Why did the chicken cross the road? I have no idea. But I do know how the chicken crossed the road—it used math to move and perhaps rotate itself from one point to another on its journey. At least that’s how the video game version of the chicken must have crossed the road. And in this video game version, I also know why it crossed the road—because you told it to!

Today, we’re talking about real world uses of complex numbers. So why in the world am I talking about chickens and video games? Because, believe it or not, you can use complex numbers to describe the motion of a chicken or anything else in both the video game and real worlds. And you can do a bunch of other useful stuff with complex numbers, too.

Of Chickens and Position Vectors

Let’s kick things off by talking about chickens and position vectors … because that’s not weird at all! You might be wondering why I’m compelled to contemplate this combination? Well, I'm thinking about how I might go about designing a chicken crossing the road video game. And one of the important parts of building such a game is figuring out how to keep track of the positions of chickens as they confront roads and other obstacles.

One way to do this is to set up an x-y coordinate system so that the position of each chicken can be labeled with x and y values. For example, at some point in time a chicken might be standing at position x=3, y=4 in this coordinate system before moving to position x=2, y=3.

But instead of thinking about ordered pairs of coordinates to keep track of a chicken’s location, you could also imagine drawing an arrow from the origin of the coordinate system to the chicken. This arrow is called the chicken’s position vector, and the changes in this vector over time tell you how the chicken is moving. Of course, the x and y components of this vector are exactly the ordered pairs of points we talked about before, but this way of thinking about locations as vectors has some advantages—so it's good to keep in mind as we go along. So, how can we make our chickens and their position vectors move around?

Adding and Subtracting Complex Numbers

To understand, we first need to talk about adding and multiplying complex numbers. I know, these two things seem to have nothing in common, but stick with me for a minute and you’ll see that they are actually related.

By adding or subtracting complex numbers…we can move the chicken anywhere in the plane.

Let’s start by thinking about the complex plane. As we’ve discussed, every complex number is made by adding a real number to an imaginary number: a + b•i, where a is the real part and b is the imaginary part. We can plot a complex number on the complex plane—the position along the x-axis of this plane represents the real part of the complex number and the position along the y-axis represents its imaginary part.

Now imagine a chicken standing at the origin—that’s the point (0, 0)—on the complex plane. If we add or subtract the real number 1, we end up at either the point 1 or –1 on the real axis. A vector from our chicken’s starting point (the origin) to either of these points represents its new position. If we instead start back at the origin and add or subtract the complex number i, we end up at either the point i or –i on the imaginary axis. So by adding or subtracting a real number from the complex number representing the position of our chicken, we can make it move to the right or left. By adding or subtracting a purely imaginary number from the chicken’s complex position vector, we can make it move up or down. And by adding or subtracting complex numbers made up of both a real and an imaginary part, we can move the chicken anywhere in the plane.

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