Question

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

Answer 1

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

Explanation:

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.

GOOD LUCK!