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A professor of mine once marveled at how nonintuitive the real numbers are. If you were going to pick a set of numbers to do math with, would you have made sure to include the square root of 17?
His casual comment points to something essential about numbers, something that every mathematician knows but that no one seems to talk about.
Numbers are in our heads. Nobody ever has, or ever will, hold 7 in their hands or look at -1 under a microscope. Among other things, number is an abstraction we use to understand the world.
Because of this, we can choose any flavor of abstraction we want to think about. Like an artist or a sci-fi writer, we can choose a universe to explore. So when I study different sets of numbers, I sometimes get the sense that I’m looking at parallel universes.
In one universe, there are integers. End of story. In the integer universe, 1/2 doesn’t exist.
In the rational-numbers universe, there are integers and fractions. If a concept doesn’t fit into one of those categories, then again, it doesn’t exist.
In the real-numbers universe, we completely filled the number line. It sounds like the real numbers contain every number possible. But as we’ll see, they don’t.
Remember we defined “the square root of 17” as “the number that, when you square it, gives you 17.” The number sqrt(17) is a real number. It’s somewhere between 4 and 5. We might not be able to write it out with perfect precision, but it’s got a spot on the real number line.
Suppose, instead of 17, we looked at -4. What number, when you square it, gives you -4?
Could it be 2? Well, 2 squared is 2 * 2, which is 4. Not -4.
Maybe it’s -2. Let’s try it. -2 squared is (-2) * (-2), which is… also 4.
What happened here? When you multiply two positive numbers, you get a positive number. But when you multiply two negative numbers, you also get a positive number. So the square root of -4 can’t be positive, and can’t be negative. And it can’t be 0, because 0*0 = 0, not -4.
In the universe of real numbers, sqrt(-4) doesn’t exist. It doesn’t fit on the number line. It doesn’t even make sense!
We’ve now reached a conceptual crossroads. We could ban sqrt(-4). It’s quite reasonable to say, “negative numbers don’t have square roots, and so sqrt(-4) doesn’t exist.” And if we do that, we stay inside the world of real numbers.
The other option is to embrace sqrt(-4), and whatever weird ramifications that number may have.
The complex numbers are the real numbers with a twist: we allow for one extra concept, sqrt(-1). Usually, sqrt(-1) is represented by the letter i.
The square root of -4, then, is 2*i, written as 2i. If you’d like to see the algebra, it’s this:
2i * 2i = 4*(i*i) = 4 * (-1) = -4.
So the complex numbers allow for square roots of negative numbers. Some complex numbers that aren’t real numbers are (1/2)i, 5 + i, and i*sqrt(2).
The complex numbers are quite powerful in their own way, in different areas than the real numbers are. Pictorially, instead of a number line, complex numbers give you a plane: