Disclaimer: Some of the terms defined below are defined differently elsewhere. The differences are mostly semantical and it’s easier to only focus on the most useful definition for each, so that is what we will do.
When we learn numbers, we first learn the counting or natural numbers, that is 1, 2, 3, and so on. It’s a good place to start because we can look down at a table and see 1 cookie, 2 cookies, 3 cookies, etc. Then when we throw a fit and lose our cookie privileges for the day, we look down at an empty table and learn a painful lesson about the number 0. Later we learn about negative numbers, which are a bit harder to picture. Getting a negative cookie is the same as losing a cookie. Getting -5 cookies is the same as losing 5 cookies.
Once this concept starts to click, we have a nice picture of all the numbers we know: the number line. The integers, … , -2, -1, 0, 1, ... can be represented as a series of equally spaced dots in a line that goes on forever in both directions.
As we get older and start to learn about division we encounter rational numbers for the first time, although we usually learn the word fraction first. A rational number is any number that can be written as the ratio (hence rational) of an integer and a natural number. More precisely, a rational number is any number that can be written in the form “A divided by B” or “A/B” where A is any integer and B is any natural number. For instance, when you and your brother are both eyeing the same cookie, you can split it fairly by cutting it down the middle and each taking half. If, like me, you have an older brother, you’ll be lucky to wind up with a third.
Keep in mind that any integer A is rational because A = A/B if B is 1 and 1 is a natural number (so A satisfies the definition of a rational number). The Ancient Greeks, having no reason to believe otherwise, generally thought that all numbers were rational. After all, rational numbers can be observed naturally by splitting cookies into pieces. The idea of numbers that can’t be written in this naturally observable way was disconcerting to them, even abominable to some. But, as with many long held mathematical understandings (and it was long held) it turned out to be wrong.
The constants π and e, the heroes of high school geometry and algebra respectively are both irrational. They’re actually super irrational or, a little more technically transcendental, numbers.
I want to focus on a friendlier irrational number, perhaps the friendliest of all: the square root of 2. To jog your memory, the square root of two is the only positive number S such that S2 (that is S x S) equals 2. There is a nice concise proof that the square root of 2 is irrational that, unlike proofs of the irrationality of π and e, requires only tools that you may well have learned in the course of your elementary and middle school education.
Be warned that just because the tools are simple, it doesn’t mean the proof is easy. If you’re not familiar with mathematical proofs, don’t be frustrated if you have to reread it a few times. Even for mathematicians, a page of proof takes much longer to understand than a paragraph of narrative or dialog.
Here’s what you need to know:
For any natural numbers A and C, we call C a “factor” of A if a pile of A objects can be split into piles that each have C objects with no leftovers. For example 3 is a factor of 6 because 6 cookies can be split into 2 separate piles that each have 3 cookies. On the other hand 3 is NOT a factor of 8 because if we try to split 8 cookies into piles that each have 3 cookies, we’ll make 2 piles and then be stuck with 2 unavoidable leftovers.
For any natural numbers A and B, a natural number C is called a common factor if it is a factor of BOTH A and B. For example 7 is a common factor of 14 and 35. However 7 is NOT a common factor of 28 and 20 because, even though 7 is a factor of 28, it is not a factor of 20.
Any positive rational number can be written in the form A/B where A and B are natural numbers whose only common factor is 1.
The square of an odd number is always odd so if A2 is even for a given natural number A, then A must be even. In other words, if 2 is a factor of A2, it also has to be a factor of A.
To prove that the square root of 2 is irrational we use a method called “proof by contradiction.” The principle is that if we assume something and show that it has inconsistent consequences then our assumption must have been wrong. In this way we have disproven our assumption. For instance, imagine that I accuse my sister of using my computer while I’m out and she responds, “Why would I use your computer? It’s too slow and the F-key sticks when you press it.”
I reason to myself, “Suppose she is being upfront with me. Then she didn’t use my computer today. The F-key only started sticking this morning so, because she has not used my computer today, she must not know. However, this contradicts her alibi. Therefore it cannot possibly be the case that she was upfront, because that assumption has two contradictory implications.”
Leaving her to wallow in shame, let’s get back to math. Suppose that S, the square root of 2, is rational. We need to show this has inconsistent consequences and therefore must be false. From Fact 3 above, there must be natural numbers A and B such that S = A/B and the only common factor of A and B is 1. If this is the case, then A = S x B so, squaring both sides, A2 = S2 x B2. Remember though, S is not just any old number, it has the very special property that S2 = 2. Putting all this together we have that 2 is a factor of A2. Then Fact 4 tells us that 2 is also a factor of A.
Because 2 is a factor of A, there is some other natural number C such that A = 2 x C. Now we return to the equation A2 = S2 x B2 and, using the facts that A = 2 x C and S
4 x C2 = 2 x B2
2 x C2 = B2
Now comes the trick. The last equation means that 2 is a factor of B2 and therefore, using Fact 4, a factor of B. Looking back, we have seen that our (ultimately doomed) assumption that S is rational led to the conclusion that 2 is a factor of both A and B. However, this contradicts the fact that the only common factor of A and B is 1, which was an important part of why we even started using A and B in the first place. This means it can’t possibly be true that S is rational.
Congratulations, you just navigated your way through one of mathematics’ most treasured proofs. Next week we’ll prove the Riemann hypothesis.