Brad DeLong's Semi-Daily Journal (2004)

If you think that’s mysterious try Riemann surfaces and branch cuts: ways to cope with multiple valued functions. First we were told functions really had to be single valued, then came complex variables where they (sort of) didn’t.

So what are Riemann surfaces? From Eric Weisstein’s Mathworld (a great web resource):

“The Riemann surface S of the function field K is the set of nontrivial discrete valuations on K. Here, the set S corresponds to the ideals of the ring A of integers of K over . (A consists of the elements of K that are roots of monic polynomials over .) Riemann surfaces provide a geometric visualization of functions elements and their analytic continuations.”

Got that?

http://mathworld.wolfram.com/RiemannSurface.html

It's easy to see why Exp(Pi i)=-1 if you think about the operations geometrically.

i is an infinitessimal rotation in a plane, a generator of rotations.
Exp takes infinitessimal changes and makes them finite.

So if we take a vector and hit it with the exponential of iPi we are rotating it by Pi, hence pointing it in the other direction, which is equivalent to multiplying it by -1.

Well, duh.

..isn't that the eventual answer to all "why" math questions? after all, proofs at root are merely formal demonstrations of tautaulogies.

..sorry -- tautologies.

The answer to the gravity question is no, the result doesn't depend on gravity. Why would it?

Pi needs no metric. But metrics can lead to pi.

A tautology is a tautology, no? But an equation, like the one above from the field of complex numbers is not trivial. ie 1=1 is a tautology. And trivially true. And boring.
But the field of complex numbers is not (and neither is Mr. Euler). The idea is to make statements that illuminate. Douglas can teach my children math anytime.

So let's try to understand the mysteries of "e" and why this is true.

1) For any real number, t, f(t)=e^{i t} is a complex number of absolute value 1. The set of values for f(t) comprise the unit circle in the complex plane.

2) d/dt (e^{a t}) = a e^{a t}. In particular, df/dt = i f(t).

3) The area of the unit circle is pi.

1) is true if you replace e by any other real number. But 2) is rather special. It tells use that df/dt is also a complex number of absolute value 1.

Now let us compute the area of the wedge of the unit circle swept out by f between "time" t and t+dt. For small enough dt, this amounts to calculating the area of the triangle whose vertices are (0, f(t), f(t+dt)). It's easy to see that the area is just dt/2. (If you can prove this at t=0, where the vertices are (0,1,1+i dt), you can prove it for any other time by simply rotating the picture.

The *rate* at which area is swept out is

dA/dt = 1/2

The area of the full unit circle is pi. The time it takes to go all the way around is 2pi.

To go halfway around the unit circle, one sweeps out half the area, and it takes half the time.
In so doing, one goes from f(0)=1 to f(pi)=-1.

Thus: e^{i pi} = -1

You have to consider a complex number as composed of two coordinates, a real coordinate, and an imaginary one. So instead of a 2 coordinate vector with real coordinates (x,y), you write a complex number as one single number x + iy. The multiple y of i is the imaginary part, the real number x is the real part. i is called imaginary, as it is defined as the square root of -1. In real number, a square number can only be positive, so having a square root for -1 is heresy. Or imaginary.

Now it is easy to 'see' a complex number, it is just a point in a system with 2 axis.

You can define the point on the unit circles [the circle centered on the origin of the axis system that goes through (1,0),(0,1),(-1,0),(0,-1) or, in complex numbers, respectively 1,i,-1,-i] by their angular coordinate: for the point on the circle such that the angle with the vector 1 is alpha, its coordinates are (cos(alpha), sin(alpha)). That is in real numbers. Just write it as cos(alpha) + i sin(alpha) or, by definition, exp(i alpha) in complex numbers.

If we accept this, then exp(i pi) = (cos(pi),sin(pi)) = (-1,0), and exp(i pi) + 1 = (1-1,0+0)= (0,0) = 0 if you keep the convention that number without parenthesis are complex numbes.

Why should we accept that exp(i alpha) = (cos(alpha),sin(alpha))? That is because one way of defining the exponential function is "the function such that exp(a+b)=exp(a)exp(b)" (+ some conditions on differentiability).

So exp(x+iy) should be equal to exp(x)exp(iy) = (exp(x)cos y,exp(x)sin(y)). Let's define it this way. exp(0)=1, so it is consistent with our previous definition of exp(i alpha).

But exp(x+x'+i(y+y'))? Let's look at the first coordinate: exp(x+x')cos(y+y')=exp(x)exp(x')cos(y+y'). And if you recall high school trigonometry, cos(a+b)=cos(a)cos(b)-sin(a)sin(b).

Now, if we look at the real term of the product: exp(x)[cos y+i siny]exp(x')[cosy' + i siny'] you get exp(x+x')[cos y cosy' + i^2 sin y sin y']. Since i^2 =-1, QED, both are equal.

And now Exp(pi i) = -1 as sequence...

There are deeper points about this equation, which I think have to do with the mysteries of 'analytic continuation'. Question: Why does the exponential function, which is monotonically increasing (and never negative) on the real axis, have to become periodic (and periodically negative) when you extend it into the complex plane? Follow-on Question: Why and how do 'topological' properties arise from local properties (like being complex-analytic)?

"Here's my question: does Euler's fundamental fact regarding complex numbers change depending on gravity?"

Absolutely not, Bradford -- nor does the value of pi change depending on gravity. Randolph Fritz gave you the correct answer in the thread below: pi is the circumference/diameter ratio of a circle ON A FLAT SURFACE -- whereas the bending of space by gravity creates non-Euclidean geometry. The fact that the circumference/diameter ratio of a circle on a curved surface does not equal pi does absolutely nothing to change the value of pi, or the truth of the equation you listed -- and, for exactly the same reason, neither does the curvature of space by gravity. Mathematics always precedes (and in fact describes) physics, not vice versa.

In this connection, have you ever read Arthur C. Clarke's story "Superiority"? It concerns a war fought with increasingly fantastic weapons whose side effects keep blowing up in the face of the side that resorts to them -- and one of those weapons is the Exponential Field, which bends 3-D space to produce a sphere around a ship which has a finite surface area but an almost infinite radius (akin to a circle surrounding the rim of an extremely deep funnel of curved 2-D space), thus making it instantly impossible for the enemy ships surrounding that ship to hit it with their weapons. But, once again, there is one of those disastrous side effects...

I should mention \pi is known to the scatterpoints, who have no recognizable differential geometry. They discovered it in order to get the right constant: 1/\sqrt{2 \pi i} as the correct normalization of Haar measures to make the Fourier-Plancherel formula valid. They needed this in the development of Quantum Mechanics which is very relevant (as I mentioned earlier) to the scatterpoint universe. They have no need for general relativity, so their Quantum Field Theories are easy.

For some of the come-as-you-are DNA meetings at Cold Spring Harbour in the fifties anybody could wander in off the street -- but there was a challenge question at the door: prove that e to the pi-i is minus on.

My favourite proof, which I am confident would have got me in the door, is simply pivoting my forearm in front of my face through 180 degrees. This, of course, is simply an illustration of Doug Kutach's sensible comments above.

To elaborate on Bruce Moomaw's excellent explanation, Euler's formula is derived in the complex plain, which has a Euclidian geometry. In any Euclidian space, the ratio of a circle's circumference to its radius will be exactly pi. In non-Euclidian spaces (i.e. the universe we live in, especially near massive objects with large gravitational fields) the ratio of a circle's circumference to radius will not necessarily be equal to 3.14159.... If you want to define pi as that ratio, you'll be in the minority, since it's a slippery definition - better to define pi as the ratio in Euclidian space, and then describe the non-Euclianness of a space in terms of how the circumference-to-radius ratio differs from pi.

Thus, the formula is correct, because pi doesn't change depending on the space.

And in case you're having trouble visualizing three dimensional space curving through a higher dimension, here's an example that's easy to visualize:

We all know that the sum of the angles of any triangle adds up to 180 degrees (or pi, if you're working in radians - there's that number again!). That's certainly true for triangles in a plane. But imagine a triangle on the surface of a the earth with one vertex at the north pole, another at the equator longitude=0 and the third at the equator with longitude = 90. Clearly, each vertex has a 90 degree angle, making the sum greater than 180.

What's the deal? Simple: the earth is a sphere, not a plane. Geometry on the surface of a sphere does not follow the rules of Euclidian geometry, since a sphere is not a Euclidian space.

To elaborate on Bruce Moomaw's excellent explanation, Euler's formula is derived in the complex plain, which has a Euclidian geometry. In any Euclidian space, the ratio of a circle's circumference to its radius will be exactly pi. In non-Euclidian spaces (i.e. the universe we live in, especially near massive objects with large gravitational fields) the ratio of a circle's circumference to radius will not necessarily be equal to 3.14159.... If you want to define pi as that ratio, you'll be in the minority, since it's a slippery definition - better to define pi as the ratio in Euclidian space, and then describe the non-Euclianness of a space in terms of how the circumference-to-radius ratio differs from pi.

Thus, the formula is correct, because pi doesn't change depending on the space.

And in case you're having trouble visualizing three dimensional space curving through a higher dimension, here's an example that's easy to visualize:

We all know that the sum of the angles of any triangle adds up to 180 degrees (or pi, if you're working in radians - there's that number again!). That's certainly true for triangles in a plane. But imagine a triangle on the surface of a the earth with one vertex at the north pole, another at the equator longitude=0 and the third at the equator with longitude = 90. Clearly, each vertex has a 90 degree angle, making the sum greater than 180.

What's the deal? Simple: the earth is a sphere, not a plane. Geometry on the surface of a sphere does not follow the rules of Euclidian geometry, since a sphere is not a Euclidian space.

Can I mention polylogarithms now?:)
Or would you like to know the universal properties of the exponential function in terms of characteristic classes?

Take a look at
http://www.sims.berkeley.edu/~hal/exp.nb.pdf
for a little "picture proof" about why $e^{\pi i} = -1$.

Just do the Taylor series expansion of exp(i pi) and you get the answer in terms of sines and cosines. If I remember correctly, this becomes sin(-pi) + i cos(pi) = -1 + 0 i

To rephrase what some commenters have said: Pi precedes geometry, and is found in the description of cycles in time as well as space; if you use sine and cosine in your descriptions of economic cycles, where you accept that d(S)=C,d(C)=-S, then you're drawing circles in phase space...S^2+C^2=1, and the rest follows.

In a Euclidean geometry [Lorentzian space-time], you can get a Pythagorean theorem, r^2=x^2+y^2, and that gives you pi -- randomly choose real numbers x,y in [-1..+1], and pi/4 will fall in the unit circle with r^2 <= 1. If you enjoy javascript, you can play with

function piApprox(N){
var res=0;
for(var i=0;i<N;i++)
if(sqr(randP())+sqr(randP())<=1)res++;
return 4*(res/N);
}
function sqr(N){return N*N;}
function randP(){return -1+2*Math.random();}

and you should find that piApprox(10000) or so is reasonable.

In a non-euclidean geometry, that won't hold generally, but our physical geometry is described as "Locally Euclidean" [Lorentzian]; your library probably has
http://www.amazon.com/exec/obidos/tg/detail/-/0716703440/

and this is section 1.4; you'll have fun and you can describe the apple (space gets flat as you look at it closely) to your kid. You can also choose a cyclic example like a simplified Lotke-Volterra, foxes/rabbits where

dF = R (foxes increase with rabbit increase)

dR = -F (rabbits decrease with fox increase)

and now graph the F/R population cycles (or do it as a spreadsheet); you can almost-sensibly interpret actual numbers as deviations from equilibrium --- and you can read an approximate value for pi from the corresponding time-series graph, with gravity simply nowhere in the system except for holding the animals down.

Cautionary Tale: twenty years ago I too (as a young computer science asst prof) gave things like this to a mathematically-adept young son, and even made him "discover" the Pythagorean theorem by measuring right triangles and writing a page with columns for x, y, r, x^2, y^2, r^2, x/y, x+y, and so on. It worked fine, but he is now taking graduate classes in classics. (End Cautionary Tale)

...and having accepted the mysterious Euler identity, we are then faced with further mysteries such as the consequence that i^i is a real number, around 0.2.

http://www.fortune-room.com/index2.html

wow thats a surprise.......

wow thats a surprise.......