division
The division of two complex number can be performed by multiplying the numerator and denominator by its conjugate value of the denominator, and then apply the FOIL Method.
(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.i^{4k+2Â }= 1Â i^{4k+3} =Â i.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})^{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})^{3}

Properties of Complex Numbers
The properties of complex numbers are listed below:
 The addition of two conjugate complex numbers will result in a real number
 The multiplication of two conjugate complex number will also result in a real number
 If x and y are the real numbers and x+yi =0, then x =0 and y =0
 If p, q, r, and s are the real numbers and p+qi = r+si, then p = r, and q=s
 The complex number obeys the commutative law of addition and multiplication.
Â Â Â Â Â z_{1}+z_{2}Â = z_{2}+z_{1}
Â Â Â Â Â z_{1}. z_{2}Â = z_{2}. z_{1}
 The complex number obeys the associative law of addition and multiplication.
Â Â Â Â Â (z_{1}+z_{2}) +z_{3 }= z_{1} + (z_{2}+z_{3})
Â Â Â Â Â (z_{1}.z_{2}).z_{3 }= z_{1}.(z_{2}.z_{3})
 The complex number obeys the distributive law
Â Â Â Â Â Â z_{1}.(z_{2}+z_{3}) = z_{1}.z_{2 }+ z_{1}.z_{3}
 If the sum of two complex number is real, and also the product of two complex number is also real, then these complex numbers are conjugate to each other.
 For any two complex numbers, say z_{1 }and z_{2}, then z_{1}+z_{2} â‰¤ z_{1}+z_{2}
 The result of the multiplication of two complex numbers and its conjugate value should result in a complex number and it should be a positive value.
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 Problems
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 i^{2}
= 46 i – 10 (1)
= 46i + 10
b) (7i)(5i) = 35Â i^{2}Â = 35 (1) = 35
c) 11i + 13i – 2i = 22i
Example 2:
Express the following in a+ib form:
(5+âˆš3i)/(1âˆš3i).
Solution:
Given:Â (5+âˆš3i)/(1âˆš3i)
Frequently Asked Questions on Complex Numbers
What is meant by complex numbers?
The complex number is the combination of a real number and imaginary number. An example of a complex number is 4+3i. Here 4 is a real number and 3i is an imaginary number.
How to divide the complex numbers?
To divide the complex number, multiply the numerator and the denominator by its conjugate. The conjugate of the complex number can be found by changing the sign between the two terms in the denominator value. Then apply the FOIL method to simplify the expression.
Mention the arithmetic rules for complex numbers.
The arithmetic rules of complex numbers are:
Addition Rule: (a+bi) + (c+di) = (a+c)+ (b+d)i
Subtraction Rule: (a+bi) â€“ (c+di) = (ac)+ (bd)i
Multiplication Rule: (a+bi) . (c+di) = (acbd)+(ad+bc)i
Write down the additive identity and inverse of complex numbers.
The additive identity of complex numbers is written as (x+yi) + (0+0i) = x+yi. Hence, the additive identity is 0+0i.
The additive inverse of complex numbers is written as (x+yi)+ (xyi) = (0+0i). Hence, the additive inverse is xyi.
Write down the multiplicative identity and inverse of the complex number.
The multiplicative identity of complex numbers is defined as (x+yi). (1+0i) = x+yi. Hence, the multiplicative identity is 1+0i.
The multiplicative identity of complex numbers is defined as (x+yi). (1/x+yi) = 1+0i. Hence, the multiplicative identity is 1/x+yi.
Learn more about the Identities, conjugate of the complex number, and other complex numbers related concepts at BYJU’S. Also, get additional study materials for various maths topics along with practice questions, examples, and tips to be able to learn maths more effectively.
there is no particular information about square root of a complex number
Please check: https://byjus.com/maths/squareroot/