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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,665

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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Dear BobbyM,

May I know from where are you getting all these facts?

Dear anonimnystefy,

The other two which we have chosen are Pythagoras and Turing.

BTW Is Turing OK?

.

.

Some of my classmates don't keep track of Mathematics so I need your advice?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,393

hi

I think he has a random fairy story generator.

Fibonacci, Turing and Pythag make a good set. You've covered a wide spread of mathematical topics and history.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,393

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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Hi

I like stories very much?

What has Turing contributed to Mathematics

*Last edited by Agnishom (2012-06-15 20:34:24)*

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,393

Quick 'google':

http://www.sciencemuseum.org.uk/visitmu … tAodyhTc0w

http://www.turing.org.uk/turing/

http://en.wikipedia.org/wiki/Alan_Turing

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,665

Hi Agnishom;

May I know from where are you getting all these facts?

Just a little math humor.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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Does it mean you were making it up?

If so, you have awesome creativity

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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The wikipedia says that the Fibonacci Sequence is a complete sequence which means any integer can be expressed as a sum of fibonacci numbers.

Can you so how it is proved?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,665

Hi Agnishom;

If so, you have awesome creativity

Trust me it is just a story. I hoped it would make you laugh.

For the Fibonacci proof, I think it is called Zeckendorf's theorem.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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Would you show it here please?

What is this induction thing?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,393

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,393

hi Agnishom

Here is a flow chart that shows the 'proof'.

Notes

(i) As every Fibonacci number is the sum of a larger and a smaller number (after the initial 1s) say a + b where b>a

=> when the value lies between b and a + b the largest F number that can be subtracted is b, and the remainder is less than b.

(ii) It will always be possible to choose the largest F whenever the value is not already an F number.

example:

53

This is not an F number so subtract 34.

Remainder is 19.

This is not an F number so subtract 13.

Remainder is 6.

This is not an F number so subtract 5.

Remainder is 1

This is an F number so stop.

List:

Number = +34 + 13 + 5 + 1

Bob

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,393

hi

This article is from IT Now, the magazine of the BCS. **See page 51.**

If you cannot scroll about you need to switch off 'full screen mode'. I did this by saving the document and then re-loading it using Adobe.

http://www.bcs.org/upload/pdf/itnow-jun12.pdf

Bob

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**phanthanhtom****Member**- Registered: 2012-06-22
- Posts: 215

Carl F. Gauss

Leonhard Euler

Euclid

René Descartes

Fermat

Pythagoras

Newton should go to the nomination for best physicists. Most of his life spent on physics, not mathematics!

*Last edited by phanthanhtom (2012-06-22 03:05:18)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,665

Hi phanthanhtom;

Welcome to the forum.

For the invention of the Calculus, the binomial theorem, Newton's iteration and the nurturing of Abraham De Moivre is the reason Newton is there.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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Bob,

What you said in post #38 shows the algorithm to find the representation. But how can you prove that that algorithm will always be possible for any number?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,393

hi Agnishom,

I've got to log out now, but I'll answer that later today.

Bob

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Hmmm....

Its 9:02 PM in our country

Feeling Sleepy!

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,393

hi Agnishom,

Night night then.

Here's the proof for when you wake.

Let's call the number N.

If N is a Fibonacci number then there's nothing to do, because you either accept that a 'sum' can have just one number or if that makes you unhappy, you write it as the sum of the two Fibonacci numbers that made it by definition.

So what if it isn't?

The algorithm requires that we find the next F number below N and subtract it.

Can we definitely do this and be sure we never repeat an F ?

Let the next Fibonacci number above N, be (a + b) where a and b are the two F numbers that made it. The Fs go on for ever so there will certainly be one that is just bigger than N.

Suppose also that a < b.

Then b is the next F number going down from N. ie. b < N < a + b

For if there was another F number squeezing between b and N, let's call it 'c', then b + c is the next F number after N, not a + b.

The algorithm requires that we compute N - b and write b in the list. (You always subtract the biggest F that you can.)

So we are left with a remainder of N-b

Now N < a + b => N - b < a.

So if we now replace N with N - b in the algorithm we will never be able to subtract b again because N - b < a < b

So b will only occur in the list once.

So on this next loop our (new N) is now the (old N) - b

So can we carry on finding smaller and smaller Fs to subtract.

Yes, because 1, 2 and 3 are F numbers so we can either find a bigger F between 3 and N or we must be looking at N = 1, 2 or 3.

So eventually we must escape from the loop.

This is essentially the proof in the Wiki article. I've just put it into words that, hopefully, make it easier to understand.

I found it was best to try some examples.

eg 1) N = 67 b = 55 new N = 67 - 55 = 12.

N = 12 this isn't an F so b = 8 new N = 12 - 8 = 4

N = 4 this isn't an F so b = 3 new N = 4 - 3 = 1

N = 1 this is an F so stop.

67 = 55 + 8 + 3 + 1

eg 2) N = 103 b = 89 new N = 103 - 89 = 14

N = 14 b = 13 new N = 14 - 13 = 1

103 = 89 + 14 + 1

Try it yourself and you'll find it always works.

Bob

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,523

Hi Bob

I have a problem with your first line:

bob bundy wrote:

or if that makes you unhappy, you write it as the sum of the two Fibonacci numbers that made it by definition.

The theorem states that you can write any number as a sum of **non-adjacent** Fibonacci numbers not including F[1] , which means that a Fibonacci number must be written as a sum of only one element - itself, as you stated in the first part of the sentence.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,393

Ok. I thought I was just proving that a sum of Fs is always possible. (post #34)

Accidentally, my proof seems to cover what you want.

Bob

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,523

I tried reading your proof, but my tiny mind couldn't comprehend it. That is why I am providing a link of something I have barely comprehended:

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,665

Hi all;

I recommended that page in post #35. No one even saw it!

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,523

I have seen it. It is worth it to try posting it again.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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