For any right triangle, the sum of the squares on the legs is equal to the square on the hypotenuse. Sometimes folks just shorten it to a^2 + b^2 = c^2.I know, right? Prove the Pythagorean Theorem!? No way.
Here's one example from @kayfayayyy's blog--she thinks she's going to be a math teacher one day--except here she is, showing her students a supposed 3-4-5 right triangle. Oh yes, very clever Miss Fayayyy (if that's even your real name).
You've learned your lessons well: just dangle some candies in front of your kids and they'll believe anything you say. You can read more about her lies at her blog if you like.
But what Miss Fayayyy doesn't know about candy is that it likes to play both sides. How about this 3-5-6 right triangle?
Go ahead, count the Skittles. 9 + 25 = 36? No way, man. There it is: a counter-example, in all its Wild Berry flavor glory. We must conclude that the Pythagorean Theorem is false.
And yet my students persist in trying to prove these lies. Take this fellow @DominicTaylor11 for example. He wants to become a math teacher too, of course, and claims on his own blog to have proven the Pythagorean Theorem in his math class.
His argument can be illustrated by this series of pictures he took.
He says you can just start with two squares with dimensions (a+b) by (a+b), cleverly composed of a bunch of squares and triangles, as shown.
Then, he says, you just keep taking the red triangles away, one at a time, until you are left with the supposed Pythagorean relationship: a^2 + b^2 = c^2.
Well that's clearly not right. I mean, you can't just take triangles away and think the area of the remaining shapes will stay the same, can you?
What's that you say? You don't see what's wrong with that? Oh, come now. You're not one of those people who believes area stays the same when you cut up a shape and move the pieces around, are you? Don't you know Euclid got in trouble by using that principle of superposition almost 2,300 years ago!?
Look, look, look. I'll show you: Cutting shapes up and moving their pieces around changes the area.
Check it out. Let's start with an 8x8 square (area 64). Then cut it up and move the pieces around, as shown below. See, you get a new shape that has area 65. Watch closely:
See? When you cut things up and move things around, area changes! Here's the final image, in case you want to look it over again.
This notion that area stays the same when you cut things up and move them around is just a bunch of garbage. There we go: another "proof" bites the dust. It all starts to fall apart as soon as we start questioning the underlying assumptions.
You may not have realized it, but it turns out this math stuff is really all one big house of cards. Folks like Euclid made a few simple assumptions a long time ago, like "If equals are subtracted from equals, then the remainders are equal", and before you know it folks are using it to prove statements that are clearly nonsense. Does anyone ever go back and question those assumptions? No, they do not.
House. Of. Cards.
Believe me, I should know. I studied this stuff in college from 1997 to 2005. What's that, like 10 years? And today they actually pay me to teach this stuff! (Not very much, mind you, but they do pay me!)
I don't know how much longer I can do it though. The hypocrisy is eating me up inside.
I'm going to go get another cup off coffee to settle my nerves. Please don't tell anyone about this post. I'd be ruined! Shh, we must keep this secret.
You won't tell, will you?