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

Aug 25, 2017

Polar coordinates $\left(r , \theta\right)$ are $\left(0 , \frac{\pi}{2}\right)$

#### 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 $\left(r , \theta\right)$ 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

$\frac{x}{r} = \cos \left(\theta\right)$

Since our x coordinate is 0, then $\frac{0}{3} = \cos \left(\theta\right) = 0$

Also from trigonometry, we know that $\frac{y}{r} = \sin \left(\theta\right)$
From our cartesian coordinates (0,3) we have y = 3 and r = 3, therefore,

$\sin \left(\theta\right) = 1$.

$\sin \left(\theta\right) = 1 \mathmr{and} \cos \left(\theta\right) = 0$ implies that $\theta = \frac{\pi}{2}$.

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!