What is most useful about this visualisation of the complex numbers is how addition and multiplication behave. Multiplication is a bit more difficult to see. Before we go on, it is useful to introduce another means of specifying points in the plane. Suppose that X is any point in a plane and suppose that O is the origin. This angle and distance are known as the polar coordinates of X .
Multiplication corresponds to adding the angles of the two points and multiplying their lengths.
Understanding multiplication helps us understand other operations such as taking the square root. For the sake of simplicity, first consider the case in which the number we are interested in has a distance of 1 from the origin. Now we can understand why we got the answer we did for the square root of i . It is easily seen that i forms an angle of 90 degrees with the real axis and has distance of 1 from the origin. A cautious reader will note that there is some ambiguity in choice of the angle in our definition of polar coordinates.