What Are Complex Numbers?
Real and Imaginary Numbers
In one sense all numbers are “real.” After all, I’m real and I’m thinking about them … so they’re certainly real to me. But as we’ve discussed before, only some numbers are really real. And that’s because the word “real” has a particular meaning in math. Real numbers are all the numbers you can find on a real number line. That means that real numbers include zero and the other integers, all the rest of the rational numbers, and each and every one of the infinity of irrational numbers, too.
Imaginary numbers can be used to solve problems that real numbers can’t deal with.
Before our discussion last week, you might have thought that this was the entire world of numbers. But as we found out, there’s an additional and altogether different type of number floating around out there called an imaginary number. What are imaginary numbers good for? Well, the interesting thing about them is that when they’re squared you get back negative numbers. Which means imaginary numbers can be used to solve problems that real numbers can’t deal with such as finding x in the equation x2 + 1 = 0.
But that’s not the end of our story because, as I mentioned at the outset, imaginary numbers can be combined with real numbers to create yet another type of number. Which is … ? We’ll get to that in just a minute—right after we take a little stroll along the number line.
A Walk Along the Number Line
As you might imagine, taking imaginary walks along the real number line (no imaginary numbers here!) is one of my favorite things to do. You and I have been here plenty of times before, so I won’t dwell too long on the details of what we’re seeing.
As you know, the real number line is just a way for us to represent the continuum of real numbers. They’re typically drawn with arrows on both ends to inform you that if we had a magical sheet of infinitely long paper, the line would extend out indefinitely in either direction towards what we call infinity. Larger and larger numbers usually head off to the right in what’s called the positive direction, and smaller and smaller numbers head off to the left in the negative direction.
But what if we’re bored walking along this real number line. What if you and I instead decide to take a sharp 90-degree turn at zero and start walking upwards perpendicular to the real number line. Is this even possible? Where might this new path lead us?
The answer is yes, it is possible. As we take this turn, we take our first steps into an exciting new dimension and we head out into what’s called the complex plane. What’s that? Well, an imaginary number can be added to a real numbers to create something more complex called, sensibly, a complex number. Every complex number can therefore be written as the sum of a real and an imaginary number: z = a + b•i. The symbol a here represents what’s called the real part of the complex number, z, while the symbol b represents the imaginary part (i is the complex unit).
You can picture the relationships between the various types of numbers like this:
This picture says that every number is actually a complex number, but there’s a certain subset of these numbers that are also real numbers (those whose imaginary components are equal to zero). Within this realm of real numbers, there is a certain subset called rational numbers (the rest are irrational numbers). Finally, within this world of rational numbers, there is a subset called integers (the rest are fractions).
It’s kind of amazing to realize that all of the everyday numbers we know and love are just a tiny part of the great big world of complex numbers to which you’ve now been introduced.
The Complex Plane
This brings us back to our walk along the number line …
Once we take that fateful turn at zero and start walking upwards perpendicularly away from the real number line, we begin walking along what’s called the “imaginary axis” of the complex plane. The old real number line that we were walking along is dubbed the “real axis” in this plane. Any point on this plane represents a complex number, z = a + b•i. The real component, a, tells you the “x-coordinate” of this complex number in the complex plane. The imaginary component, b, tells you the “y-coordinate.”
Take a minute and let this soak in … because it’s a pretty incredible new idea we’ve stumbled upon. When we took that turn and first wandered off the real number line, we suddenly had our eyes opened to an entirely new world of numbers. In this world, the numbers are no longer confined to exist along a 1-dimensional line. But suddenly they can live anywhere in this 2-dimensional plane!دانلود نسخه قابل چاپ اعداد مختلط به انگلیسی
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