#math What is the Imaginary Number i ? What is the geometrical meaning of Imaginary Number i?

Hello friends, this is Mahesh Ivlekar and you are watching NextGenMaths. In this video we are going to discuss the geometrical meaning of complex number i. Any complex number z can be expressed as a a plus i b with the property i square equal to minus 1 where a is called real part of the complex number and b is called imaginary part of complex number. Now we plot that number on a complex plane where horizontal x-axis is called real axis and vertical y-axis is called imaginary axis. Then that number is shown as a radius vector where real part lies on x-axis and imaginary part b lies on a y-axis. Now to understand geometrical meaning of complex number i, we consider complex number z is equal to 1. We plot that number on a complex plane which lies on x-axis. Now we multiply z is equal to 1 by i then we have a i. Now we plot which lies on positive y-axis. Now we multiply z is equal to i by i then we have a i square and i square equal to minus 1. Now we plot z is equal to minus 1 on the complex plane which lies on negative x-axis. We multiply z is equal to minus 1 by i then we have minus i and minus i plot on a complex plane then it lies on a negative y-axis. Further we multiply z is equal to minus i by i then we have a minus i square and i square equal to minus 1. So minus 1 into minus 1 equal to plus 1 and if we plot on a complex plane then we come back to the original point z is equal to 1. Then what is happening when we multiply any number by i. So look at here when we multiply z is equal to 1 by i then the resultant vector z is equal to i it lies on a y-axis. Now we multiply z is equal to i by i then the resultant vector z is equal to minus 1 lies on a negative x-axis. Now z is equal to minus 1 multiplied by i then the resultant vector lies on a negative y-axis. That means when we multiply any number by i then that number is rotated by 90 degree in the anticlockwise direction. Now to understand this concept we will see one more example. Now consider the complex number z is equal to 1 plus i. Now plot on the complex plane which is lies in a first quadrant. Now we multiply z is equal to 1 plus i by i then we have a minus 1 plus i. Now minus 1 plus i lies on a second quadrant. Further we multiply minus 1 plus i by i then we have a minus i plus i square, i square equal to minus 1. So we have minus 1 minus i. Now minus 1 minus i plot on the complex plane which is lies on a third quadrant. One more time we multiply minus 1 minus i by i then we have minus i minus i square, i square equal to minus 1 so minus minus plus 1 so 1 minus i and 1 minus i is lies in a fourth quadrant. Now we multiply 1 minus i by i then we have a i minus i square, i square equal to minus 1 so minus minus plus 1 and we have 1 plus i and we come back to the original vector z is equal to 1 plus i. Here we can easily observe that when we multiply z is equal to 1 plus i by i then the number is rotated by 90 degree. And similarly when we multiply each and every time number by i then that number is rotated by 90 degree in the anticlockwise direction. That means the geometrical meaning of complex number i is if any number is multiplied by i then that number is rotated by 90 degree in the anticlockwise direction on the complex plane. Thanks for watching. For more such video please like, share, subscribe and press the bell icon of NextGenMath.