What are the E, pi, phi constants relations

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In summary, the conversation revolves around the relationship between various mathematical constants, including pi, e, phi, and i. Some suggest that there may be a deeper significance to these relationships, while others attribute them to "number mysticism". The conversation also touches on methods for calculating pi, including power series and the use of trigonometric functions. There is also mention of phi being an algebraic number and e and pi being non-algebraic. A script for calculating pi and e using power series is shared, along with a suggestion to use software for calculating millions of digits of pi.
  • #1
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do those constants have any relation to each other?
does something like pi-e or pi/e has any significance?
 
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  • #2
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.)
 
  • #3
yes, there are a few identites in maths such as

ii = e-π/2 and -1 = eπi
 
  • #4
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.
 

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Last edited:
  • #5
e(pi)i=-1
 
  • #6
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.
 
  • #7
Originally posted by mathman
e(pi)i=-1
i forgot about this equation.
any significance to it?
 
  • #8
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.
 
  • #9
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.
 
  • #10
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?
 

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  • #11
The golden ratio, (1 + 5^(1/2)) / 2 = 1.618... is often denoted by the symbol φ.
 
  • #12
I've heard of the golden ratio, but what is it used for and why is it golden?
 
  • #13
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.

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


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:

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)
 
  • #14
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.
 
  • #15
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.
 
  • #16
What about values such as 0 and 2?
 
  • #17
The general solution to the recurrence f(n+2) = f(n) + f(n+1) is:

f(n) = A * φ^n + B * (1 - φ)^n
 
  • #18
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)
 
  • #19
Oh, A and B aren't supposed to be terms 0 and 1; they're constants for which you need to solve.
 
  • #20
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.
 
  • #21
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!
 
  • #22
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.
 
  • #23
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?
 
  • #24
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
 
  • #25
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
 
  • #26
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.
 
  • #27
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
 
  • #28
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
 
  • #29
Originally posted by synergy
phi+phi^2 which is phi^3

Did you mean phi*phi^2 here.
 
  • #30
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 I am still in high school and we havnt touched this sort of stuff yet. I just like reading maths sites :)
 
  • #31
Nope, there just identities of cos x and sin x.
 
  • #32
The hyperbolic and circular trig functions are related through complex numbers. E.G.

cosh ix = cos x
sinh ix = i sin x
 
  • #33
The hyperbolic functions do have simlair identities though:

sinh x = ½(ex - e-x)

cosh x = ½(ex + e-x)

ex = cosh + sinh x
 
  • #34
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
 
  • #35
Weird, I've never come across that before. I've never even thought about that before.
 
<h2>1. What is the relationship between the constants E, pi, and phi?</h2><p>The constants E, pi, and phi are all mathematical constants that are related to circles and curves. E, also known as Euler's number, is the base of the natural logarithm and is approximately equal to 2.718. Pi, denoted by the symbol π, is the ratio of a circle's circumference to its diameter and is approximately equal to 3.14159. Phi, also known as the golden ratio, is a mathematical constant that appears in nature and is approximately equal to 1.618.</p><h2>2. How are E, pi, and phi used in mathematics and science?</h2><p>E, pi, and phi are used in a variety of mathematical and scientific applications. E is commonly used in calculus and exponential functions. Pi is used in geometry, trigonometry, and physics to calculate the circumference, area, and volume of circles and spheres. Phi is used in art, architecture, and design to create aesthetically pleasing proportions and patterns.</p><h2>3. Are there any real-world examples of the relationship between E, pi, and phi?</h2><p>Yes, there are many examples of the relationship between E, pi, and phi in the natural world. For instance, the spiral patterns found in seashells and galaxies follow the golden ratio. The human body also exhibits the golden ratio in the proportions of its limbs and facial features. Additionally, pi is used in the design of circular structures such as wheels and gears.</p><h2>4. Can you explain the significance of E, pi, and phi in mathematics?</h2><p>E, pi, and phi are significant in mathematics because they are fundamental constants that appear in many mathematical equations and formulas. They are also important in understanding the properties of circles, curves, and exponential growth. Furthermore, the relationship between these constants has been studied and appreciated by mathematicians and scientists for centuries.</p><h2>5. How can the relationship between E, pi, and phi be visualized?</h2><p>The relationship between E, pi, and phi can be visualized in various ways. One common visualization is the Fibonacci spiral, which is created by connecting arcs with radii based on the Fibonacci sequence, a series of numbers related to the golden ratio. Another visualization is the unit circle, which shows the relationship between E and pi by representing the values of sine and cosine on the unit circle. There are also many other geometric constructions and diagrams that can illustrate the relationship between these constants.</p>

1. What is the relationship between the constants E, pi, and phi?

The constants E, pi, and phi are all mathematical constants that are related to circles and curves. E, also known as Euler's number, is the base of the natural logarithm and is approximately equal to 2.718. Pi, denoted by the symbol π, is the ratio of a circle's circumference to its diameter and is approximately equal to 3.14159. Phi, also known as the golden ratio, is a mathematical constant that appears in nature and is approximately equal to 1.618.

2. How are E, pi, and phi used in mathematics and science?

E, pi, and phi are used in a variety of mathematical and scientific applications. E is commonly used in calculus and exponential functions. Pi is used in geometry, trigonometry, and physics to calculate the circumference, area, and volume of circles and spheres. Phi is used in art, architecture, and design to create aesthetically pleasing proportions and patterns.

3. Are there any real-world examples of the relationship between E, pi, and phi?

Yes, there are many examples of the relationship between E, pi, and phi in the natural world. For instance, the spiral patterns found in seashells and galaxies follow the golden ratio. The human body also exhibits the golden ratio in the proportions of its limbs and facial features. Additionally, pi is used in the design of circular structures such as wheels and gears.

4. Can you explain the significance of E, pi, and phi in mathematics?

E, pi, and phi are significant in mathematics because they are fundamental constants that appear in many mathematical equations and formulas. They are also important in understanding the properties of circles, curves, and exponential growth. Furthermore, the relationship between these constants has been studied and appreciated by mathematicians and scientists for centuries.

5. How can the relationship between E, pi, and phi be visualized?

The relationship between E, pi, and phi can be visualized in various ways. One common visualization is the Fibonacci spiral, which is created by connecting arcs with radii based on the Fibonacci sequence, a series of numbers related to the golden ratio. Another visualization is the unit circle, which shows the relationship between E and pi by representing the values of sine and cosine on the unit circle. There are also many other geometric constructions and diagrams that can illustrate the relationship between these constants.

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