Can you suggest six names of great mathematicians quickly?

bobbym · 2012-06-13 07:10:44

Agnishom · 2012-06-15 20:23:08

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?

Bob · 2012-06-15 20:29:33

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

Bob · 2012-06-15 20:33:07

ps.

The clue is that his name was not Frankie.

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

Bob

Agnishom · 2012-06-15 20:33:47

Hi
I like stories very much?
What has Turing contributed to Mathematics

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

Bob · 2012-06-15 20:38:23

Quick 'google':

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

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

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

Bob

bobbym · 2012-06-15 21:46:00

Hi Agnishom;

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

Just a little math humor.

Agnishom · 2012-06-16 15:02:55

Does it mean you were making it up?
If so, you have awesome creativity

Agnishom · 2012-06-16 15:22:18

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?

bobbym · 2012-06-16 22:31:32

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.

http://en.wikipedia.org/wiki/Zeckendorf%27s_theorem

Agnishom · 2012-06-17 00:17:59

Would you show it here please?
What is this induction thing?

Bob · 2012-06-17 00:26:02

http://www.mathsisfun.com/algebra/mathe … ction.html

Bob

Bob · 2012-06-17 03:50:50

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

Bob · 2012-06-17 18:57:51

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

phanthanhtom · 2012-06-22 03:04:07

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)

bobbym · 2012-06-22 03:08:54

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.

Agnishom · 2012-06-22 03:22:57

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?

Bob · 2012-06-22 03:31:08

hi Agnishom,

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

Bob

Agnishom · 2012-06-22 03:32:50

Hmmm....
Its 9:02 PM in our country
Feeling Sleepy!

Bob · 2012-06-22 04:45:27

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

anonimnystefy · 2012-06-22 04:50:28

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.

Bob · 2012-06-22 07:41:41

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

anonimnystefy · 2012-06-22 07:54:13

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:

Proof of the Zeckendorf's Theorem

bobbym · 2012-06-22 13:22:48

Hi all;

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

anonimnystefy · 2012-06-22 13:25:47

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

Math Is Fun Forum

#26 2012-06-13 07:10:44

Re: Can you suggest six names of great mathematicians quickly?

#27 2012-06-15 20:23:08

Re: Can you suggest six names of great mathematicians quickly?

#28 2012-06-15 20:29:33

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#29 2012-06-15 20:33:07

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#30 2012-06-15 20:33:47

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#31 2012-06-15 20:38:23

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#32 2012-06-15 21:46:00

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#33 2012-06-16 15:02:55

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#34 2012-06-16 15:22:18

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#35 2012-06-16 22:31:32

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#36 2012-06-17 00:17:59

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#37 2012-06-17 00:26:02

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#38 2012-06-17 03:50:50

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#39 2012-06-17 18:57:51

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#40 2012-06-22 03:04:07

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#41 2012-06-22 03:08:54

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#42 2012-06-22 03:22:57

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#43 2012-06-22 03:31:08

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#44 2012-06-22 03:32:50

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#45 2012-06-22 04:45:27

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#46 2012-06-22 04:50:28

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#47 2012-06-22 07:41:41

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#48 2012-06-22 07:54:13

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#49 2012-06-22 13:22:48

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#50 2012-06-22 13:25:47

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