Imaginary Numbers in the Real World
Although every complex number has a so-called “imaginary” part, in many ways they’re just as “real” as real numbers. And complex numbers have many real uses in the real world.
Jason Marshall, PhD
Listen
Imaginary Numbers in the Real World
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.
Multiplying Complex Numbers
While moving our chickens and their position vectors around like this is great, it’s not really the magical part of doing all of this with complex numbers. For that, we need to think about what happens when we multiply complex numbers instead of add them.
To see, imagine a chicken standing at the location 1 on the real-axis. What happens if we multiply this position by the imaginary unit, i? Well, 1 • i = i. What if we then multiply this by the imaginary unit again? Well, i • i = –1 (that’s the definition of the imaginary unit). If we now multiply this by i again, we end up with –1 • i = –i. And then one final time brings us back to ‑i • i = –(i2) = –(–1) = 1 … right back where we started from.
Each time we multiply by the imaginary unit, i, the chicken jumps from one axis to another.
Why is this interesting? Take another look at what’s happening. Each time we multiply by the imaginary unit, i, the chicken jumps from one axis to another—first from the positive real axis, then to the positive imaginary axis, then the negative real axis, the negative imaginary axis, and finally back to the positive real axis. If you think about it (and maybe try out a couple of other examples for yourself), you’ll see that this is telling us that multiplying by i has the effect of rotating the chicken and its position vector 90 degrees around the origin of the complex plane.
This is the real magic of our use of complex numbers here—not only can we move the chickens around side-to-side and up-and-down using addition and subtraction, we can actually rotate their position vectors simply by multiplying by complex numbers. We won’t go into the details right now, but it turns out we’re not stuck with 90 degree rotations either. With a bit of cleverness, we can actually rotate our vector by any arbitrary angle with one multiplication by a complex number.
If we put all of this together, we now have everything needed to build our video game. Namely, if we construct our roads, obstacles, and chickens in the complex plane, we can move all of them around and even rotate them in the plane using relatively simple complex addition and multiplication.
More Complex Numbers in the Real World
While this is all very cool, you might be wondering if anybody would actually choose to build their chicken crossing the street video game this way? The answer is probably not—most people would likely use another approach. But the point is that this is a viable solution to the problem—and it goes to show you that complex numbers can absolutely have real world applications.
And the real world applications don’t stop there. It turns out that complex numbers show up all over the place in physics. From electronic circuits to quantum mechanics and all sorts of other things, complex numbers are a very real part of the real world we live in.
Wrap Up
OK, that’s all the math we have time for today.
For more fun with math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of The Math Dude on Facebook, where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Chicken image courtesy of Shutterstock.
About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.