Slippery debaters and the square root of 2

It may seem odd, but I am going to introduce an ancient clever mathematical argument with a story about teenage promiscuity.

Some years ago, a local radio host was interviewing a woman who wanted school sex education programs replaced with abstinence-only pledges. (I am reminded of a quip by Newfoundland comedian Mary Walsh that went something like this: “Using aspirin to treat a real headache is like expecting a ‘just say no’ campaign to eliminate teenage pregnancy.”)

The host made the reasonable suggestion that an abstinence recommendation could be used to complement the birth control information in sex education.

The abstinence-only advocate’s response? “Condoms are not 100 per cent effective.”

The effect of this non-sequitur was to allow her to slide away from a losing argument by changing the subject. Unfortunately, the host fell for the trick and the discussion moved to the failure rate  of condoms. When the guest found this topic too uncomfortable, she changed the subject again, this time to the contention that sex education promotes promiscuity.

Changing the subject to avoid a commitment to an argument is an old rhetorical trick. Used repeatedly, it can turn a discussion into a  pointless round of the same old arguments.

Now. How is all that relevant to the square root of two? It will take some time, so follow me.

The ancient Greeks had established the existence of two kinds of numbers, the natural numbers (1, 2, 3, …) and the rational numbers, which are quotients of the natural numbers. 1/2, 7/5 and 707/500 are examples of rational numbers.

(The natural numbers are actually rational numbers themselves. 3, for example, can be written as 3/1.)

The rational numbers made excellent sense to the Greeks. In fact, they were necessary.

“Odysseus’s house is 3/4 of a stadion down the road” or “I’ll take a 1/2 cotyla of wine.”

Furthermore, it was commonly thought that all geometrical distances were commensurable. That means that given two lines, there was a measuring stick that would measure each line a whole number of times. For example, suppose the length of one line is 3/4 and the length of a second line is 2/3. Then the common measuring stick is a line of length 1/12. (Which I got by multiplying 4 and 3.) Call that length a stadion. Then line one is 9 stadions and line two is 8 stadions. They are commensurable. Equivalently, the relationship between the two lines can be expressed as a rational number: line one 9/8 times the length of line two; or line two is 8/9 times the length of line one.

Everything seemed to be going smoothly until some Pythagorean burst in and spoiled the party by announcing the discovery of a bunch of new numbers that were decidedly unnatural, irrational and incommensurable with any rational number. Everyone was embarrassed, particularly because the new numbers had been lying around for all to see for centuries.

The Greeks, remember, loved geometry and one of their most beloved theorems was the one, usually attributed to Pythagoras, about the square on the hypoteneuse of a right-angled triangle being equal to the sum of the squares on the other two sides. In the diagram, the two sides adjacent to the right angle are length a and b; the hypoteneuse has length c. So, the theorem guarantees that

$c^2 = a^2 + b^2$

No irrational numbers in sight yet, you say. Wrong. Watch this.

To make things simple, suppose a and b are both 1, so that

$c^2 = 2$            or              $c = \sqrt{2}$

This is exactly what you get if you slash a diagonal across a square of side 1.

But what about $\sqrt{2}$ and 1. What is the common measure? Is there a rational number m/n such that $\sqrt{2}$ is m/n times as big as 1? Or, putting into algebraic form, is there a whole number fraction m/n such that

${\frac {m} {n}} = {\sqrt{2}}$

I am going to demonstrate that such a rational number is impossible. And I am going to use a clever logical technique, well-known since ancient times, called “proof by contradiction”.

I will assume that the number m/n exists and then let pure logic lead me to a contradiction. This technique runs counter to intuition. If I want to prove that m/n exists, why would I start by assuming that it does? Wait and see.

Okay, suppose there is a rational number m/n that is equal to $\sqrt{2}$.

Now here’s the crunch, the topic I have been leading up to. I am taking the position the radio host should have adopted with the abstinence lady. I want a commitment from you and I don’t want to reach the end of my argument only to have you reopen your commitment.

The commitment. I don’t want m and n (both whole, natural numbers) to have any common factors. I don’t want them both to be divisible by 2 or by 3 or by 5 or by any other whole number. For example, it is unacceptable if m = 140 and n =100. They are both divisible by 2. And by 5. We can reduce m/n from 140/100 to 14/10 (dividing both by 5) and then to 7/5 (dividing top and bottom by the common factor 2). Same number as 140/100, but with m = 7 and n = 5, we have no common factors.

Do you agree with the commitment? If so, then I will hold you to it. If not, stop reading and go join a strange religion.

So now we have

${\frac {m} {n}} = {\sqrt{2}}$

Our first step is to square both sides of the equation.

${\frac{m^2} {n^2}} = 2$

Now multiply both sides by $n^2$.

$m^2 = 2n^2$

This means that $m^2$ is even. Now, if m were odd, $m^2$ would also be  odd. So m must be even. And it can be written as m = 2k (where k is a whole number) in the last equation.

$(2k)^2=2n^2$

$4k^2=2n^2$

We can divide both sides by 2.

$2k^2=n^2$

But this implies that n is even.

We have come to the conclusion that both m and n are even. But that contradicts the fact that we reduced m/n so that m and n have no common factors. There’s no wriggling out of that commitment. Don’t change the subject; don’t renege. The only conclusion we can arrive at is that the square root of 2 was never a rational number to begin with.

That is, there are no whole numbers m and n such that

$\sqrt{2}= \frac {m} {n}$

End of proof.

It may seem that your calculator contradicts this. After all, asking for root 2 gives 1.414213562, which is a rational number: 1414213562/1000000000. In fact, the decimal expansion of root 2 — or any other irrational number — is infinite. The poor Greeks, however, did not understand this. They did not have decimals.