What is the significance of [e^([pi]i)]+1=0 in mathematics?

[Muon12]

I know that the proof for "there are no solutions to the equation x^(n)+y^(n)=z^(n) when x,y,z and n are non-zero integers and where n is greater than 2" is of definite importance, since the typed out proof (which in full is over 100 pages) unifies modular forms with elliptic equations. I read about it in a book entitled Fermat's Last Theorem. In the greater scope of mathematics, this now proven theorem takes on an immense weight of conjectures and important logical arguements that some say has "revolutionized number theory". I would say they're right, based on my understanding of its significance. Some of the proof (the first four pages?) can be found at- http://www.ams.org/notices/199507/faltings.pdf -I can't even grasp the first page of mathematics. But if any of you can follow it through those first pages, I would suggest looking at the full version. It would definitely take a while to read through of course. Plus, comprehending all of the material...

 

[mathwonk]

I do not think there are any really super important equations. Ideas are more important than formulas.

maybe the riemann roch formula is important, but how you understand it is more important, and how you use it.

the formula by the way is dimL(D) = 1-g + deg(D) + dim(K-D), I think [I.e. no matter how important a formula is, people still have trouble remembering it], where D is a divisor on a riemann surface (algebraic curve), and deg(D) is the number of points in the divisor, L(D) is the space of meromorphic functions with pole divisor dominated by D, and g is the topological genus of the surface.

More important is the meaning of the formula. E.g. when you write it this way:

dimL(D) - dim(K-D) = deg(D) + 1-g, you see that on the left side we have a number that depends on the analytic structure of the riemann surface, while on the other side we have anumber that only depends on the topology.

Now that is an idea. I.e. certain combinations of analytic invariants are actually topological invariants. This leads one to the realization of how to generalize this formula, as hirzebruch did.

here for example is the generalization to algebraic surfaces: on the left again we have the alternating sum of the dimensions of certain spaces associated to the divisor, and on the right we have some topological invariants:

dimL(D)-dimH^1(D) + dim(K-D) = (1/2)[D.(D-K)] + (1/12)(K^2 + chi(top))

again i am not sure I have the formulas on the right correct, but who cares. I can check it on an example when I need to. I.e. formulas are not important, what they mean is important.

In the same way, I think e^(ipi) +1 = 0, has no importance at all, beauty maybe but not importance, but the formula that gives it meaning: e^(it) = cos(t)+isin(t), that has some importance.

 

[Hyperreality]

Sorry for interrupting, but I just can't help it when i saw the title.

e^ix = cos x + i sinx

If you you plot it, it's a circle which can have a parametric equation

e^i(theta) = cos (theta)

so, when theta=pi, cos pi = -1, therefore

e^(i*(pi)) + 1 = 0.

I'm surpsied nobody mentioned that... it's quite simple (I wonder if there other ways of proving it, I found out this during a boring calculus period)

 

[SGT]

[e^([pi]i)]+1=0 :I had a friend with a T-shirt displaying this deceptively simple equation. I know it to be true, but I have no real understanding of the relationship between these three apparently unrelated, irrational numbers (e,pi and i) within this equation. While I know how significant this discovery is in that it relates and brings a sort of whimsical unification between e, pi, and i, I fail to understand the true nature of this statement, or how it exists for that matter. Could anyone here explain, in somewhat detailed terms, how to create this proof/equation based on prior knowledge of the numbers it involves, but not based on the knowledge that it is infact true. Plus, what does this equation mean in relation to the greater body of mathematics? How can it (if at all) be applied beyond number theory?

It seems nobody has really given a proof based on prior knowledge of mathematics.
I suppose you are familiar with the Taylor series, that represents any function by a possibly infinite sum of powers of the independent variable.
The Taylor series for e^z is:
e^z = 1 + z + z^2/2! + z^3/3! + ...
If we let z = ix, the series becomes:
e^ix = 1 + ix - x^2/2! - ix^3/3! + ...
Separating the real and imaginary parts of the series we get:
e^ix = 1 - x^2/2! + x^4/4! -+ ... + i(x -x^3/3! + x^5/5! -+ ...)
The real part is the series for cos x and the module of the imaginary part is the series for sin x, so we get Euler's formula:
e^ix = cos x + i sin x
If we make x = π
e^iπ = cos π + i sin π = -1 + 0i = -1
so
e^iπ+ 1 = 0

 

[Muon12]

Oh, okay. That helps make some more sense of it. Thanks.

 

[mathwonk]

Are some posters unaware of the previous posts? The same comments and proofs are occurring three or four times, as if they had not already been presented. indeed in the very first answer to this question i both gave the equation e^ix = cos x + i sin x, and proved it, using uniqueness of solutions of differential equations. the second answer or so gave the taylor series explanation. and yet it is all cycling over again like e^z. As i predicted, people like answering this question, apparently much more than reading previous answers.

If something new is forthcoming, besides the taylor series or diff eq answer, I would be interested. perhaps a path integral. since e^z is inverse to the path integral of 1/z, i guess we could ask why the path integral if 1/z from 1 to -1, equals i <pi>. but that integral has an exact real part, and an imaginary part equivalent to dtheta, so one does get arg(-1) = i<pi> + 2n<pi>.

i admit that one is not so original either. any more?

 

[Gonzolo]

Perhaps we could compare with \pi^{ie}...

Is i ever used in the power of other numbers than e? Any use to to doing this?

 

[mathwonk]

Look. pi^(ie) is just e^(ie ln(pi)), so NO number is ever used as an exponent for bases other than e. That is to say, e is the universal base for all exponents.

i.e. [(haha) perhaps I should say in russian: "tau yest" instead of "id est"]\\anyway: for any a, we have a^b = e^(bln(a)).

Let's start with e^i. Note that since e^(it) = cos(t) + isin(t), that then e^i =
cos(1)+i sin(1), not at all an interesting or elementary number at least not to me.

Thus in the same spirit, pi^(ie) = e^(ie ln(pi)) = cos(e ln(pi)) + i sin(e ln(pi)), apparently a similarly ugly number. If anyone can give a nice interpretation of this number, my hat is off, and I would enjoy seeing it.

 

[arildno]

Thus in the same spirit, pi^(ie) = e^(ie ln(pi)) = cos(e ln(pi)) + i sin(e ln(pi)), apparently a similarly ugly number. If anyone can give a nice interpretation of this number, my hat is off, and I would enjoy seeing it.

Would you still consider it an UGLY number?

 

[polack]

tan(x) vs. e^(x) - pi^(-x)

Maybe this is getting off on a tangent (horrible pun intended ), but I'm exploring how similar the graphs of tan(x) and e^{x}-\pi^{-x} are... this may weave back to the earlier observations about path integrals from mathwonk:

To see this, just notice that e^z is the inverse of ln(z), which is the path integral of 1/z
which means the value varies according to how the path winds around 0 and infinity.
On the other hand tan(z) is the inverse of arctan(z) = the path integral of 1/(1+z^2),
which is determined by how many times the path winds around i and -i. I.e.
1/((1+z^2) is actually continuous at infinity and single valued there, so the two
functions (if I got this right) seem to differ only by a mobius transformation which
interchanges the pair 0 and infinity, for i and -i.

...so I'm curious if this goes anywhere new.

 

[waht]

Are some posters unaware of the previous posts? The same comments and proofs are occurring three or four times, as if they had not already been presented. indeed in the very first answer to this question i both gave the equation e^ix = cos x + i sin x, and proved it, using uniqueness of solutions of differential equations. the second answer or so gave the taylor series explanation. and yet it is all cycling over again like e^z. As i predicted, people like answering this question, apparently much more than reading previous answers.

If something new is forthcoming, besides the taylor series or diff eq answer, I would be interested. perhaps a path integral. since e^z is inverse to the path integral of 1/z, i guess we could ask why the path integral if 1/z from 1 to -1, equals i <pi>. but that integral has an exact real part, and an imaginary part equivalent to dtheta, so one does get arg(-1) = i<pi> + 2n<pi>.

i admit that one is not so original either. any more?

I actually attempted to prove Euler's identity using a different way. This was discussed recently.

https://www.physicsforums.com/showthread.php?t=174527&highlight=euler's+identity

 

[HallsofIvy]

Maybe this is getting off on a tangent (horrible pun intended ), but I'm exploring how similar the graphs of tan(x) and e^{x}-\pi^{-x} are... this may weave back to the earlier observations about path integrals from mathwonk:



...so I'm curious if this goes anywhere new.

I actually attempted to prove Euler's identity using a different way. This was discussed recently.

https://www.physicsforums.com/showthread.php?t=174527&highlight=euler's+identity

How in the world did you find this thread? All previous posts were from three years ago!

 

[matheinste]

Hello all.

Just a bit of trivia but in someway I feel descriptive of the power, beauty and simplicity of the formula. I once saw it referred to as A Mathematical Poem.

Matheinste.

 

[K.J.Healey]

I alwyas thought that this equation was much more beautiful and mysterious:

i^2+j^2+k^2=i j k = -1

 

[rbj]

Just so you can see it in latex:

e^{(pi)i}+1=0

Paden Roder

wouldn't it be

e^{i \pi} + 1 = 0

?

 

[matheinste]

I alwyas thought that this equation was much more beautiful and mysterious:

i^2+j^2+k^2=i j k = -1

Thanks for your reply Healey01.

Your equation is certainly mysterious. When I find out what it means it might also be beautiful.

Mateinste

 

[ZioX]

It's just the defining equation of quaternions.

The algebra of it is a lot more interesting...

 

[polack]

How I found this three-year-old thread...

How in the world did you find this thread? All previous posts were from three years ago!

Why, with a Google http://www.google.com/search?hl=en&safe=active&q=+tan+e+pi+equation+theory"? ) I was curious if someone had already invented the wheel I was working on... so I searched for it.

This sort of rediscovery isn't too unusual... there's Wile's 1994 rediscovery of "[URL last theorem[/URL] from 1637--only 357 years later, but he didn't use the http://en.wikipedia.org/wiki/Internets_(colloquialism)" .

 

[mathwonk]

exponentiation is a homomorphism from a ddition to multiplication, and surjects onto the positive reals for sure, and neither the value nor the derivative is ever zero. hence i claim it is a topological covering map from the complexes to the non zero complkexes, hence must wrap around the origin, and be periodic for some value. i.e. it must take the values 1 and -1 infinitely often.

it remains only to find the value x such that e^x = -1. that follows from trig (eulers style via power series expansions of sin, cos).

i think this is a new answer to an old question.

 

[polack]

It keeps going and going...

exponentiation is a homomorphism from a ddition to multiplication, and surjects onto the positive reals for sure, and neither the value nor the derivative is ever zero. hence i claim it is a topological covering map from the complexes to the non zero complkexes, hence must wrap around the origin, and be periodic for some value. i.e. it must take the values 1 and -1 infinitely often.

it remains only to find the value x such that e^x = -1. that follows from trig (eulers style via power series expansions of sin, cos).

i think this is a new answer to an old question.

So it wraps like a http://www.sciam.com/article.cfm?chanID=sa003&articleID=D55BA3C5-E7F2-99DF-3B0A5A55D41B63FB&ref=rss" then? No edges or ends... periodic, as you say. I feel the flow; circular. So \pi is to circle/sphere as e is to exponentiation? I'm still working on the graph of tan(x) compared to e^{x} - \pi^{-x}... the "-" implies an i in there somewhere.

 

[mathwonk]

im just saying there is essentially no other way to map the plane onto the punctured plane with derivative non zero, except to go around and around, so it has to hit the same point more than once, i.e. it hits 1 in finitely often, and also -1.

i see it like a spiral staircase.

 

[widewombat]

Beautiful and useful too, phasors for AC circuit analysis are based on Euler's identity.