And I found a fourth proof of it 😀 Here it comes.

So I start off at the moment when we showed the following:

Which should be true for all for some and with coprime. Furthermore, assume , which we may, but to completely justify that now is tedious, so think of it as a homework exercise :). Since divides the RHS, it should also divide the LHS. And since and are coprime, has to divide for all , in particular . Let’s say . Now we use induction; assume for some . We then have:

So is either or it equals . But if it is the latter, it’s not a multiple of . So it must be the former. And with induction we proved the following:

Which is true for all . Now, let be the fractional part of (so , say, with and let be the smallest integer such that . We now have:

And all we can hear is the sweet sound of contradiction..

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October 11, 2010 at 07:10 |

I think Theorem Pythagoras has many diffrent proof:) I like this theorem so much

Dear Prof Woett

To day, when I walking, It’s rain,my friend has an arugument with me, he said: If we faster,we will not get wet

I don’t agree

In past, I read in Chinese mathematics book, I don’t remember name book but It’s say “If you don’t want get wet, you should has velocity run= velocity rain”,I don’t remember prove it

Can you help me

October 11, 2010 at 12:35 |

I think Theorem Pythagoras has many diffrent proof:) I like this theorem so much

Did you know it’s possible, in principle, to construct infinitely many proofs of the Pythagorean Theorem? I think that’s kinda neat 🙂

Dear Prof Woett

To day, when I walking, It’s rain,my friend has an arugument with me, he said: If we faster,we will not get wet

I don’t agree

In past, I read in Chinese mathematics book, I don’t remember name book but It’s say “If you don’t want get wet, you should has velocity run= velocity rain”,I don’t remember prove it

Can you help me

Well, your friend is right, to some extent 🙂 The amount of rain that hits your front doesn’t depend on your speed, while the amount of rain that hits the top of your head (slightly) does, with more rain on your head when you walk slowly. But, I think, the difference is often negligible. The thing you cannot neglect is the fact that wetness when walking (or running) in the rain is essentially a function of time; the longer you’re outside, the wetter you are. And since running gets you home faster, you should run!

October 11, 2010 at 14:04 |

I want run but I can not, My velocity is very slow 🙂

thank you so much 🙂