A couple of simple proofs, made easier to grasp

I’ve always felt a certain undercurrent of awe at how elegant and beautiful the idea of a mathematical proof is.

Don’t get me wrong, I’m not a math geek, but I can’t help but admire the beauty of the proving process. (While, on paper, I fit the bill for a math geek, I’m too results oriented and learned to resent theoretical math because it was “drudgework my computer is supposed to be doing but the teacher would mark me down for that”.)

That said, I wanted to see if I could help people feel that sense of minor awe without all of the “First, you must complete three quests learn more math” that turns people off… so, here we go. I’ve decided to put my communication skills to the test on a couple of simple proofs. (Sorry for the lack of proper math typesetting)

If I’ve done my job, even someone who didn’t finish high school and doesn’t remember their high-school math courses should have a fair shot at understanding this.

TL;DR: If you’re not willing to try reading this, at least scroll down to the bottom where I’ve provided some very short, but awe-inspiring YouTube videos.

Both of these proofs use a technique called “proof by contradiction”. The idea is that, if you can make math that’s the right kind of nonsense, you can say “Statement X must be false because, if it was true, we’d get nonsense”.

It’s a lot easier to explain by example, so let’s get started:

1. Why you cannot divide by zero

Dividing by zero is one of those things everyone seems to wonder about, so let’s use it to demonstrate proof by contradiction.

  1. Suppose that anything divided by zero is zero. (Makes intuitive sense. Cut something into zero pieces and you have none of it.)
  2. That would mean that 1/0 = 0 and 2/0 = 0.
  3. Ok, an equals sign is just a fancy way of saying “these two things are the same, so let’s use that zero to combine the two equations:
    1/0 = 0 = 2/0
  4. …which means that 1/0 = 2/0
  5. Now, since we can do anything to an equation as long as we do the same thing to both sides, let’s get rid of that “divided by zero” by multiplying both sides by zero. (Remember, 5 * 2 / 2 is the same as 5. The twos cancel each other out.)
  6. Oops. Now we’ve got 1 = 2

Well, the only thing we assumed was that something divided by zero equals zero, so that must be what caused us to get nonsense.

That’s proof by contradiction. Assume one (and only one) thing, then show that it causes nonsense.

You also can’t use infinity (or anything else where you always get the same result) but infinity has the additional problem that it isn’t a number in the same way that “east” and “west” aren’t places.

If you’re still curious, I recommend the Numberphile video Problems with Zero, which shows multiple ways to explore this “dividing by zero” thing as well as other math quirks zero has. (And, unlike me, they draw graphs too)

2. How we know that √2 is irrational

One of the things you might have been told by your high-school math teacher is that √2 (the square root of 2) is irrational.

Refresher (square root):
If n*n = x, then “n” is the square root of “x“. For example, 2*2 = 4, so 2 is the square root of 4.
Refresher (irrational number):
When a number can be represented as a ratio (fraction), such as 0.25 = 1/4, we call it “rational”. An irrational number is a number that can’t be represented as a ratio (fraction). Irrational numbers have an infinite number of decimals which don’t get stuck in a repeating loop. (Pi is the most famous irrational number)

Here’s how you prove that √2 is irrational (that the number you have to multiply by itself to get 2 has an infinite number of decimal places without repeating, like Pi):

  1. For the purpose of our proof, lets assume that √2 is rational.
  2. If √2 is rational, that means that we can find some ratio (fraction) a/b that’s equal to √2.
  3. Before we start, we’ll say that a and b have already been put in simplest terms. (eg. There’s no number that both a and be can be divided by without getting decimals)
    √2 = a/b
  4. Algebra says that you can do anything you want to an equation as long as you make the same change to both sides of the equals sign, so they stay equal.
  5. Let’s multiply both sides by b
    √2 * b = a/b * b
  6. Dividing by b and then multiplying by b is the same as doing nothing, so a / b * b is the same as a
    √2 * b = a
  7. Now, let’s get rid of that symbol. First, since √n * √n = n (that’s the whole point of square roots), we’ll just square both sides (multiply both sides by themselves):
    (√2 * b) * (√2 * b) = a * a
  8. Now, to actually get rid of the symbol, we use another rule that’s already been proven called the associative property, which says that you can reorder the numbers and move the parentheses around in a sequence that’s all multiplication or all addition:
    (√2 * √2) * (b * b) = a * a
    2 * (b * b) = a * a
  9. This can also be written:
    2 * b² = a²

    Now for the clever bits:

  10. Since the left-hand side is 2 times something and both sides are equal, the right hand side must also be 2 times something (eg 2*8 = 4*4 is the same as 2*8 = 2*2*4).
  11. Let’s say that a = 2 * c and replace a in the equation. (We can do this because we’re not adding any new limits on what a can be. We’re just saying that c is 1/2 of whatever a is.)
    2 * b² = (2 * c)²
  12. OK, now we use the definition of what an exponent is (2² = 2 * 2) and the associative property to get rid of those brackets:
    2 * b² = (2 * c) * (2 * c)
    2 * b² = (2 * 2) * (c * c)
    2 * b² = 2² * c²
    2 * b² = 4 * c²
  13. Now let’s divide both sides by 2
    b² = 2 * c²
  14. OK, we’re half-way around to step 9. One side can be divided by 2, so the other side must also be divisible by 2. If we were to repeat steps 10, 11, and 12, we’d be back to step 9 exactly.
  15. The only way you get this behaviour is if a and b are even numbers, which means that a and b are both divisible by 2.
  16. …but, in step 3, we already assumed we’d divided them until they weren’t divisible by 2 anymore!

This means that we’ve got a nonsense equation and the only thing we assumed without proving it was that √2 is rational (that there exist integers (numbers without decimals) a and b where √2 = a/b)

Therefore, √2 must not be rational.

And, indeed, if you’ve got an irrational number, that would explain why step 3 failed. If you’ve got an infinitely complex number, then you’d have to divide a and b by 2 an infinite number of times to prepare it for this proof and, as I said earlier, infinity is a direction, not a destination.

3. Awe-Inspiring YouTube Videos

So, yeah. If you really don’t want to read the proofs, get your dose of awe from these instead, please:

CC BY-SA 4.0 A couple of simple proofs, made easier to grasp by Stephan Sokolow is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

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