 Hello and welcome to the session. In this session we will represent conjugation and multiplication of complex numbers on complex plane. Now we know that any number of the form x plus number y, where x and y are real numbers is called a complex number and here x is called the real part and y is called the imaginary part of the complex number x plus iota y. Now we know that the complex number x plus y iota can be represented by the ordered pair x y and this complex number can be represented in this complex plane where x axis is called the real axis and y axis is called the imaginary axis as real part of the complex number is represented as a point on the x axis and imaginary part of the complex number is represented by the point on the y axis. Thus the complex numbers are represented on this complex plane. Now let us discuss conjugation of complex numbers in complex plane. Now if z is equal to x plus iota y is any complex number then its conjugate is given by z bar or z star which is equal to x minus y iota or x minus iota y. Thus in conjugate the sign of imaginary part is changed. Now we know the complex number and its conjugate on the complex plane. Now let the complex number z is equal to 2 plus 3 iota and its conjugate that is z bar is equal to and here we change the sign of imaginary part. So its conjugate will be 2 minus 3 iota. Now here we have both z and z bar on the complex plane. First of all let us draw the complex number 2 plus 3 iota. Now from a region we will move 2 minutes right and 3 minutes up and we reach at this point. So this is the point p with coordinates 2, 3 and it represents the complex number 2 plus 3 iota. Now let us represent its conjugate on the complex plane that is z bar is equal to 2 minus 3 iota. Now here from a region a we will move 2 minutes right and 3 minutes down and reach at this point. Let this point be q with coordinates 2 minus 3 which represents conjugate of complex number z that is z bar which is equal to 2 minus 3 iota. Now let us see the 2 complex numbers and you can see that the point q is the reflection of point p in x axis that is the real axis. Thus when we have to draw conjugate of complex number in the complex plane then we simply reflect h in real axis. Now let us discuss multiplication of complex numbers in complex plane. Now we will use the real form of the complex numbers to show multiplication. Now we know that the real form of a complex number is given by z is equal to r into cos theta plus iota sin theta the whole where r is the modulus of the complex number and theta is the argument of the complex number. Now let z bar is equal to r1 into cos alpha plus iota sin alpha the whole and z2 is equal to r2 into cos theta plus iota sin theta the whole the 2 complex numbers. Now let us find the product z1 into z2 or z1 z2 which is equal to r1 into cos alpha plus iota sin alpha the whole into r2 into cos theta plus iota sin beta the whole to be equal to r1 r2 into cos alpha plus iota sin alpha the whole into cos beta plus iota sin beta the whole. Now let us open the brackets so this is equal to r1 r2 into cos alpha cos beta plus iota cos alpha sin beta plus iota sin alpha cos beta plus iota into iota will be iota square into sin alpha sin beta the whole. Now we know that iota square is equal to minus 1 so this will be equal to r1 r2 into cos alpha cos beta plus iota cos alpha sin beta plus iota sin alpha cos beta minus sin alpha sin beta the whole as iota square is minus 1 now combining real and imaginary parts this will be equal to r1 r2 into cos alpha cos beta minus sin alpha sin beta now for these 2 terms taking iota common it will be plus iota into cos alpha sin beta plus sin alpha plus beta the whole and this complete whole. Now using this trigonometric formula cos alpha cos beta minus sin alpha sin beta will be equal to cos alpha plus beta so this is equal to r1 r2 into cos of alpha cos beta the whole and using this trigonometric formula cos alpha sin beta plus sin alpha cos beta will be equal to sin alpha plus beta so here we have plus iota sin alpha plus beta the whole and this complete whole thus we have obtained the product z1 z2 which is again a complex number with modulus r1 r2 and amplitude or argument alpha cos beta plus we see that product of two complex numbers is again a complex number whose amplitude is sum of the arguments of the two complex numbers and modulus is the product of modulus of two numbers that is the two complex numbers so generically its representation will be like this this is the point p representing complex number that one its modulus is r1 and argument or amplitude is alpha point q represents complex number z2 whose modulus is r2 and argument is beta point r will represent complex number given by z1 z2 and will have modulus r1 r2 and argument alpha plus beta so this is the symmetrical representation of multiplication of complex numbers in complex plane so in this session we have discussed how to represent conjugation and multiplication of complex numbers on complex plane and this completes our session hope you all have enjoyed the session