An exact equation for pi as a function of e and pi - are e and pi correct?

[Semjase]

I discovered this equation with a proof through mathematical trial and error.

 

Here's the equation pi=e^(2*(((ln(pi))^2+2*ln(pi)+1)/(2*ln(pi)+2))-1)

 

Using ttmath's online calculator with 1024 bit mantissa the equation gave me the exact

same value for pi as the constant value provided for pi by the calculator.

 

When using the calculator with a 2048 bit mantissa the 2 values of pi were

different, the last least significant digit there was a difference in 1 between the

the 2 digits.

 

If someone has access to a calculator that can calculate too more precision

I am curious to find out if the discrepancy is due to calculating error or

either the current value of e or pi is incorrect.

 

[John Cuthber]

Are you taking the piss?

Calculating pi in terms of pi is quite easy.

pi = pi

[D H]

I discovered this equation with a proof through mathematical trial and error.

 

Here's the equation pi=e^(2*(((ln(pi))^2+2*ln(pi)+1)/(2*ln(pi)+2))-1)

 

You don't need a calculator to verify this. It's a trivial tautology. What you wrote is just a convoluted way of saying [imath]\pi=e^{\ln \pi}[/imath], or even more succinctly, [imath]\pi=\pi[/imath].

[John Cuthber]

"An exact equation for pi as a function of e and pi"

 

e = pi * e / pi

Gosh!

[Semjase]

Are you taking the piss?

Calculating pi in terms of pi is quite easy.

pi = pi

 

 

I'm not completely stupid If you change the value for

e and/or pi in the equation it alters the value for

pi from the equation, you have to have 100% accurate

values for e and pi in the equation to get the 100%

accurate value for pi out of the equation. So if both sides

of the equation are equal then the values for e and pi

are 100% accurate.

 

 

You don't need a calculator to verify this. It's a trivial tautology. What you wrote is just a convoluted way of saying [imath]\pi=e^{\ln \pi}[/imath], or even more succinctly, [imath]\pi=\pi[/imath].

 

Such is not the case in that equation if you change the value of pi on one side of the equation you get the same value for pi on the other

side of the equation. In my equation if you keep the value for e constant and change the value for pi in the equation you get a different

value for pi on the other side of the equation.

 

I apologize it's wrong I tested on a Microsoft calculator

program it gave me a different value double checking it's

the same value.

[x(x-y)]

What are you talking about in that last post when you say "if you change the value for e and/or pi" and other variations of this statement? You cannot just change the value of e and pi, they are constants - the term is self-explanatory. And the others are correct anyway, your original post is just a long winded version of saying π = π

[uncool]

I discovered this equation with a proof through mathematical trial and error.

 

Here's the equation pi=e^(2*(((ln(pi))^2+2*ln(pi)+1)/(2*ln(pi)+2))-1)

 

Using ttmath's online calculator with 1024 bit mantissa the equation gave me the exact

same value for pi as the constant value provided for pi by the calculator.

 

When using the calculator with a 2048 bit mantissa the 2 values of pi were

different, the last least significant digit there was a difference in 1 between the

the 2 digits.

 

If someone has access to a calculator that can calculate too more precision

I am curious to find out if the discrepancy is due to calculating error or

either the current value of e or pi is incorrect.

Let's unwind this.

 

e^(2*(((ln(pi))^2+2*ln(pi)+1)/(2*ln(pi)+2))-1)

 

e^(2*((ln(pi) + 1)^2)/(2 * ln(pi) + 2) - 1)

e^(2 ((ln(pi) + 1)^2)/(2 * (ln(pi) + 1)) - 1)

e^(((ln(pi) + 1)^2/(ln(pi) + 1)) - 1)

e^((ln(pi) + 1) - 1)

e^(ln(pi))

pi

You've written a tautology. In other words, if you used any number instead of pi (in every position), the equation would remain true. e.g.

 

3 = e^(2*(((ln(3))^2+2*ln(3)+1)/(2*ln(3)+2))-1)

=Uncool-

Edited February 9, 2013 by uncool

[John Cuthber]

I'm not completely stupid If you change the value for e and/or pi ...

Discuss.

[Cap'n Refsmmat]

 

 

I'm not completely stupid If you change the value for

e and/or pi in the equation it alters the value for

pi from the equation, you have to have 100% accurate

values for e and pi in the equation to get the 100%

accurate value for pi out of the equation. So if both sides

of the equation are equal then the values for e and pi

are 100% accurate.

Not true. You can change the value of e and still get the right answer -- so long as you change the base of the natural logarithm to be whatever new value you chose.

[Crimson Sunbird]

An exact equation for pi as a function of e and pi

 

How about …

 

[latex]\pi\,=\,(-\pi)e^{i\pi}[/latex]

[Amaton]

Semjase, your identity is a special case of a general equality that holds true for all real numbers. Working backwards...

 

[math]x=x[/math]

 

[math]x=\exp[\ln x][/math]

 

[math]x=\exp[(\ln x+1)-1][/math]

 

[math]x=\exp\left[\frac{(\ln x+1)^2}{\ln x+1}-1\right][/math]

 

[math]x=\exp\left[\frac{2(\ln x+1)^2}{2(\ln x+1)}-1\right][/math]

 

[math]x=\exp\left[2\times\frac{\ln^2 x+2\ln x+1}{2\ln x + 2}-1\right][/math]

 

We can complicate it more and more if we'd like, but as you can see it's quite trivial.

 

Another cool but less obvious identity: [math]\pi=-2i\,\ln(i)[/math] which does the same thing but in the guise of imaginary arguments.

[Semjase]

Here's an exact equation for e in terms of pi

 

e=1/i^(i*2/pi)

 

I was wondering on Google calculator when entering

this equation if it used an infinite series value for pi

to determine the value of e. If you compared the

infinite series value for e and compared it to

the value of e generated by this equation and

they weren't equal then there would have to be an

error somewhere either in the value for e, pi or the i calculation.

[John Cuthber]

Seriously?

You were wondering if Google used something infinite?

I mean, I know they are big, but infinite?

 

In all sensible probability they have just stored the numbers for pi and e in the calculator.

 

All you are looking at are rounding errors.

[Semjase]

I'll rephrase that, if they actually use the known value of pi

at all in that calculation which therefore would of have to been

to be derived from part of an infinite series for the pi value.

There's a prediction that one day an error will be found

in the longest calculated value of pi.Would this be a calculating

error or an equation error or would this error occur be in all

known equations for pi, for the pi value calculated to many decimal places?

Here's part of the Enoch prophecy pertaining to pi

 

"the exposure and rectification of an error in the Pi-number calculation."

e also could be expressed this way

 

e=i^(2/(i*pi))

[John Cuthber]

"There's a prediction that one day an error will be found in the longest calculated value of pi."

There may well be a prediction that it will rain frogs in London tomorrow.

But the predictions are probably not correct.

 

​On what basis is the prediction made?

[Semjase]

It's an old prophecy that was discovered and

been published, and it will be interesting to see

if an error will be found in the known value of

pi.

[John Cuthber]

Seriously, and "old" prophecy about e?

 

And this is now in the wrong section, it's not maths any more.

[Amaton]

It's an old prophecy that was discovered and

been published, and it will be interesting to see

if an error will be found in the known value of

pi.

 

Why does this sound so elemental and mystic? It didn't clarify the question.

 

What specific "prediction" are you referring to and by what reason is it made?

[Semjase]

Here's an exact equation for pi in terms of i and base e.

The length of a section of the unit circle can be expressed

as the

 

integral of ( x^2/(1-x^2)+1)^.5 which equals i*ln((x^2-1)^.5+x)

 

from which letting x=1 and x=.5^.5 you can derive the equation

 

pi= -4*i*ln((-.5)^.5+.5^.5)

 

This equation is an alternative to Euler's equation

 

e^(i*pi)=-1

 

This pi equation can be used to check the accuracy of the current

value of pi. Maybe the prophecy for a mistake in the current value

of pi revealed by a virtually unknown prophecy will be confirmed.

[EdEarl]

There are open source arbitrary precision calculators that you may download and use for free. One for windows is at http://preccalc.sourceforge.net/

[ajb]

There are many known expressions for pi in terms of series, limits, geometric constructions and so on. Many of them look quite messy in my opinion and I am not sure how useful many of them actual are. Some of the constructions can be used to calculate pi to a large number of decimal places.

Anyway, have a look at MathWorld here for some examples.

[Semjase]

With the an exact equation for e I've derived an exact equation for pi

without e

 

pi=(-2*i)/(log,base(i),((cos(1/2)+i*sin(1/2))^(2/i)))

 

does this also mean that pi and e are both not transcendental numbers?

 

 

e=(cos(1/2) +i*sin(1/2))^(2/i)

Edited October 11, 2013 by Semjase

[ajb]

does this also mean that pi and e are both not transcendental numbers?

N0!

[imatfaal]

With the an exact equation for e I've derived an exact equation for pi

without e

 

pi=(-2*i)/(log,base(i),((cos(1/2)+i*sin(1/2))^(2/i)))

 

does this also mean that pi and e are both not transcendental numbers?

 

 

e=(cos(1/2) +i*sin(1/2))^(2/i)

 

 

That is just mindless manipulation - not a formula

 

cosx +i sinx is e^x

 

so your fomula give you e^-i which log base i will always be log(e^-i)/log(i)

 

the top is just -i and the bottom is where your pi comes in -> log(i) is ipi/2 as we know e^(ipi/2) = i

 

now it is just cancelling out and you are left with pi. no new ideas just time wasting manipulation of well understood relations and some algebra