### Look What I've Done

I think I've just wrecked math... again. I know a few people who are going to be upset!

Here's how: I've discovered three numbers that do not satisfy the "Pythagorean Theorem", namely 20, 10, and 25. Go ahead, try to do it. It just won't work. 20

You may argue that the numbers are only supposed to work for the sides of a right triangle. I'm not an idiot, I know that people claim this. Let's pretend that I grant you this, are you trying to tell me that I can't construct a right triangle with sides of length 10 cm, 20 cm, and 25 cm? I don't believe you. The burden of proof, I do believe, is on you.

Here's how: I've discovered three numbers that do not satisfy the "Pythagorean Theorem", namely 20, 10, and 25. Go ahead, try to do it. It just won't work. 20

^{2}+ 10^{2}= 500 and 25^{2}= 625.You may argue that the numbers are only supposed to work for the sides of a right triangle. I'm not an idiot, I know that people claim this. Let's pretend that I grant you this, are you trying to tell me that I can't construct a right triangle with sides of length 10 cm, 20 cm, and 25 cm? I don't believe you. The burden of proof, I do believe, is on you.

## 16 Comments:

I find your blog repeatedly amusing.

Thank you, and I yours, and I eagerly await your new one.

I'm low on ideas. I think the end is near.

Something tells me you're actually being serious with this post.... In a space where the Pythagorean Theorem holds, 10, 20, and 25 cannot be the lengths of sides of a right triangle. If you cannot prove that, you might want to hold back from trying to find major results in geometry. You can construct a space where lengths 10, 20, and 25

doform a right triangle, however, the Pythagorean Theorem (as you used it) no longer holds.Dear anonymous: You are assuming your conclusion. If a^2 + b^2 = c^2, then my lengths can not be the lengths of a right triangle. Well, obviously. I am making the ground breaking statement that I'm pretty sure the lengths I've given can form a right triangle, and therefore, the Pythagorean theorem is wrong. Are you following now? No offense, but you're sounding kind of dumb.

sorry, but youre wrong. your lengths do not make a right triangle. they just dont.

No, my lengths do. Or at least I think they should. I can imagine in my mind that I can form a right triangle with these lengths. What more do you want? Probably for me to rebuild mathematics, yes? I admit, it is easier to tear down the false edifice of mathematics than it is to rebuild it properly. This should take some time.

Just because a triangle should be able to be made with three different side lengths, doesn't mean one can. You're obviously not a mathematician, are you?

Dave, you've displayed the precise problem with mathematics. Arrogance. I've said it before, and I'll say it again, there is never one "right answer" in mathematics. My opinion clearly differs from yours, but does that make it less valid?

No, your opinion isn't less valid, however, I think you're missing a few things.

Let's suppose there is a triangle with sides 10, 10, and 25. I'm not saying there is or isn't. I'm just saying let's assume there is. By the law of cosines, we can find the biggest angle by

25^2=10^2+10^2-2(10)(10)(cos(theta))

425=-200cos(theta)

-2.125=cos(theta)

Cosine's range is -1<=x<=1 so there's no such triangle.

OOPS you said 10, 20, and 25, my mistake.

By the same logic, we have

25^2=10^2+20^2-2(10)(20)(cos(theta))

125=-400cos(theta)

theta = 108

So that triangle can't possibly be a right triangle, but it is obtuse.

Dave, I have a link you may want to check out. It may help you understand what I am trying to say a little better. Link

I should've known *blush*

You're being just as stubborn as you're accusing others of being--except you're not correct.

Mathematics has prospered many times from people questioning basic principles. Thing is--what you're pondering has already been pondered.

Yes, it is possible to imagine a right triangle with sides 10, 20, and 25. You simply need to redefine either "right triangle" or "length" from how it is used in standard Euclidean geometry. The Pythagorean Theorem is a theorem applying to right triangles in a standard Euclidean plane, where a right triangle is defined as a triangle where one of the angles is pi/2 radians, and the length between two points is defined as the distance in "unit intervals" of the line segment connecting them.

Now, this is dumbing it down a bit, for there are much more formal and exact ways of discussing angles and length.

Your triangle, as you are describing it, can certainly exist as you are claiming. But, inarguably, it simply will not "fit" in a standard 2-D space in which the Pythagorean Theorem holds.

I'm not "assuming my conclusion." You're just not reading carefully, nor are you grasping the theorem you're claiming to refute.

Why not "imagine a world" where 3^2 is 10 and not 9? Then, 3^2 + 4^2 does not equal 5^2, and the Pythagorean Theorem fails for what "should" be a right triangle. Have I disproved the theorem? No, I'm just working in an "imaginary world" in which the Pythagorean Theorem no longer holds.

Dear anonymous: I invite you to check out the link I gave to dave a few comments back.

*Sigh* with so many crackpots legitimately attempting to tear apart math, I can't distinguish those who are serious from those who are not anymore. Sorry, but who posted this on sci.math anyway?

lol, I like the link A spoonful weighs a ton. And people, If a person want to make a right triangle with sides of 20, 10, and 25. Its fine by me... I want to find a analitic solution of Navier-Stocks and ppl pay me for it... so... Whos the crasy?

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