E, pi, phi

MathematicalPhysicist

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

HallsofIvy

Homework Helper
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

Gold Member
yes, there are a few identites in maths such as

ii = e-&pi;/2 and -1 = e&pi;i

lavalamp

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

Code:
         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.

Attachments

• 915 bytes Views: 104
Last edited:

e(pi)i=-1

Integral

Staff Emeritus
Gold Member
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&Pi;i+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

Gold Member
Originally posted by mathman
e(pi)i=-1
any significance to it?

MathematicalPhysicist

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

Code:
         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:

Code:
    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?

Attachments

• 1.1 KB Views: 100

Hurkyl

Staff Emeritus
Gold Member
The golden ratio, (1 + 5^(1/2)) / 2 = 1.618... is often denoted by the symbol &phi;.

lavalamp

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

Hurkyl

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

&phi; : 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.

Code:
+---+--+
|   |  |
|   |  |
|   |  |
+---+--+

&phi;, 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( &phi;^n / sqrt(5) )

Where "round" means round to the nearest integer.

The exact formula, incidentally, is:

Fn = (&phi;^n - (1 - &phi;)^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

Staff Emeritus
Gold Member
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

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

f(n) = A * &phi;^n + B * (1 - &phi;)^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 = (&phi;^n - (1 - &phi;)^n) / sqrt(5)

Hurkyl

Staff Emeritus
Gold Member
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

Staff Emeritus
Gold Member
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

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

Code:
    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

Gold Member
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

Gold Member
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-&pi;/2

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving