How do you convert the following complex number into its polar representation: 3i?
Complex numbers are written as a + bi, where a & b are real numbers.
For this example, a = 0.
Complex numbers are often interpreted as 2 dimensional Cartesian coordinates. The complex number a+bi would be represented as an x-coordinate of a, and a y-coordinate of b. For 0 + 3i, x would be 0, and y would be 3.
The polar representation of these coordinates would be a pair of numbers
For this example, I will choose the "zero" angle to be due east, pointing towards the right of the graph, or in the same direction as positive x on the Cartesian plane.
And we're going to convert Cartesian coordinates (0,3) to polar coordinates.
The radius r is easily calculated from the Pythagorean theorem:
Now, picture a point (x, y) drawn on the cartesian plane.
Since our x coordinate is 0, then
Also from trigonometry, we know that
From our cartesian coordinates (0,3) we have y = 3 and r = 3, therefore,
This is due "north", or straight up on the graph. Plot a point 3 units along this direction. It should be easy to see that this is the same point as (0,3) as plotted on the Cartesion version of the graph.