Common Core Standards: Math
4. Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
Up until now, the odds are pretty good that all your students' graphing has been on the Cartesian plane. (Their graph paper has always looked like a grid of little blue squares on a white background, with the x and y axes representing real numbers.)
We know that any point in a plane can be represented in terms of an x and a y coordinate. Those coordinates are real. Well, that's all over. We're going to change the types of numbers and then change the whole graph.
Instead of the y-axis representing real numbers, it's going to represent imaginary numbers. In other words, we can label it i, 2i, 3i, etc., going up from the origin, and -i, -2i, -3i, etc. going down.
Adding i's to the y-axis has made it possible to take any complex number in a + bi form and to locate it on the plane. The x-coordinate is the "a" and the y-coordinate covers the "bi" part. Tell your students to hold their applause, please.
So, to plot 3 + 7i, we start at the origin. We go over 3 units on the x-axis, and up 7i units on the y-axis. Our point would be in the first quadrant, right here.
If our point was -4 – 11i, we would go 4 to the left and 11i down, and plot the point in the third quadrant. (And yes, we do expect students to know the quadrants: they start with I in the top right and go counterclockwise.)
But, wait, it gets even better. Think back to every single corny airplane movie you've ever seen. You know the scene where the air traffic controller realizes there's a problem, and they show his screen as he's running around the room in a panic? Think of his screen. There were no tiny little blue boxes on a white background. No, his graph was cool. It wasn't at all square; it was circular.
The different dots on this screen are each represented by a two things: an angle and a radius. The coordinates are (r, θ). The coordinate r is the distance away from the pole. (Not the North Pole. The "pole" is essentially the same as the origin.) The coordinate θ is the angle made by the point compared to a horizontal line.
These are called polar coordinates and have no relation to polar bears. Complex numbers can be represented using polar coordinates (not so much polar bears).
We won't lie to you. The conversion process from Cartesian to polar might look kind of scary until your students get the hang of it. Don't despair, though. It's not easy, but it's not rocket science, either (although they are applicable to airplanes, so…).
Let's start with a nice, easy point. How about 4 + 3i? It looks friendly.
The first step is to find the magnitude (its distance from the pole, or origin). The magnitude formula is a take-off from our good old distance formula.
In our example, the magnitude would be , or , or 5. We call this coordinate r, for radius. For the record, they don't always turn out to be whole numbers. We told you 4 + 3i looked friendly, didn't we?
Okay, great. So we have our first coordinate. We know how far out the number is. Now we just need to find the angle that would get the direction down. How on Earth do we do that?
To find the angle in question, θ, we use the following formula:
In the case of 4 + 3i, we get:
We'll let our calculator do its magic. If it's being cooperative, we should get θ ≈ 37 degrees.
So, our rectangular coordinates of 3 + 4i become (5, 37°) when translated to polar coordinates. Piece of (airplane-food) cake.