Complex numbers are numbers which are expressed in the form of a+ib where i is an imaginary number calledÂ iotaÂ and has the value of (âˆš1). 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. The key concepts are highlighted in this lesson will include the following:
 Introduction
 Algebraic Operation on Complex numbers
 Formulas
 Power of Iota (i)
 Complex Number Identities
 Modulus and Conjugate of a Complex number
 Examples of Complex Numbers
 Argand Plane & Polar Representation of Complex Number
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 Numbers Definition
The complex number is basically the combination of a real number and an imaginary number.
Real Numbers Re()
Any number which is present in a number system such as positive, negative, zero, integer, rational, irrational, fractions, etc. are real numbers. For example: 12, 45, 0, 1/7, 2.8,Â âˆš5 are all real numbers.
Imaginary Numbers Im()
The numbers which are not real are imaginary numbers. When we square an imaginary number, it gives a negative result. 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.
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
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 Number 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 of a Complex number
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 of Complex Number
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 Example Problem
Example: Express the following in a+ib form.
\(\frac{5+ \sqrt{3}i}{1\sqrt{3}i}\).And then find the Modulus and Conjugate of the complex number.
Solution:
Given \(\frac{5+ \sqrt{3}i}{1\sqrt{3}i}\)
z= \(\frac{5+ \sqrt{3}i}{1\sqrt{3}i} \times \frac{1+\sqrt{3}i}{1+\sqrt{3}i} = \frac{53+6\sqrt{3}i}{1+3}=\frac{1}{2} + \frac{3}{2}i\)
Modulus, \(\bar{z} = \sqrt{\left ( \frac{1}{2}\right )^{2} + \left (\frac{3}{2}\right )^{2}}\) \(\bar{z} = \sqrt{\left ( \frac{10}{4}\right )}\) \(\bar{z} = \frac{\sqrt{10}}{2}\)
Conjugate, \(\bar{z} = \frac{1}{2} – \frac{3}{2}i\)
Related Links:
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.