Assume the square root of 2 is a fraction, and watch the assumption collapse

0:00Calling a number irrational is a much bigger claim than it sounds. It does not say the number is strange, or awkward, or hard to write down. It says: there is no fraction that equals it. Not that nobody has found one. That there is not one. One whole number over another, with the bottom not zero. None of them. Ever. And there are infinitely many fractions, so you cannot check them.

0:29You could test a million and learn nothing, because the next one is still out there. So the claim has to be settled some other way entirely, and that is what makes this proof worth watching. Start with the obvious idea, and watch it fail. Take a calculator, get the square root of 2 as a decimal, and look for a pattern. Here is how that goes. 99 over 70 is 1.414285 and so on.

1:03Square it. 99 squared is 9801, and 70 squared is 4900. Twice 4900 is 9800. So the top is 9801 and it needed to be 9800. It is out by one. One, out of nine thousand eight hundred. Try harder. 577 over 408 is 1.4142156. 577 squared is 332929, and twice 408 squared is 332928. Out by one again. You can keep going and the gap keeps shrinking, and it is never zero.

2:00But notice what that actually tells you: nothing. Getting closer is not the same as arriving, and no amount of closeness is evidence. There is a second version of the same mistake, and it is worth killing now. The decimal goes on forever, so it must be irrational. No. One third goes on forever too. Nought point three three three, with no end to it, and one third is a perfectly ordinary fraction.

2:32One seventh goes on forever as well: one four two eight five seven, and then the same six digits again, and again. So going on forever is not the test. Repeating is. Every fraction's decimal either stops or falls into a repeating block. Every single one. Take every fraction with one on top and a bottom up to 400. That is 399 of them. 22 stop dead. The other 377 settle into a repeating block.

3:08Not one of them wanders forever without a pattern. So to use this test on the square root of 2, you would have to show its digits never repeat. Which is a statement about infinitely many digits, and you are back where you started. The proof that follows never looks at a single digit. So here is a completely different kind of attack. Instead of hunting for a fraction, suppose there is one.

3:37Not because we believe it. We are going to test the idea to destruction. Suppose the square root of 2 equals a over b, where a and b are whole numbers and b is not zero. Now follow the consequences honestly, step by step, and see where they lead. If they lead somewhere impossible, then the only thing that can be wrong is the supposing. That is the whole method, and it has a name we will come back to.

4:10Before touching the algebra, there is one move to make, and it is free. Any fraction can be reduced. If a and b share a factor, cancel it. If what is left still shares one, cancel that too. This stops, because the numbers only get smaller, and it costs nothing, because the value of the fraction never changes. So assume from the start that a and b share nothing except 1.

4:38Write that down and leave it on the board, because we are going to come back to it and it is going to matter enormously. It looks like tidying up. It is not tidying up. It is the trap the rest of the proof walks into. Now the algebra, and there is very little of it. The square root of 2 equals a over b. Multiply both sides by b. So b times the square root of 2 equals a.

5:09Square both sides. The root disappears, which is the whole reason for squaring. On the left, b squared times 2. On the right, a squared. So 2 b squared equals a squared. That is the entire equation this proof runs on, and everything from here is reading it carefully. Look at what that equation is saying. The left side is 2 times a whole number. So the left side is even.

5:39Therefore a squared is even. Which is to say: 2 divides a squared. And here is another way to feel how strange this equation is. Write out the squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. And the doubled squares: 2, 8, 18, 32, 50, 72, 98, 128, 162, 200. The equation is demanding a number that appears on both strips. Slide them past each other as far as you like and nothing ever lines up.

6:21That is not a proof either, but it tells you what is being asked for. Now comes the only interesting step in the whole argument. 2 divides a squared. Therefore 2 divides a. That looks like it needs no justification at all, and it is the one line that does. Because the general version is false. 4 divides 36. 36 is 4 times 9, so that is beyond argument. But 4 does not divide 6, and 36 is 6 times 6.

6:57So a divisor of a square is not automatically a divisor of the number. It works here for exactly one reason: 2 is prime. A prime cannot be assembled out of pieces of two copies, because it has no pieces. So write it down properly. 2 divides a squared, 2 is prime, therefore 2 divides a. Remember where that was used, because it is about to happen a second time.

7:272 divides a. So a is 2 times something. Call it 2c. Put that back into the equation. 2 b squared equals a squared, and a squared is 2c all squared, which is 4 c squared. So 2 b squared equals 4 c squared. Divide both sides by 2. b squared equals 2 c squared. Now look at the shape of that. It is the same equation we started with, with b where a was.

7:57Which means the same reasoning applies, word for word. The left side is 2 times a whole number, so b squared is even, so 2 divides b squared. And 2 is prime, so 2 divides b. That is the second use of the same theorem, and it is why the proof repeats rather than needing a new idea. So put the two conclusions side by side. 2 divides a. 2 divides b.

8:27Both of them are even. They share a factor of 2. And now look back at what we wrote on the board at the start. a and b share nothing except 1. Those two statements cannot both be true. Not unlikely. Not surprising. Impossible. That is the collision, and it is the end of the proof. Now be careful about what has just been shown, because this is the part that gets garbled.

8:57Every line after the assumption is valid. Reducing the fraction was legitimate. Squaring both sides was legitimate. Reading off that a squared is even was legitimate. Both uses of the prime theorem were legitimate, because 2 is prime. Not one step is in doubt. So the impossible thing at the end cannot be blamed on any of them. The only thing left to blame is the supposing, right at the beginning.

9:28There is no fraction equal to the square root of 2. It is irrational. But now a question that ought to bother you. That argument barely mentioned the number 2. Would it not prove every square root is irrational? It would not, and this is the best thing in the topic, so run the identical script on 4 and watch. Suppose the square root of 4 is a over b, with a and b sharing nothing.

9:58Multiply up and square: 4 b squared equals a squared. The left side is even, so a squared is even, so 2 divides a. Same line, same theorem. So a is 2c, and a squared is 4 c squared. Substitute: 4 b squared equals 4 c squared. Divide by 4. And here is where the two runs come apart. For the square root of 2, this line read: b squared equals 2 c squared, and that 2 is what let the argument repeat.

10:32For the square root of 4, it reads: b squared equals c squared. There is no factor of 2 left. Nothing to hand on. The argument just stops. And it has to, because the square root of 4 is 2, which is 2 over 1, and 2 and 1 share nothing at all. The proof was never about square roots being awkward. It was about the 2 out in front being prime.

10:59One more thing to take away and see what breaks. Go back and delete the step where we reduced the fraction. Everything else still works. You still reach: 2 divides a, and 2 divides b. And now ask what is wrong with that. Nothing whatever. Take every fraction with a top and a bottom up to 200. That is 40000 of them. 10000 have two even parts. Four over six, ten over twelve, and so on, endlessly.

11:32Two even parts is not a contradiction. It is completely ordinary. It is only a contradiction because we banked the opposite first. And here is the check that makes the point sharpest: of those 10000 fractions with two even parts, the number that share nothing is zero. Which is exactly why the free move at the beginning was not free at all. It was the whole trap. So stand back and look at what the argument actually did.

12:06It never searched. It never computed a decimal. It never examined a single digit. It took the thing it wanted to disprove, assumed it, and followed it until it contradicted itself. That method is called proof by contradiction, and it is worth knowing by name because you will meet it again. It feels like cheating the first time you see it, and it is not. You are not claiming the assumption. You are testing it to destruction, and it broke.

12:39And notice what it bought you. Not a fact about one fraction, or a million. A fact about every fraction there will ever be, settled once and for all. That is what this kind of argument is for.