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On a Whim by Mat in Vadnais Heights.

I seem to be hearing it more and more these days: “You can’t prove a negative.” Usually by folks who go on to use the statement as justification for “you can’t prove Bigfoot doesn’t exist” or “you can’t prove flying saucers don’t exist.” Sometimes I try to be polite, but in truth this absolutely sets my teeth on edge. Not because the specific examples cited above are silly, but because the assertion that “you can’t prove a negative” is totally, unequivocally, demonstrably wrong. And I will demonstrate this to you right now.

So, what am I going to do? Maybe prove that there isn’t a silver teapot full of mashed bananas orbiting a red dwarf star somewhere in the Andromeda galaxy? No, I shall leave that one alone. Instead, I will prove to you—big dramatic pause —that the square root of two is not a rational number.

A good number of you are tempted to roll your eyes and stop reading at this point, but really, It’s the same kind of question, isn’t it? What is a rational number? Simply a number that can be expressed as the ratio of two integers— like 2/3 or 5/6 or 22/7. I claim that the square root of two (that is, the number that, multiplied by itself, equals two) cannot be represented by a ratio of any two integers. Stated that way, proving it sounds like a tall order. After all, there is an infinite number of integers. How can I say for certain that there does not exist a pair of them somewhere up in the squillions—let’s call them p and q—such that p/q is exactly equal to the square root of two (which I will henceforth refer to as “root-two” for brevity)? My extragalactic teapot problem seems almost preferable.

To spoil the ending, we do it with a tried-and-true style of mathematical proof called a reductio ad absurdum, or “reduction to absurdity.” We start by assuming the opposite of what we want to prove and show that doing so always leads to a logical contradiction. Since contradictions are the ultimate no-no in mathematics, any assumption that generates one must be false.

So, let’s begin. I promise this will involve nothing more advanced than arithmetic concepts you learned in high school or earlier … and an ability to focus for a few minutes even if you don’t like the subject matter.

As I stated above, we start by assuming the opposite of what we want to prove. That is, we assume that there are two integers p and q such that p/q equals root-two. Now we work through the implications.

Step one: If our p and q have any common factors, remove them before continuing. For example, say integers p = 9 and q = 6 make up our potential p/q, yielding 9/6. Since 9 and 6 are both divisible by three, we divide both p and q by that common factor, redefining them as p = 3 and q = 2. We can get away with this because removing common factors always leaves a ratio with the same value (9/6 is precisely equal to 3/2, in our example). That’s a long-winded explanation of common factors, but I want to avoid any cries of foul play later on.

Step 2: Square both sides of the equation: p/q = root-two becomes p2/q2 = 2.

Step 3: Multiply both sides by q2, in other words p2/q2 = 2 becomes p2 = 2q2. With me so far? Let’s pause here and take stock. p2 = 2q2 means that p2 is an even number (because it can be represented as some other integer—q2 in this case—multiplied by two). And if p2 is even, then p must also be even, because any even integer squared is always another even integer, and any odd integer squared is always odd. So … . Interim conclusion No. 1: p is an even number.

Step 4: Since p is even, we can define it as 2r, where r is another integer (equal to half p’s value, duh). This seems obvious and rather pointless, but it’s important. Trust me. So… p = 2r.

Step 5: Square that last equation: p = 2r becomes p2 = 22r2 or, more simply, p2 = 4r2.

Step 6: Look back at step 3 above. From that we see that p2 = 2q2, right? But from step 5 we also know that p2 = 4r2. So, we know two different ways to define p2. Cutting out the p2 middleman, we can say 2q2 = 4r2. Got that?

Step 7: Really simple. Divide both sides of that equation by two: 2q2 = 4r2 becomes q2 = 2r2.

Similar to what we did after step 3 above … q2 = 2r2 means that q2 is an even number. And since q2 is even, we know that means q is even. So … .

Interim conclusion No. 2: q is an even number.

Well, so what? All we’ve shown is that if p/q = root-two then both p and q must be even. But… way back in step 1 we specified that p and q have been reduced to have no common factors. If they are both even, they have at least one common factor (namely 2). We have arrived at contradictory statements: “p and q have no common factors” and also “p and q have at least one common factor.” These statements cannot both be true. That means that our initial assumption, that root-two can be represented as the ratio of two integers p and q, must be false.

Conclusion: root-two cannot be represented as the ratio of any two integers. QED.

There, I have successfully proven a negative. And it’s not like I invented this all by myself. Hippasus of Metapontum proved that root-two is not rational way back in the fifth century BCE. Legend has it that Pythagoras, who was a big believer in ratios, had Hippasus sentenced to death by drowning for his heretical discovery. Some people really can’t handle being wrong.