Good morning students, today, we will study Complex Numbers. Those numbers which are not real,

what do we call those numbers, Complex Numbers. Everybody of you know that the solution to this

type of equation is not real. So, this solution is called complex numbers. There are two solutions of

this equation, one is square root of minus 1 other is minus time square root of minus 1. Square root

of minus 1 is called iota. What is square root of minus 1? Iota square is minus 1, iota cube is iota

square into iota that is minus one into iota that is minus iota. And iota raised to par 4 minus 1 whole

square, iota whole square that is 1. So, what is iota, let me repeat again. Square root of minus 1, iota

square is minus 1, iota cube is minus iota, iota raised to par 4 is 1. Iota raised to par 5 again becomes

iota. Because iota raised to par 5 is iota raised to par 4 into iota and iota raised to par 4 is 1, hence

iota raised to par 5 is iota. So, you know what is iota? It is square root of minus 1. You have studied

briefly about iota in 11th standard. Now, we will use this and study about complex numbers. All

numbers which can be written in the form x plus iota y where x and y are real numbers are complex

numbers. All numbers which can be written in the form x plus iota y, where x and y are real numbers

are complex numbers. Where x is called as real part of complex number and y is called as imaginary

part of complex numbers. x is real part and y is imaginary part. Real part is also a real number and

imaginary part, which is y, it is also a real number. The real and imaginary part of complex numbers

both are real numbers.

Let us talk about Equality of two complex numbers. So, there are two complex numbers z1 which is

equal to x1 plus iota y1 and z2 which is equal to x2 plus iota y2, then z1 equals to z2 implies, z1 will

be equal to z2 when x1 is equal to x2 and y1 is equal to y2. When will two complex numbers be

equal when their real parts is also equal and their imaginary parts is also equal. Say, x minus 3 plus

iota y is equal to 4 plus 7 iota, when this type of equation is there, then we can directly say, x minus

3 is equal to 4 and y is equal to 7, that is x is equal to 7 and y is also equal to 7. We will equate real

parts separately and will also equate imaginary parts separately. This was example of equations of

complex number. In complex numbers, equations means real parts are equal as well as imaginary

parts are also equal.

Next thing that you know is Conjugate of a complex number. Say z is a complex number which is x

plus iota y, then z conjugate will be x minus iota y. Change the sign of imaginary part, whatever

complex number you get, it is called z conjugate. It is to be denoted by z bar, how do you denote it,

by z bar. Z conjugate is x minus iota y. Replace minus iota in the place of iota.

Let us solve few easy questions. Find the conjugate of z equals to 3 plus 4 iota. What will be the iota

of 3 and 4, 3 minus 4 iota. In place iota what you should replace? minus iota. Second question, find

the conjugate of z equals to 1 upon 1 plus iota. Z conjugate will be 1 upon 1 minus iota. What you

will replace in place of iota, minus iota. We get real value of conjugate. So similar question was

asked in AIEEE examination few years back, it was – find the conjugate of z equals to 1 upon iota

minus 1. That is first question of your worksheet. Very easy, z conjugate will be,