# Thread: √2 as a sum of infinite series

1. Now I just want the sequence αν that says how many pieces can be accomodated into the diagonal. That means, when I divide the side of lenght 1 into ν pieces, then αν pieces can fit into the diagonal. a1=1, a2=2, a3=4, and obiously aν=(non-decimal part of ν√2), but I don't want aν to contain an irrational number, and no "non-decimal part of".

2. Originally Posted by ThodorisK
Now I just want the sequence αν that says how many pieces can be accomodated into the diagonal. That means, when I divide the side of lenght 1 into ν pieces, then αν pieces can fit into the diagonal. a1=1, a2=2, a3=5, and obiously aν=(non-decimal part of ν√2), but I don't want aν to contain an irrational number, and no "non-decimal part of".
Look, you might understand what the real question is but, from what you're posting, the rest of us don't. None of your posts in this thread have made sense - due to lack of clarity in what the real question is. The above makes no sense. We are not mind readers and we do not have the time or inclination to continually stab in the dark.

So for the last time, please think carefully about what the question is your working on and then post it in such a way that we can clearly understand it. We are not mind readers.

3. I asked for the sequence αν, where:

ν=(to how many pieces I divided the side of the square)
αν=(how many pieces of length 1/ν can the diagonal accommodate)

ν<->αν
1<->1
2<->2
3<->4
4<->5
5<->7
6<->8
7<->9
8<->11

You can think of the pieces as little-squares that can sit on the diagonal.
The little-squares that can fit on the square are ν^2.
And 2ν^2 little squares can fit on the (literal) square of the diagonal if you cut some of them in half making them little-triangles, but then they cannot sit on the diagonal.

4. Originally Posted by ThodorisK
Was it really non-lucid what I asked? I doubt so. I asked for the sequence αν, where:

ν=(to how many pieces I divided the side)
αν=(how many pieces of length 1/ν can the diagonal accommodate)

ν<->αν
1<->1
2<->2
3<->4
4<->5
5<->7
6<->8
7<->9
8<->11

CB

5. Ok, I won't instist much to get an answer, it seems nobody is interested to know the meaning of √2 or you all know it and it's so obvious that I don't deserve an answer. However, I will make one more try, as if the reason was that I wasnt' lucid enough, I should try again. How about this, still not lucid?:

The sequence αν says how many pieces can be accomodated into the diagonal. That means, when I divide the side of lenght 1 into ν pieces, then αν pieces can fit into the diagonal.
ν=(to how many pieces I divided the side of the square)
αν=(how many pieces of length 1/ν can the diagonal accommodate)
ν<->αν
1<->1
2<->2
3<->4
4<->5
5<->7
6<->8
7<->9
8<->11
You can think of the pieces as little-squares that can sit on the diagonal.

αν=(non-decimal part of ν√2), so what's αν?

6. Originally Posted by ThodorisK
Ok, I won't instist much to get an answer, it seems nobody is interested to know the meaning of √2 or you all know it and it's so obvious that I don't deserve an answer. However, I will make one more try, as if the reason was that I wasnt' lucid enough, I should try again. How about this, still not lucid?:

The sequence αν says how many pieces can be accomodated into the diagonal. That means, when I divide the side of lenght 1 into ν pieces, then αν pieces can fit into the diagonal.
ν=(to how many pieces I divided the side of the square)
αν=(how many pieces of length 1/ν can the diagonal accommodate)
ν<->αν
1<->1
2<->2
3<->4
4<->5
5<->7
6<->8
7<->9
8<->11
You can think of the pieces as little-squares that can sit on the diagonal.

αν=(non-decimal part of ν√2), so what's αν?
I will repsond to what is posted in red. Quite a few people were interested and tried their very best to get you to post the question in a way that made sense to us. You have consistently failed to do so. To say that it's so obvious to us that we have decided not to tell you is false, churlish and uncalled for

The fact is that your question is incomprehensible. You talk about a diagnol. What diagnol? You have been asked to post the question execatly as it is written in the source it came from. Either the source is the most badly written textbook I have encountered in a long while or you refuse to do as requested.

You have had plenty of opportunity to post the question in a way that can be understood.

7. Originally Posted by ThodorisK
Ok, I won't instist much to get an answer, it seems nobody is interested to know the meaning of √2 or you all know it and it's so obvious that I don't deserve an answer. However, I will make one more try, as if the reason was that I wasnt' lucid enough, I should try again. How about this, still not lucid?:

The sequence αν says how many pieces can be accomodated into the diagonal. That means, when I divide the side of lenght 1 into ν pieces, then αν pieces can fit into the diagonal.
ν=(to how many pieces I divided the side of the square)
αν=(how many pieces of length 1/ν can the diagonal accommodate)
ν<->αν
1<->1
2<->2
3<->4
4<->5
5<->7
6<->8
7<->9
8<->11
You can think of the pieces as little-squares that can sit on the diagonal.

αν=(non-decimal part of ν√2), so what's αν?
If you take a square of side $1$ and a set of smaller squares of side $1/n$, $n$ in $\mathbb{N}$ then how many of these smaller squares can be fitted on the diagonal.

This is still ambiguous, and what we mean by "fitted on the diagonal" is not clear.

If they are placed so the small square diagonals lie on the unit squares diagonal obviously $n$ of them will fit.

If we place the small squares so that their bases are on the diagonal, then we are asking for given $n$ to find the largest $k_n$ such that $(k_n/n)^2\le 2$

Then we may consider the sequence $k_n,\ n=1...$.

The following calculates the first few terms, which seems to agree with what you post

Code:
>function seq(n)
$s=zeros(n,2);$   for idx=1 to n
$s(idx,2)=floor(sqrt(2)*idx);$      s(idx,1)=idx;
$end$   return s
$endfunction > >seq(10) 1 1 2 2 3 4 4 5 5 7 6 8 7 9 8 11 9 12 10 14 > Note this is still largely guess work since your clarification still makes little sense. CB 8. Originally Posted by CaptainBlack If you take a square of side $1$ and a set of smaller squares of side $1/n$, $n$ in $\mathbb{N}$ then how many of these smaller squares can be fitted on the diagonal. This is still ambiguous, and what we mean by "fitted on the diagonal" is not clear. If they are placed so the small square diagonals lie on the unit squares diagonal obviously $n$ of them will fit. If we place the small squares so that their bases are on the diagonal, then we are asking for given $n$ to find the largest $k_n$ such that $(k_n/n)^2\le 2$ Then we may consider the sequence $k_n,\ n=1...$. The following calculates the first few terms, which seems to agree with what you post Code: >function seq(n)$   s=zeros(n,2);
$for idx=1 to n$      s(idx,2)=floor(sqrt(2)*idx);
$s(idx,1)=idx;$   end
$return s$endfunction
>
>seq(10)
1             1
2             2
3             4
4             5
5             7
6             8
7             9
8            11
9            12
10            14
>
Note this is still largely guess work since your clarification still makes little sense.

CB
In other words we are looking for the numerator of the best rational approximation m/n to the square root of 2 from below for a given denominator n.

CB

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