# Pi, phi, e



## Dim7 (Apr 24, 2009)

Vote for your favo(u)rite.


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## elgar's ghost (Aug 8, 2010)

Is the 'e' eta or epsilon?


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## musicrom (Dec 29, 2013)

No "all of the above"? So hard to choose...

Phi has some really cool properties, but at least for me, hasn't come up very often, not sure how useful it is.

Pi and _e_ are used everywhere, and it's often surprising the places they come up. Hard to choose between the two for me, but I chose _e_ figuring pi would get more votes. Maybe not..?

What about _i_?


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## Huilunsoittaja (Apr 6, 2010)

Golden Ratio!

Pi isn't really useful in music, but the Golden Ratio can be used. Bartok is famous for his use of the Golden Ratio.


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## Figleaf (Jun 10, 2014)

Pi is the only one I've ever heard of, and I'm proud to say that I can recite it to three decimal places. I know it's rude to boast, but I also know several times tables (two, five and ten).  In spite of this mathematical prowess, I voted 'death to all numbers'.


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## mmsbls (Mar 6, 2011)

One of the most amazing equalities I've ever encountered involves the numbers e, pi, and i (square root of -1, an imaginary number).

e exp(i*pi) = -1

i.e. e raised to the power of i times pi equals -1.

That a simple formula involving 2 irrational numbers and an imaginary number could equal -1 is beautifully wonderful.

As a physicist I would have to choose e. e is everywhere and not just in flat space.


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## Strange Magic (Sep 14, 2015)

Ip


mmsbls said:


> One of the most amazing equalities I've ever encountered involves the numbers e, pi, and i (square root of -1, an imaginary number).
> 
> e exp(i*pi) = -1
> 
> ...


Euler's Identity: it's usually written as e to the i pi plus one equals zero. Euler touches all the bases here, but if he was any good, he would have gotten phi in there also. I'd vote for all three on the list if I could.


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## Balthazar (Aug 30, 2014)

I voted _e_, but I would vote _i_ were it an option.


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## Manxfeeder (Oct 19, 2010)

I chose the last option. I credit this to my overly impatient and overly bad-tempered sophomore-year algebra teacher for turning this promising young mathematician from the glories of numbers. Of course, I also blame my six-grade cafeteria superintendent for making pudding so repulsive that to this day I gag at the thought of it. 

School should come with a warning label.


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## Guest (Apr 14, 2016)

I voted for the most Scrabble points.


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## Taggart (Feb 14, 2013)

e may be everywhere, and I voted for it, but the fibonacci sequence also crops up in a large number of places and from the extension of Binet's theorem:










implies phi.

Why no option for "all numbers are interesting and deserve a long and happy lifetime"?


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## SixFootScowl (Oct 17, 2011)

I recall reading a book about math where some mathematician had noted, "Numbers are your friends." This is very true. I love math, not very good at it, but fascinating topic, and underlies everything in all creation. God's hand for sure.


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## Strange Magic (Sep 14, 2015)

I like the Prime Number Theorem as evidence that e is a number of quite some potency. It states that the number of primes less than x is approximately given by x/log x, where log x is the natural log, or ln x. Actually, the number of primes less than x falls between x/ln x and x/(ln x-1). Cicadas know all about prime numbers.


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## Poppy Popsicle (Jul 24, 2015)

I voted "I wish a slow, painful and lingering death to all numbers" (option 4, above) but on reflection I should have voted for Euler's "e" as I now realise he had a concept of _gradus suavitatis_ ('degree of [musical] pleasure') thus:

The formula is G(p/q)=1+Σei(pi−1)
where p,q are relatively prime, the pi are the prime factors of pq and ei is the multiplicity of pi

Here's the link for more detail. Don't ask me for any explanation: http://mathoverflow.net/questions/136572/number-theory-underlying-eulers-theory-of-music


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## omega (Mar 13, 2014)

Balthazar said:


> I voted _e_, but I would vote _i_ were it an option.







​
123456789012345


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## musicrom (Dec 29, 2013)

Taggart said:


> e may be everywhere, and I voted for it, but the fibonacci sequence also crops up in a large number of places and from the extension of Binet's theorem:
> 
> 
> 
> ...


Indeed. All numbers are interesting.

Even the number 1729:

G.H. Hardy on the ridiculously-good-with-numbers Ramanujan and the taxicab number:



> I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."


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## SeptimalTritone (Jul 7, 2014)

mmsbls said:


> One of the most amazing equalities I've ever encountered involves the numbers e, pi, and i (square root of -1, an imaginary number).
> 
> e exp(i*pi) = -1
> 
> ...


Yeah, it's a great formula.

It arises from the fact that exponentiation is repeated multiplication, and that repeated complex multiplication is like a rotation in the 2D plane. If repeated real multiplication makes a number "grow", repeated complex multiplication makes a number "rotate in a circle", and when you have arcs of circles, you have pi. Hence, exp(i a) = lim(n->infinity) (1+a*i/n)^n = cos(a) + i sin(a)


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## Belowpar (Jan 14, 2015)

Taggart said:


> e may be everywhere, and I voted for it, but the fibonacci sequence also crops up in a large number of places and from the extension of Binet's theorem:
> 
> 
> 
> ...


There will be two RHS Chelsea gardens this year based around the occurrence of the fibonacci sequence in nature.

Personally I've never been convinced that the fact you can find examples of something having the following no's of features means anything at all when it's 1,1,2,3,5,8,13 etc We shall see.


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## Chris (Jun 1, 2010)

I thought of a way of making e comprehensible to non-mathematicians. 

Say a bank offers you 100% annual interest, and promises to reinvest the interest in the account the moment it is earned, so that the interest earns interest. If you put one pound in the account, after one year it will have grown to e pounds.


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## Taggart (Feb 14, 2013)

Chris said:


> I thought of a way of making e comprehensible to non-mathematicians.
> 
> Say a bank offers you 100% annual interest, and promises to reinvest the interest in the account the moment it is earned, so that the interest earns interest. If you put one pound in the account, after one year it will have grown to e pounds.


Hmm.

The formula for compound interest is:










where t is the time in tears and n is the number of compounding periods per year.

If the compounding period is infinitesimally small, achieved by taking the limit as n goes to infinity then the formula becomes:










People who do mathematics are a bit like the Queen in Alice:



> Alice laughed. "There's no use trying," she said: "one can't believe impossible things."
> "I daresay you haven't had much practice," said the Queen. "When I was your age, I always did it for half-an-hour a day. *Why, sometimes I've believed as many as six impossible things before breakfast.*"


whereas those who can't see the beauty are a bit like Alice.


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## Pat Fairlea (Dec 9, 2015)

It has to be _e_, because natural logs provide a home for so many tiny forest creatures.


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## Taggart (Feb 14, 2013)

Pat Fairlea said:


> It has to be _e_, because natural logs provide a home for so many tiny forest creatures.


I thought log tables were made from trees with square roots?


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## Guest (Apr 15, 2016)

Do the decent thing, and infract yourself.


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## Dr Johnson (Jun 26, 2015)

Taggart said:


> I thought log tables were made from trees with square roots?


At least you could offer to get your coat.


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## TurnaboutVox (Sep 22, 2013)

Taggart said:


> Hmm.
> 
> The formula for compound interest is:
> 
> ...


I believe that this is a called a Freudian slip, Taggart!


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## Pat Fairlea (Dec 9, 2015)

Taggart said:


> I thought log tables were made from trees with square roots?


Excellent, Taggart. And they are made using twigonometry, if I may go off at a tangent.


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## Richannes Wrahms (Jan 6, 2014)

Fibonacci is not as omnipresent as they say, most logarithmic spirals in nature are probably not the golden spiral.


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