What are the E, pi, phi constants relations

[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 = e^πi

 

[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

e^Πi+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 tendency 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:

F_n = round( φ^n / sqrt(5) )

Where "round" means round to the nearest integer.

The exact formula, incidentally, is:

F_n = (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 e^x), but it's significance is that it is the special case of x = π in Euler's formula:

e^ix = cos x + i sin x

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

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

cos x = (e^ix + 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 = (e^ix - e^-ix)/2i

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

Arn't those the hyperbolic functions?
cosh sinh?

I might be terribly wrong I am still in high school and we havnt touched this sort of 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(e^x - e^-x)

cosh x = ½(e^x + e^-x)

e^x = 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

 

[lavalamp]

Weird, I've never come across 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]

Relation between pi, phi and e

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 doesn't 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