Number Flavors Part III: The Reals Again


Image credit: “Raindrops on a red apple,” by Patrik Nygren on flickr.

 

In the last post we started to define the real numbers. Real numbers fit onto a continuous line. What this means is that if we take the number line and fill it in so it has absolutely no holes, we’ll have captured the real numbers.

"NumberLineIntegers" by Ezra Katz - Own work. Licensed under CC BY-SA 3.0 via Wikimedia Commons.

How do we get rid of all the holes? Let’s start with the integer number line, then fill in some of the missing numbers and see where it takes us.

First, we might notice that there’s a gap between 0 and 1, and add in the number 1/2:

NumberLineIntegers3Is the line continuous yet? No way. Now we’ve got a gap between 0 and 1/2, and between 1/2 and 1. We could try to remedy this by adding in a few more numbers to our number line:

NumberLineIntegers4

By now you probably see that adding numbers one by one isn’t going to work. There will always be a smaller gap to fill.

What if we put all the rational numbers into our number line? That sounds promising: after all, there are infinitely many* rational numbers between 0 and 1 (or between 2 and 3, or between 26 and 27…). Surely that will be enough to make the number line continuous.

*If it’s not clear why, feel free to ask in the comments and I’ll explain.

Suppose we stick all of the rational numbers into our number line. Let’s call that the rational number line. Strangely enough, the rational number line still has holes.

Screen Shot 2015-11-29 at 6.52.24 PM

Here I drew a curve (in blue) that crosses the number line (black and horizontal). The curve is designed to cross the number line in a very particular place.

My curve crosses the number line at the number sqrt(2), or “the square root of 2.” The abbreviation sqrt(2) means the number that, when you multiply it by itself, gives you 2.

A quick refresher on some terminology:

  • When I say “3 squared” or “3^2,” I mean 3 times 3, or, written differently, 3*3.
  • Since 3 times 3 is equal to 9, I can say 3^2 = 9. Equivalently, the “square root” of 9 is 3, which I abbreviate sqrt(9) = 3.
  • Similarly, sqrt(4) = 2 means that 2 * 2 = 4.
  • Technically, sqrt(4) can refer to 2 OR to -2, because -2 * -2 = 4 also. But for simplicity, we’re just going to talk about positive square roots.

Let’s go ahead and check that sqrt(2) is a sensible idea for a number. We can see that sqrt(9) is a number, because 3 is a number, and sqrt(9) equals 3. Same thing with sqrt(4), because sqrt(4) is another name for the number 2. Even sqrt(2.25) is a number. You can see that sqrt(2.25) = 1.5, because 1.5 times 1.5 is 2.25.

So what is sqrt(2) equal to, written in a familiar way? Well, 1 * 1 = 1, which is too small. So sqrt(2) is bigger than 1. And 2 * 2 = 4, which is too big, so sqrt(2) is less than 2. And 1.5 * 1.5 = 2.25, so sqrt(2) is smaller than 1.5.

We don’t know what sqrt(2) is equal to yet, but we know it’s somewhere between 1 and 1.5. You can actually see that in the picture. You can try guessing a few more familiar numbers and checking to see if they equal sqrt(2).

But don’t try too hard. You can plug in numbers for the rest of your life, and you still won’t find sqrt(2).

That’s because there is no way to write sqrt(2) as an integer, fraction, or decimal. Never. Not even with thirteen thousand decimal places. You might find the square root of 1.999396 or the square root of 2.002225. But you won’t find the square root of 2.

Why? Because that sqrt(2) is irrational:  it’s a number, but it’s not a rational number. This was proven by the Ancient Greeks. (This was actually quite controversial at the time: some legends have it that the person who divulged the secret of the irrationality of sqrt(2) was punished by the gods with drowning.)

In other words, sqrt(2) is not on the rational number line. So my curve would pass through a “hole” in the rational number line! To make the line continuous, I would need to include numbers like sqrt(2).

Another “hole” in the rational number line exists at the number pi— the number that equals the circumference of any circle divided by its diameter. It’s a number that comes up in geometry along with loads of other mathematical disciplines. But there’s no way to represent pi as an integer divided by an integer. Pi is irrational too.

As it turns out, these “holes” are far from rare. We can pick out some specific ones by looking at square roots or mathematical constants, but there are infinitely many more that we have no way to even describe. The set of real numbers is the set of all of the numbers on that continuous line– all of the infinitely many, weird, and complicated numbers that you need to include so that the real line has no holes.

It’s a hard concept to even think about. There are more numbers than we can possibly specify– more numbers than we can even imagine or describe– in the set of real numbers. And yet it’s a critically important category. For one thing, the real numbers are a fundamental starting point for calculus.

In the next Number Flavors post we’ll talk about a set of even stranger, but also quite useful, set of numbers: the complex numbers.

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