 Hello and welcome to the session. In this session we are going to discuss how to convert polar form into j operator form and voice versa. In complex numbers we have studied that a complex number is of the form a plus iv where a is the real part and v is the imaginary part. We also know that i is equal to under root of minus 1 is called the imaginary unit. The j operator is same as i that is the imaginary unit. The people in science and technology use j for the imaginary unit thus in technical mathematics the imaginary unit is represented by the symbol j where j is equal to under root of minus 1 and it is known as the j operator. So a complex number is of the form a plus jb where a is the real part and v is the imaginary part. Thus we shall make a note of the following things. j is used instead of i in technical mathematics, i is used in pure mathematics and j is used in electronics. Thus a plus jb is same as a plus iv. Also the representation of a plus jb on argon plane is same as that of a plus iv. The rectangular form of a complex number is given by a plus jb. For example 2 plus j4 is a complex number with real part 2 and imaginary part 4. Now let us discuss the representation of a complex number on argon plane. We know that the representation of a plus jb on the argon plane is same as that of a plus iv. So here on the argon plane x axis is the real axis and y axis is the imaginary axis. The complex number is represented by the ordered pair ab on the complex plane. It is drawn in the form op where op is equal to a plus jb. Now next we shall discuss the polar form of a complex number a plus jb. The polar form of a plus jb is same as the polar form of a plus iv. We can write a plus jb in polar form using trigonometric ratios. For this we put a equal to r cos theta and b equal to r sin theta. So a plus jb is equal to r cos theta plus jr sin theta. So here if we take r common this is equal to r into cos theta plus j sin theta the whole where r is the modulus theta is the amplitude. Also r is equal to under root of a square plus b square where r is strictly greater than 0 and theta is equal to tan inverse of b by a. Another method of representing polar form is r angle theta which means same as r into cos theta plus j sin theta the whole. So here on the argon plane the horizontal axis is the real axis, the vertical axis is the imaginary axis which represents the j operator. The vector represents the complex number a plus jb which is r angle theta or r into cos theta plus j sin theta the whole in the polar form. Now we shall discuss how to convert polar form into j operator form and vice versa. We have already discussed conversion of polar form of a complex number of the form a plus ib under the topic moron complex numbers. We will follow the same procedure and steps to convert complex number of form a plus jb into polar form and vice versa. Let us consider the following example write 1 plus j root 3 in polar form. On comparing it with a plus jb form we get a equal to 1 and b equal to root 3. So we first find modulus r for this. Since r is equal to under root of a square plus b square where r is strictly greater than 0 on putting values of a and b r is equal to under root of 1 square plus root 3 square r is strictly greater than 0. So r is equal to under root of 1 plus 3 which is equal to under root of 4 which is equal to plus minus 2. But we will take r equal to 2 as r is strictly greater than 0. Now next we shall calculate the amplitude theta. We know theta is equal to tan inverse b by a. Again putting values of a and b theta will be equal to tan inverse root 3 upon 1 which is equal to tan inverse root 3. And as we know tan inverse root 3 is equal to 60 degrees. So theta is equal to 60 degrees. So we have found that the modulus r is equal to 2 and the amplitude theta is equal to 60 degrees. So the polar form of 1 plus j root 3 is equal to r into cos theta plus j sin theta the whole. When we put the values of r and theta the polar form of 1 plus j root 3 is equal to 2 into cos 60 degrees plus j sin 60 degrees the whole. Now 60 degrees is equal to pi by 3 in radians. On writing angles in radians we get radians 1 plus j root 3 equal to 2 into cos pi by 3 plus j sin pi by 3 the whole or it can also be written as 2 angles 60 degrees. Now we see how to convert polar form in j operator form. For this let us consider the example convert 3 into cos 120 degrees plus j sin 120 degrees in j operator form. We know a plus jb is equal to r into cos theta plus j sin theta the whole. So on comparing r into cos theta plus j sin theta the whole with 3 into cos 120 degrees plus j sin 120 degrees the whole we get r is equal to 3 and theta is equal to 120 degrees. Now since a is equal to r cos theta and b is equal to r sin theta putting the values of r and theta we get a equal to 3 cos 120 degrees and b is equal to 3 sin 120 degrees. Now since cos 120 degrees is equal to minus half and sin 120 degrees is equal to root 3 by 2 we get a is equal to 3 into minus half which is equal to minus 3 by 2 and b is equal to 3 into root 3 by 2 which is equal to 3 root 3 by 2. So a plus jb is equal to minus 3 by 2 plus j into 3 root 3 by 2 which is the required form. So in this session we discussed the j operator form and how to convert polar form into j operator form and vice versa. This completes our session. Hope you enjoyed the session.