e, pi, phi

[MathematicalPhysicist]

do those constants have any relation to each other?
does something like pi-e or pi/e has any significance?

[HallsofIvy]

Well, they are real numbers! Any other relationship I suspect is more "number mysticism" than mathematics. (Phi, in any case, is an algebraic number while e and pi are not.)

[jcsd]

yes, there are a few identites in maths such as

ii = e-π/2 and -1 = eπi

[lavalamp]

If it's any help these are the power series for [pi] and e:


r=[oo]
[pi] = 4 * [sum] ((-1)^r) = 4 - 4 + 4 - 4 + 4
r=1 (------) - - - - ... etc.
( 2r-1 ) 3 5 7 9

And

r=[oo]
e = [sum] ( 1 ) = 1 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1
r=1 (------) -- -- -- -- -- ... etc. - - -- ... etc.
((r-1)!) 0! 1! 2! 3! 4! 2 6 24


[pi] can also be obtained like this:

x * Sin (180/x) where x is a very large number and 180/x is in degrees.

I've attached a script to calculate pi and e using the above power series', however I have not been able to calculate pi using the Sin method as JavaScript assumes that the angle is measured in radians and it does not have a built in Math.pi method to allow me to convert the angle from radians into degrees.
Be careful if you are calculating pi to 1,000,000 iterations, I have an Athlon 1800+ and it caused my PC to hang for a couple of seconds, although I was listening to music at the time.

If you want to view the source, generally in Windows browsers, you can go View > Source.

[mathman]

e(pi)i=-1

[Integral]

Originally posted by mathman
e(pi)i=-1
A few years back I took Complex Analysis from Dr. King, then Chairman of the Lehigh U Math Department. He spent a fair amount of time with this relationship. He preferred to write it

eΠi+1=0
This expression relates 5 of the most important numbers of mathematics, Pi, e, i, 1 and 0 using all of basic mathematical operations, exponentiation, multiplication, and addition. On top of this it is an astounding, nearly unbelievable result.

He considered it poetry in Mathematics.

[MathematicalPhysicist]

Originally posted by mathman
e(pi)i=-1
i forgot about this equation.
any significance to it?

[MathematicalPhysicist]

Originally posted by lavalamp
If it's any help these are the power series for [pi] and e:


r=[oo]
[pi] = 4 * [sum] ((-1)^r) = 4 - 4 + 4 - 4 + 4
r=1 (------) - - - - ... etc.
( 2r-1 ) 3 5 7 9

And

r=[oo]
e = [sum] ( 1 ) = 1 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1
r=1 (------) -- -- -- -- -- ... etc. - - -- ... etc.
((r-1)!) 0! 1! 2! 3! 4! 2 6 24


[pi] can also be obtained like this:

x * Sin (180/x) where x is a very large number and 180/x is in degrees.

I've attached a script to calculate pi and e using the above power series', however I have not been able to calculate pi using the Sin method as JavaScript assumes that the angle is measured in radians and it does not have a built in Math.pi method to allow me to convert the angle from radians into degrees.
Be careful if you are calculating pi to 1,000,000 iterations, I have an Athlon 1800+ and it caused my PC to hang for a couple of seconds, although I was listening to music at the time.

If you want to view the source, generally in Windows browsers, you can go View > Source.
the condition for the summations in both cases is the same, ie r=infinity r=1.

[dcl]

What is phi exactly?
I though it was just another unknown like 'x' 'theta' etc etc


The above formula can also be expressed as

e^(i*x) = cos(x) + i*sin(x)



also 'e' can be derived from

(1 + (1/k))^k

as k approaches infinity, the value of 'e' is more accurate.


Also, if you would like a few million digits of pi, download PiFast and SuperPi and you can calculate them with relative ease :) . Alot of people use these programs to benchmark their overclocked computers and to test stability.

[lavalamp]

Originally posted by loop quantum gravity
the condition for the summations in both cases is the same, ie r=infinity r=1.And I put that, what do you think this is:


r=[oo]
e = [sum]
r=1
It's just that if I were to make a script that would run forever you'd never get an answer so what would the point of it be?

Anyway I've re-posted the script if anyone's interested, it includes the (1 + (1/k))^k way to calculate e.

By the way, does anyone know the formula for finding the decimal places of [pi]? I have heard of a formula that when you put in a number (say n, for the nth decimal place), you get an answer. I assume there is one for e as well, so does anyone have that?

[Hurkyl]

The golden ratio, (1 + 5^(1/2)) / 2 = 1.618... is often denoted by the symbol φ.

[lavalamp]

I've heard of the golden ratio, but what is it used for and why is it golden?

[Hurkyl]

The ancient greeks thought that the most visually pleasing rectangles had their side lengths in the proportion

φ : 1


Such a rectangle, called a golden rectangle, has the property that if you cut a square out of it as follows, the new rectangle has the same proportions as the original rectangle.


+---+--+
| | |
| | |
| | |
+---+--+



φ, like some other constants, has a tendancy to appear in unexpected places. One of the most interesting is the fact that for n >= 0, the n-th Fibbonachi number can be written as:

Fn = round( φ^n / sqrt(5) )

Where "round" means round to the nearest integer.

The exact formula, incidentally, is:

Fn = (φ^n - (1 - φ)^n) / sqrt(5)

[lavalamp]

Is that assuming that the first two starting numbers are 0 and 1? Is there a formula for finding the nth term for the Fibbonacci sequence that doesn't start with 0 and 1?

I also thought that the sequence was one of those things that didn't have a formula, I wonder where I got that idea from.

[Hurkyl]

Yes, I was using F0 = 0 and F1 = 1.


If you want a different starting point, just substute n with n + k for some k.

[lavalamp]

What about values such as 0 and 2?

[Hurkyl]

The general solution to the recurrence f(n+2) = f(n) + f(n+1) is:

f(n) = A * φ^n + B * (1 - φ)^n

[lavalamp]

Hmmm, sorry about chasing you around with this but, if you put in 0 and 1, for A and B respectively, you don't get:

Fn = (φ^n - (1 - φ)^n) / sqrt(5)

[Hurkyl]

Oh, A and B aren't supposed to be terms 0 and 1; they're constants for which you need to solve.

[lavalamp]

So I would need the first few terms of the sequence before I could find A and B. OK, fair enough. Maybe I'll find a pattern for the values of A and B for various starting values.

Thanks for the help.

[Hurkyl]

Any two would do, actually. Two equations in two unknowns. You could write down an explicit formula for A and B in terms of f(0) and f(1) if you wanted!

[lavalamp]

I'll think that I'll save that little treat for another time. Just like last night, I'm tired and I don't work well (or at all) when I'm tired.

[MathematicalPhysicist]

Originally posted by lavalamp
And I put that, what do you think this is:


r=[oo]
e = [sum]
r=1
It's just that if I were to make a script that would run forever you'd never get an answer so what would the point of it be?

Anyway I've re-posted the script if anyone's interested, it includes the (1 + (1/k))^k way to calculate e.

By the way, does anyone know the formula for finding the decimal places of [pi]? I have heard of a formula that when you put in a number (say n, for the nth decimal place), you get an answer. I assume there is one for e as well, so does anyone have that?
is there any reason why this condition applies in both of them?

[MathematicalPhysicist]

Originally posted by mathman
e(pi)i=-1
another way to write this (which i hope no one has yet written it) is:
e^(i*pi)=-1
e^[(i*pi)/2]=-1^0.5
e^[(i*pi)/2]=i

[jcsd]

Originally posted by loop quantum gravity
another way to write this (which i hope no one has yet written it) is:
e^(i*pi)=-1
e^[(i*pi)/2]=-1^0.5
e^[(i*pi)/2]=i

Be careful when doing those sorts of operations with imaginery numbers, but yes that is correct, if you look right back to the start where I gave you a couple of identities you can then put the last term to the power of i which leaves you with the well-known and proved identity of:

ii = e-π/2

[lavalamp]

Originally posted by loop quantum gravity
i forgot about this equation.
any significance to it?

If you would like I can post how it is possible to arrive at that solution (by that solution, I mean this - e^(i[pi])+1=0).

It uses the power series of e^x, but replaces x with i[pi], and you wind up with the power series for cos and sin, then when substituting in [pi], you get the equation mentioned above.

[jcsd]

Originally posted by lavalamp
If you would like I can post how it is possible to arrive at that solution (by that solution, I mean this - e^(i[pi])+1=0).

It uses the power series of e^x, but replaces x with i[pi], and you wind up with the power series for cos and sin, then when substituting in [pi], you get the equation mentioned above.

It's quite easy to derive (the orginal dervitaion comes from considering the series for cos x, sin x and ex), but it's significance is that it is the special case of x = π in Euler's formula:

eix = cos x + i sin x

Which is one of Euler's identities, the others being:

sin x = (eix - e-ix)/2i

cos x = (eix + e-ix)/2

[synergy]

If you start with 1 and 3 the ratio of consecutive terms approaches phi the fastest (for integers). In fact, each term after the first is round[(phi)^n] : 3 is phi^2, 4 is phi^3, etc. rounded to the nearest integer. If you start with (1, phi) as the first two terms instead of just integers, then the next term is 1+phi which is phi^2, next is phi+phi^2 which is phi^3, etc. (1+5^.5)/2 * (1+5^.5)/2 = (1+2*5^.5+5)/4 = (3+5^.5)/2 = 1+phi. It works!
Aaron

[lavalamp]

Originally posted by synergy
phi+phi^2 which is phi^3

Did you mean phi*phi^2 here.

[dcl]

Originally posted by jcsd

sin x = (eix - e-ix)/2i

cos x = (eix + e-ix)/2

Arn't those the hyperbolic functions?
cosh sinh?

I might be terribly wrong Im still in highschool and we havnt touched this sorta stuff yet. I just like reading maths sites :)

[jcsd]

Nope, there just identities of cos x and sin x.

[Hurkyl]

The hyperbolic and circular trig functions are related through complex numbers. E.G.

cosh ix = cos x
sinh ix = i sin x

[jcsd]

The hyperbolic functions do have simlair identities though:

sinh x = ½(ex - e-x)

cosh x = ½(ex + e-x)

ex = cosh + sinh x

[synergy]

No, lavalamp, I meant phi+(phi)^2 = phi^3, and it is the only number that has this property. Phi is one of the roots to y=x^2-x-1 and so x^2=x+1, x^3=x^2+x, etc. It's rather a cool property. Start with phi=(1+root(5))/2 and construct a fibonacci sequence with 1 and phi as your starting numbers. Your sequence will be 1,phi, 1+phi, 1+2phi, 2+3phi, 3+5phi, etc.
Your sequence will also be phi^0, phi^1, phi^2, phi^3, etc.
so phi+phi^2=phi^3
Aaron[a)]

[lavalamp]

Weird, I've never come accross that before. I've never even thought about that before.

[Hurkyl]

I meant phi+(phi)^2 = phi^3, and it is the only number that has this property.

Actually, three numbers have that property...

0, φ, and (1-φ)

[HallsofIvy]

Dang, Hurkyl, you got in before me!

[synergy]

Actually, I knew about (1-phi), I meant to say that the roots of the equation x^2-x-1 were the only numbers with that property, only I didn't think of zero (I added a root when I multiplied that eqn by x).
Aaron
[:)]

[lavalamp]

I assume that you guys are American, since HallsofIvy & Hurkyl effectively don't have profiles.
So I want to know, at what age did you guys learn about the golden ratio, the sin, sinh, cos, and cosh identities and the e^i[pi] thing? And did you learn it in school or just reading in and around Maths?

[jcsd]

Originally posted by lavalamp
I assume that you guys are American, since HallsofIvy & Hurkyl effectively don't have profiles.
So I want to know, at what age did you guys learn about the golden ratio, the sin, sinh, cos, and cosh identities and the e^i[pi] thing? And did you learn it in school or just reading in and around Maths?

Well I'm English, I took maths, physics, chemistry and further maths as A-levels before going onto to physics at university. I didn't learn any of this stuff at A-level (They didn't even teach us imaginery numbers or hyperbolic functions at A-level)

[synergy]

golden ratio and fibonacci stuff: reading math stuff as a child.
rest of it: a little in high school and mostly in college.
I figured out the stuff about phi+phi^2=phi^3 on my own, though.
Aaron

[lavalamp]

I must say, I've never really wondered when x^2 + x = x^3 myself. But then I've never really read around maths or physics before now, so this stuff is all kind of new to me. I'm just sending out a few feelers to see what situation other people are in.

I'm at A Level now, I've just dropped further maths, but I've kept math, physics and chemistry. The very next day after I dropped further maths, they (the teacher and one other pupil), went on to do P5 and the first thing they did was hyperbolic trig ratios. Shame really because I really wanted to learn about those, I'll just have to borrow his notes.
I'm hoping to do aeronautics next year at University.

[MathematicalPhysicist]

Originally posted by HallsofIvy
Well, they are real numbers! Any other relationship I suspect is more "number mysticism" than mathematics. (Phi, in any case, is an algebraic number while e and pi are not.)
i found that there is an approximation that 4/(phi)^0.5=pi.
i checked in my calculator and it's precise only from 2 place after the point of a decimal.

i stumbled upon this equation in this webpage:http://www.innerx.net/personal/tsmith/Gpyr.html

is it a reliable approximation?

[MathematicalPhysicist]

Originally posted by loop quantum gravity
i found that there is an approximation that 4/(phi)^0.5=pi.
i checked in my calculator and it's precise only from 2 place after the point of a decimal.

i stumbled upon this equation in this webpage:http://www.innerx.net/personal/tsmith/Gpyr.html

is it a reliable approximation?

no one can say anything about this approximation?

[HallsofIvy]

What do you mean "is it a reliable approximation"? You, yourself, said that it was correct only to two decimal places.

To 9 decimal places 1/√(φ)= 3.144605511 while π= 3.141592654. The error is 0.003012857. That's precisely how reliable it is!

[Bob C]

Just came across this thread and happen to be discussing the same thing in the Google sci.math forum:

If you list these numbers out to 12 decimal places along with the Fibonacci values and the sequence for these numbers (I start the sequence with zero), you will see that the 12th decimal place is the first position where all digits are equal (9). And at this point the Fibonacci value is the square of its position in the sequence. Probably means nothing... just another interesting relation with these numbers.

col 1 is the Fibonacci value, col 2 is the sequence, col 3 is pi, col 4 is PHI, Col 5 is e:

Fib Sequence pi PHI e
Val
0 0 3 1 2
1 1 1 6 7
1 2 4 1 1
2 3 1 8 8
3 4 5 0 2
5 5 9 3 8
8 6 2 3 1
13 7 6 9 8
21 8 5 8 2
34 9 3 8 8
55 10 5 7 4
89 11 8 4 5
144 12 9 9 9 *****

[moshek]

pi ^ (phi^2)=pi * pi ^( phi)

[MathematicalPhysicist]

pi ^ (phi^2)=pi * pi ^( phi)
this feature can be attribute to any base (different than 1 and 0) because it is simply pi^(phi^2)=pi^(phi+1) and then phi^2-Phi-1=0 which is ofcourse the equation for calculating the solution to the golden number, it doesnt say they have any relations which can only be attributed to these number only.

[KingNothing]

Is that golden ratio constant the actual number, or an approximation?

[moshek]

Than you should reply to me
in some different way like:

Now i have more exact way
to ask my original question !

Best
Moshek

I invite you to read:

www.physicsforums.com/showthread.php?t=17243

[HallsofIvy]

Is that golden ratio constant the actual number, or an approximation?

The golden ratio constant is the actual number. In fact, it is exactly
\frac{1+\sqrt{5}}{2}.

I can't imagine any mathematical constant being an "approximation to an actual number"!

[moshek]

It was Hipasus of metapontum who discover the existence of irrational numbers. he find self similarity on the pentagon so he prove by this that the golden ratio is not a rational number . But for some reasoned Euclid wrote on his 10 th book only about the irrationality of the root of 2 by the classical prove but this Euclid make a confusion with what really happened
with irrational numbers.


Moshek

[MathematicalPhysicist]

Than you should reply to me
in some different way like:

Now i have more exact way
to ask my original question !

Best
Moshek

I invite you to read:

www.physicsforums.com/showthread.php?t=17243
i think i did it in first two pages of this thread.

[moshek]

I look but i did not see that you did it
anyhow it is really not the main point here.

Moshek

[epii10]

Thought you guys might appreciate a little math I found concerning the numbers; Pi, e, and Phi as they relate to three unique right triangles. I call it the "Triple-Triangle-Theory" (TTT). I'm no mathematician so if anyone could tell me "why" the TTT works please feel free. (I think it has to do with the imaginary unit ie. "sqrt -1").

Here's the link:

http://www.gizapyramid.com/rick_howard%20research.htm

Rick Howard

[lavalamp]

That's a dead link.

Edit: Just try pasting the URL into the page, the forum should make it into a link automatically.

[MathematicalPhysicist]

Thought you guys might appreciate a little math I found concerning the numbers; Pi, e, and Phi as they relate to three unique right triangles. I call it the "Triple-Triangle-Theory" (TTT). I'm no mathematician so if anyone could tell me "why" the TTT works please feel free. (I think it has to do with the imaginary unit ie. "sqrt -1").

Here's the link:

http://www.gizapyramid.com/rick_howard%20research.htm

Rick Howard
he doesnt say how he arrived at the e proportion.
can someone tell me about it?
the pi and phi proportions are clear to me.

[epii10]

Loop,

I found the e proportion just messing around with the angles of the Pyramid with my HP graphic calculator. Sort of an accident really, only (and this is gonna sound weird) I loved the number e and would talk about it to anyone who would listen and had it in the back of my mind that I would find it in the Pyramid. Boy was I surprised!

My TTT paper (linked above) goes over the derivation of the e proportion but the first paper I wrote before discovering the TTT was all about the e proportion.. here is that link:

http://www.gizapyramid.com/ricks-e-proportion/rick-howards-research.html

I once had a real mathematical guy analyze the TTT and he said it was just a trick through the sustitutions.. (again, I'm no mathemetician) but simply because the equation embodies all three nearly congruent angles, and that coupled with the tetrahedral angle of 60 degrees being the only other solution to a specific question I pose along the way such that the output remains in the realm of "real" numbers, I tend to think it's just a little weird and still can't wrap my mind around it.

Pretty cool huh?

Rick

[MathematicalPhysicist]

howard have you tried plugging other numbers for the base of the triangle other than 1?

perhaps it's just a mistake but i notice that when the base is 1 then the side of the square (the base of the pyramid) is 2 and this number is in the proportion of e: beta/theta=e/2 perhaps there is a connection between the side of the base of the pyramid square and this proportion (and perhaps im hallucinating who knows (-: ).

[matt grime]

there's nothing special at all in those calculations, and the thesis contains many unjustified statements.

1) what does "optimal" mean, other than "supports me theory"
2) why do you say irrational numbers are impossible to construct with any accuracy in the real world, with the implicit assumption that somehow rationals are constructible to some degree of accuracy, and if so that negates your statement as then phi is realizable as a diagonal of a pentagon?
3) The mathematical presentation of the paper is shockingly bad, such as anyone who looks at the first diagram is given the impression that the betas are all the same when they certainly aren't.
4) why on earth is 2 so special to the point where you say no other integer will do it better?
5) the idea that you're going to produce a polynomial with phi as a root and 'wth the minimal number of terms' must be considered an ill advised boast since I'm sure i can think of one of degree 1 with phi as a root, and two if you must make me have integer coeffs, which must have fewer terms than the one you derive (a mathematically provable fact), though i didn't see it highlighted later in the text, but then that's because there's so much unnecessary waffle that it's not an easy task to extract information from the article.
6)the ideas of eqn 5.4 indicate a lack of understanding about raising complex numbers to exponents; again nothing special is going on there.
7) it would be beneficial to learn about polar representations of complex numbers to see why these coincedences occur.

[epii10]

Matt,

Your feedback of my TTT paper is appreciated. I'm always glad to have a "math smart" person give it gander. Forgive me as I realize it is a poorly done paper full of too much extranious opnion.. what can I say, I'm an artist/musician and my approach to math is aesthetic. I never claimed to be a mathematician or even "good" at math for that matter, fact is, I pretty well suck at math.

end of apology...

I didn't set out in the TTT to solve a problem. Mathematicians do that, I am not one of them. I found a ratio in the outer geometry of the Pyramid ie. (the e proportion). And there was no denying that it was there. I thought it was interesting. I looked at the Pi and Phi proportions and had long been awed by their coincidental occurence in the Pyramid. Yes, I do know that coincedence is a reality. When I started playing with the angles I began substituting terms and ended up with an equation that was true. How could it be not true? My method was elementary and I was only dealing with identities and those identities led to an overall identity as opposed to a solution to a problem.

In the end I was looking at an equation with three unique angles embedded in it such that the equation was equal to "1". If the three angles in the equation could be anything else than what they are for the thing to equal "1" then I'd be interested to know what they would be.

That's all it is, just a curious correlation of three independent right triangles living together in a simple equation. But (and this is the kicker) three right triangles that are very nearly congruent.. only ever so slightly different in shape.. and that is aesthetic if nothing else.

Rick

[matt grime]

wow... so if i spot that things are accurate to some completely artificial level of accuracy that makes it note worthy in such eulogising prose? the square root of two is 14 tenths, if you accept certain tolerances...

[epii10]

I don't know how to respond to you Matt because I don't understand your thinking. Good luck in your endevours.

Rick

[matt grime]

but the number of days in a week divided by the number of fingers (and thumb) on one hand is almost equal to the square root of two, doesn't that suggest something to you about the people who set the number of days in a week?

[lavalamp]

I know that you're not going to like this, but you brought it up so I'm going to say it anyway.

Take the number of fingers (and thumbs) most people have, and divide it by the number of days in a week.

10/7 = 1.4286

Also as you said

7/5 = 1.4

But get this, find the mid-point between the two, and you get even closer:

(10/7 - 7/5)/2 + 7/5 = 1.4143

It turns out than now you're only 7.215 * 10-5 out! Spooky! :biggrin:

[Gokul43201]

Or, the error is VERY NEARLY equal to:

(no. of days in week)/[(no. of fingers on both hands) ^ (no. of fingers on one hand)]

Oooooooooh !!