Same theorem, different proof

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

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

$p[\beta n] = q[\beta n \dfrac{p}{q}]$

Which should be true for all $n$ for some $\beta \notin Z$ and with $p, q$ coprime. Furthermore, assume $q \geq 2$, which we may, but to completely justify that now is tedious, so think of it as a homework exercise :). Since $q$ divides the RHS, it should also divide the LHS. And since $p$ and $q$ are coprime, $q$ has to divide $\beta n$ for all $n \in \aleph$, in particular $n = 1$. Let’s say $[\beta] = mq$. Now we use induction; assume $kmq = [\beta k] < \beta k < kmq + 1$ for some $k \geq 1$. We then have:

$[\beta (k+1)] < \beta (k+1) < (k+1)mq + \dfrac{k+1}{k}$

So $[\beta (k+1)]$ is either $(k+1)mq$ or it equals $(k+1)mq + 1$. But if it is the latter, it’s not a multiple of $q$. So it must be the former. And with induction we proved the following:

$[\beta k] = kmq$

Which is true for all $k \in \aleph$. Now, let $\delta$ be the fractional part of $\beta$ (so $\beta = b + \delta$, say, with $b \in \aleph$ and let $N$ be the smallest integer such that $\delta N > 1$. We now have:

$Nmq = [\beta N] = [(b + \delta)N] = bN + 1 = b(N-1) + b + 1 = [(b(N-1) + \delta (N-1)] + b + 1 = [\beta (N-1)] + b + 1 = (N-1)mq + mq + 1 = Nmq + 1$

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

3 Responses to “Same theorem, different proof”

1. Nhung Says:

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

• Woett Says:

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!