# Best video lecture on Complex Numbers from JEE Mains 2018 Mathematics section

### Let us start reading the preparation article Regarding JEE Main

Complex Number is one tricky chapter and a pain for most JEE main and JEE advanced aspirants. It has a weightage of 7% that means, out of 30 questions in JEE Mains, you will find 2-3 questions from this chapter. Given the vastness of Complex Numbers, it is practically impossible to grasp its concepts in 1-2 classroom course. We encourage all our readers to go online and search for the best video lectures on Complex Numbers so that they can revise or prepare for the chapter.

Many times our students or their parents tell us that they find it difficult to spare time to find the right or best video lecture as they have to go through each video and then decide which one is most comprehensible, has enough depth and is engaging. Don’t worry, we are here to help you.

We are starting a new section, where our experts will go through the most viewed video lectures on youtube, pick the ones which are best and share here with you.

Before we start, here are a few tips on how to approach Complex Numbers:

1. In most exams like JEE Mains, the questions from Complex Numbers are usually simple. So, we should try to learn the basic concepts and not go in much depth.
2. To understand Complex Numbers prequisites are basic understanding of straight line, circles, parabola, ellipse, hyperbola, mod of x, you should know co ordinate geometry, how the point is moving, as well as basic knowledge of quadratic equations is a must.

After going through hours and hours of videos on youtube, the best video lecture on Complex Numbers for JEE Mains the best video we found was "Complex Number of IIT JEE Mathematics by Manoj Chauhan (MC) Sir"

Manoj Chauhan Sir is an IIT Delhi alumni. He has been teaching since 2007. He is one of the most famous teacher of Kota. Previously he has taught for Bansal Classes and Etoos India.

### Ratings

Comprehensible 4.5 out of 5: The video is easily comprehensible even without headphones.

Depth 4 out of 5: Concepts are explained well followed by solving examples and problems for practice

Engaging 4 out 5: The lecture can be made more engaging.

### Script

This is the introductory lecture of Complex number in Hinglish (Hindi+English) language. The sound of the video lecture is very clear and audible. The speed of teaching is good enough and all the concepts have been explained clearly too. Although few concepts could have been explained better for weak students.

Complex number is represented by Z and it is the superset of all other numbers

Like  $N \subset W\subset I\subset Q\subset R\subset Z$

Where  N is set of natural number, W is set of whole number, I is set of integers, Q is set of rational number, R is set of real numbers,Z is set of complex  number

Suppose we are solving $x^2+2x+2=0$

then $x=\dfrac{-2\pm \sqrt{-4}}{2}=-1 \pm \sqrt{-1}=-1\pm i$

Here  $\sqrt{-1}$ is the fundamental unit of complex number. And we represent it by i called iota.

So, $\sqrt{-1}=i$.

A complex number is a combination of real number and imaginary number. For example in above case -1+i , -1 is real number and i is imaginary number.

Or lets take a complex number 7+3i. Here 7 is real part and 3i is imaginary part.

Every complex number can be written in the form of x+iy where x is real part and y is imaginary part.

Argand plane: It is similar to coordinate plane.In argand plane, x-axis is real axis and y-axis is imaginary axis. So every point on argand plane represents a complex number. Like any point A(x,y) represents a complex number z=(x,iy).

So in any complex number z=a+ib  a is real part and b is imaginary part. If  b=0 then z becomes a purely real number and when a=0  then z becomes purely imaginary number and when "b is not equal to 0" then z is imaginary number but not purely imaginary.

So, z=0+i0 is purely real because b=0. It is also purely imaginary because a=0. But it is not imaginary because b is not "not equal to 0".

Algebra of complex number

Ex $z_1=a_1 + ib_1$  and  $z_2=a_2 + ib_2$

then $z_1+z_2=(a_1+a_2) + i(b_1+b_2)$

Multiplication:

$z_1.z_2=a_1.a_2 + ia_1b_1+ia_2b_2+i^2b_1. b_2=a_1.a_2 + ia_1b_1+ia_2b_2-b_1. b_2=(a_1.a_2 -b_1. b_2)+ i(a_1b_1+a_2b_2)$

Powers of iota

$i=\sqrt{-1}$

$i^2=-1$

$i^3=i^2.i=-\sqrt{-1}=-i$

$i^4=i^2.i^2=-1.-1=1$

$i^{13}=(i^4)^3.i=1^3.i=i$

$i^{4n}=1$

$i^{4n+1}=i$

$i^{4n+2}=-1$

$i^{4n+3}=-i$

Inequality of complex number : Two complex number can be equal to each other or inequal to each other but there is no concept of greater than or smaller than other number. Since a real number is a point on the real number line. So we say that one point is greater than or smaller than other depeding upon their relative position i.e the number on right side is called greater number. But a complex  number is a point on plane in stead of a line so we can't say which is smaller or greater than other but we can say when they are equal.

Two  complex number are equal when their real part are equal and their imaginary part are equal . Example a+ib and c+id are equal if a=c and b=d.

If $z_1.z_2=0$ then either $z_1 =0$ or $z_2=0$.

$\sum_{1}^{2006}i^r=?$

Since, $i+i^2+i^3+i^4=i-1-i+1=0$ and $i^{4n+1}=i$

$\sum_{1}^{2006}i^r=\sum_{1}^{2004}i^r+i^{2005}+i^{2006}=0+i+(-1)=i-1$

Modulus and Conjugate of a  Complex number

Modulus of a complex number is basically the distance of that  complex number from the origin on argand plane.

Mathematically, Modulus of a complex number z=a+ib is represented by $\left | z \right |$ and  $\left | z \right |=\sqrt{a^2+b^2}$.

Conjugate of a complex number is represented by $\bar{Z}$ and in conjugate of a complex number sign of imaginary part changes. Example $z=a+ib$ then $\bar{z}=a-ib$

Argument gives angle between the positive real axis and the  line joining the point to the origin.

Q) Find $arg(-1-\sqrt{3}i)$

A)$4\pi+\dfrac{4\pi}{3}$

B)$-9\pi+\dfrac{\pi}{3}$

C)$-7\pi+\dfrac{4\pi}{3}$

D) all of the  above

Right answer is A and B.

Amplitude is principal value of argument.