How do you convert the following complex number into its polar representation: 3i?

1 Answer
Aug 25, 2017

Polar coordinates #(r,theta)# are #(0, pi/2)#


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 #(r,theta)# such that r represents the distance of the point from the origin, and #theta# represents the angle from the chosen "zero" direction.

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:

#r = sqrt(x^2 + y^2)#

# = sqrt(0 + 9)#

= 3

Now, picture a point (x, y) drawn on the cartesian plane.

we know #r = sqrt(x^2 + y^2)#, and, from trigonometry, we also know that

#x/r = cos(theta)#

Since our x coordinate is 0, then #0/3 = cos(theta) = 0#

Also from trigonometry, we know that #y/r = sin(theta)#
From our cartesian coordinates (0,3) we have y = 3 and r = 3, therefore,

#sin(theta) = 1#.

#sin(theta) = 1 and cos(theta) = 0# implies that #theta = pi/2#.

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.