[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?
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?
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!
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.
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. 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
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.
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.
All you need to know is Euler's equation : [tex]e^{ix} = cos(x) + i sin(x)[/tex] which is visible from the Taylor expansions of [tex]e^{x}[/tex], cos(x) and sin(x), then replacing x -> ix. Now insert [tex]x = \pi[/tex] and you're done.
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...
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"..
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
Well, if you take the vector interpretation of complex numbers, it means that if you rotate through 180^{o}, you will be facing the other way.
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.
e^(ipi)=-1 -------> My professor always said it was the most significant equation out of all of mathematics.