How does [e^([pi]i)]+1=0?

[Muon12]

[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?

[mathwonk]

Everyone here will probably leap to answer this as it is such a beautiful chapter of mathematics, but i got here first, so i will try my turn. Basically one needs to understand the connection between exponential functions and trig functions.

One way is via differential equations. e.g. the equation f'' + f = 0, has 2 independent solutions. that measn that given any two numbers a and b, you can find a unique solution f such that f(0) = a and f'(0) = b. This is also true for imaginary numbers.

So let a = 1 and let b = i. Then f(x) = e^(ix) is the desired solution, since then f'(x) = ie^(ix), and e^(i0) = 1, and ie^(io) = i.

But also f(x) = cos(x) + isin(x) has f'(x) = -sin(x)+ icos(x), hence again f(0) = 1, and f'(0) = i.

But there is only one solution with these initial conditions so we are forced to conclude that e^(ix) = cos(x) + isin(x). Now set x = pi and what do you get?

[HallsofIvy]

One thing that means "in relation to the greater of mathematics" is that, from the point of view of complex numbers, exponential, sine, and cosine are all the same function!

[mathwonk]

Good point!
I may be wrong but as I recall the tangent function is also essentially equivalent to the exponential function. I.e. in the complex sphere, both are functions which wrap the sphere around itself infinitely many times, with exactly two "branching points". The two branching points are just in different places, with e^z branched around 0 and infinity while tan(z) is branched around i and -i.

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.

Maybe this is standard, but I did notice know it until fairly recently, while making up a complex analysis prelim.

[Muon12]

Wow. That's all quite a bit to absorb, but I appreciate your responces none-the-less. There are so many connections and relationships to be understood within equations like this one, that an amateur like myself can be quite easily overwhelmed. Mathwonk, your mention of "spheres" has thrown me off a bit. What kind of spheres are we discussing. Or from what branch of mathematics do your "spheres" originate. I.e. in the complex sphere, both are functions which wrap the sphere around itself infinitely many times, If you have a way of answering this question, then please go about it with a 'late high school'-'early college' level of complexity. Thanks

[Tom McCurdy]

A random note, an older computer genius friend of mine had a girlfriend with a tatoo of that equation on her ankle... I enjoyed learning about it, however I was young and this is helpful in reminding me... and bringing up good memories.

[recon]

How does [e^([pi]i)]+1=0?

Some would say God made it that way.

[PRodQuanta]

hehe, cool thread. Thanks for the info.

Paden Roder

[mathwonk]

in order to bring infinity into focus, like when function values go

'off to infinity" we often introduce on extra point, in addition to all the points of the plane, and call that extra point
'infinity. if you try to envision how the plane looks, when going out in any direction leads to that one point, it looks like a sphere. so we speak of the enlarged plane as the complex sphere. then any two points on that sphere are basically equivalent. so on the sphere, e^z treats zero and infinity the same way tan(z)treats i and minus i.

[PRodQuanta]

Just so you can see it in latex:

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

Paden Roder

[JasonRox]

I'm posting here just to remember this post. I'll come back after I learn complex numbers.

[kronecker]

e^(PI*i) == cosPI + isinPI == -1

[HallsofIvy]

If you are going to use latex, why not

e^{\pi i}+ 1= 0

[Gonzolo]

All you need to know is Euler's equation :

e^{ix} = cos(x) + i sin(x)

which is visible from the Taylor expansions of e^{x}, cos(x) and sin(x), then replacing x -> ix.

Now insert x = \pi and you're done.

[Muon12]

Well, from the algebraic perspective, Euler's equation does bring closure to this mathematical statement. Thanks for the clarification (all who responded). I suppose that from my perspective though, there is a sort of cosmic irony in gazing at an equation that looks like it should add up to so much more than nothing (zero). Who says there isn't humor in mathematics? Anyhow, just seeing an example where the sine and cosine functions, e, pi, i, and 0 are all brought together in such an amazing way, I can't help but wonder what this means in a larger sense (bringing up complex spheres and complex number planes for example). I understand how i exists with rational and irrational numbers, forming complex numbers and an entirely new number plane (connected to fractals and chaos theory principles), but trying to envision within my mind, as I usually do, the interactions between these different principles and concepts within the larger, incredibly diversified realm of mathematics, just leaves my head spinning. It's alright though, I like it that way sometimes. Trying to see the big picture isn't always the best way to look at complicated concepts in life, especially in something as detailed and precise as mathematics, but with so much room for imaginative thought given, I just can't help myself. As long as I don't head down too many dead ends... :rolleyes:

[arildno]

I think it was Hardy who at a lecture after proving this statement exclaimed:
"And here we are, gentlemen. We know it is true because we have proven it, but we cannot fathom it"..

[Epsilon Pi]

Did you know that Euler relation

i(theta)
e = cosine(theta) + i sine(theta)

already explained in this thread, and from which your expression comes from, is the only mathematical equation that remains with the same form with those mathematical operations that represent change, i.e., with differentiation and integration? In fact, in electrical engineering, it is used to convert its differential equations into algebraic equations reducing in this way complexity into minus one. Since a long time ago I asked myself if this equation contains both a duality as that one of wave-particle and this isomorphic property, that make it an ideal unification mathematical tool, why it was not used to express the fundamental equations of physics? Is not this a way to apply it beyond number theory?

Regards

EP

[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?

[selfAdjoint]

I think it was Hardy who at a lecture after proving this statement exclaimed:
"And here we are, gentlemen. We know it is true because we have proven it, but we cannot fathom it"..

Well, if you take the vector interpretation of complex numbers, it means that if you rotate through 180o, you will be facing the other way.

[mathwonk]

That's a good point. Why is this equation considered so mysterious? There is nothing strange about the relation e^(z+w) = e^z e^w, (at least for students whose education is not confined to calculator math). Moreover we accept that for complex numbers, multiplication involves rotation. If you combine those two facts, the relation we are puzzled by is a corollary. Perhaps it all goes back to our practiced amazement that i^2 = -1.

This suggests we should teach complex numbers much earlier in school than we often do. I know I appreciated it greatly when my high school teacher presented them as ordered pairs of reals with a new multiplication. The mystery of "imaginary" numbers disappeared. The wonderful book we used was "Principles of Mathematics" by Allendoerfer and Oakley, and it contained logic, propositional calculus, theory of countability and uncountability, and definitions of groups, rings, fields. This was a terrific introduction to things all math students should know as early as possible. I do not know of a single high school today (maybe Andover, Exeter?, Bronx high shool of science?) where such material is now taught. Does anyone?

Maybe it is time for another revolution in high school teaching of math. There is too much domination of the curriculum by proponents of higher SAT scores, and AP nonsense. All my honors calc students have had AP calc, and only a few of them know what a proof is, or a quantified statement.

My advice to high schoolers out there is, even if you take AP calc, go to the best college you can, and when you get there, do not skip calc, but take the honors version from the beginning. At any decent college it will be very different from your AP class. I.e. "advanced placement" should mean entry to an honors level version of the material, not skipping the college course entirely. High school courses that really replace college courses are extremely uncommon. College courses are often taught by researchers, high school courses almost never. But choose your college course carefully, as not all college courses are the same.

[gravenewworld]

e^(ipi)=-1 -------> My professor always said it was the most significant equation out of all of mathematics.

[mathwonk]

did you ask him why?

[Epsilon Pi]

That's a good point. Why is this equation considered so mysterious?... Moreover we accept that for complex numbers, multiplication involves rotation.

Not only rotation but most importantly a sum when expressed in polar form, and a difference for division what makes it possible to convert differential equations in algebraic equations as is done in EE.


This suggests we should teach complex numbers much earlier in school than we often do. I know I appreciated it greatly when my high school teacher presented them as ordered pairs of reals with a new multiplication. The mystery of "imaginary" numbers disappeared...


Yes, it's quite important to make the "imaginary" connotation to disappear as the symbol i, is just a symbol for introducing in physics and engineering a minimum structure, a notion that does not appear in conventional physics; a notion that even ancient taoists expressed by their ying and yan principle and that is not quite understood by a dualistic mind.


Maybe it is time for another revolution in high school teaching of math...


Yes, agree, and when that revolution is done I am quite sure there will be a great revolution in modern physics too, but for it the need of a minimum structure must be accepted first: a minimum structure for differentiating time and space.

Regards
EP

[arildno]

As to those who dismiss the "mystery" of the equation:
1.In one sense, this is very important to do, since there is no "special logic" which is needed to understand it; as self-adjoint among others has said, as long as we choose the proper perspective, it is not hard to understand why it is true.
2. Unfortunately, I haven't been able to dig up Hardy's original comment, but effectively it expressed a sense of wonder at how the equation weaves together the 5 most important number in maths, along with the operations of sum, multiplication and exponentiation.
Hence, while we should not regard Euler's equation as some piece of esoteric wisdom, impenetrable to ordinary logic, I think we (at least myself) are fully justified in expressing wonder at the sheer beauty and elegance of the equation.

[Gonzolo]

In a sense, it not complete. Generally speaking, we should write :

e^{i \pi (1 + 2n)}+ 1= 0, n being an integer.

[Tom Mattson]

I think the coolest thing about the Euler relation eix=cos(x)+i sin(x) is its use in Laplace transforms. Not only is the exponential easier to integrate than either sine or cosine, but you also get 2 Laplace transforms for the price of one.

[Epsilon Pi]

Yes, good point, as it is a rotating vector in the complex plane
Regards
EP
In a sense, it not complete. Generally speaking, we should write :

e^{i \pi (1 + 2n)}+ 1= 0, n being an integer.

[Tom McCurdy]

most important equation in mathmatitcs
1=1

sorry bit of a random post-- but it seems like the most important equation would be one that defines that numbers are unique and exact values...

[Tom McCurdy]

I of course also have heard that e^{i\pi}+1=0 is the most important equation in mathmatics... of course it seems like that owuld be up for debate by many... What are some other important equations you guys know about.

[Epsilon Pi]

What about uncertainty in the reality "out there"? Is there any chance to represent it in your case?
Regards
EP
most important equation in mathmatitcs
1=1

sorry bit of a random post-- but it seems like the most important equation would be one that defines that numbers are unique and exact values...

[matt grime]

I of course also have heard that e^{i\pi}+1=0 is the most important equation in mathmatics... of course it seems like that owuld be up for debate by many... What are some other important equations you guys know about.

The meaning of 'important' that might be used to describe that equation isn't the one I infer you to have.

[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) :biggrin:

[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]

Maybe this is getting off on a tangent (horrible pun intended :tongue2:), 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.

http://www.physicsforums.com/showthread.php?t=174527&highlight=euler%27s+identity

[HallsofIvy]

Maybe this is getting off on a tangent (horrible pun intended :tongue2:), 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.

http://www.physicsforums.com/showthread.php?t=174527&highlight=euler%27s+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 in the world did you find this thread? All previous posts were from three years ago!

Why, with a Google search (http://www.google.com/search?hl=en&safe=active&q=%2Btan+e+pi+equation+theory) of course! (or is that Googol (http://graphics.stanford.edu/~dk/google_name_origin.html)? :confused:) 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 Fermat's last theorem (http://en.wikipedia.org/wiki/Fermat's_last_theorem) from 1637--only 357 years later, but he didn't use the internets (http://en.wikipedia.org/wiki/Internets_(colloquialism)) [sic] (http://en.wikipedia.org/wiki/Sic) :tongue2:.

[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]

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 mobius strip (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.