"Most remarquable formula of Mathematics" (R. Feynman)

[The french tourist]

Richard Feynman, one of the pioneer of quantum electrodynamics, said the following formula was "the most remarquable of Mathematics".

It is a simple formula, anyone having studied the rudiments of complex numbers will find it obvious. Personally, I find it beautiful because all the symbols of mathematics I love the most (for now, i'm a beginner ) are gathered together : the transcendent [math]\pi[/math] and napier constant, [math]i[/math] : square root of the polynomial [math]x^2+1=0[/math] in C, the equal sign and the two neutral elements : 0 and 1 for the binary operators addition and multiplication.

 

[math]e^{i\times\pi}+1=0[/math]

 

What do you think about it ?

[imatfaal]

for some reason I would show it with the one shifted to the other side [math] e^{i\pi}=-1[/math]. I know a group of physicists who play football with that as a slogan on their kit. And frankly I prefer euler's formula - the general form [math]e^{ix}=cosx +isinx[/math]

[D H]

For some reason I would show it with the one shifted to the other side [math] e^{i\pi}=-1[/math].

Expressing this as [math]e^{i\pi}+1=0[/math] connects the two identity elements (0 and 1), the two most important transcendental numbers \pi and e), and [math]i[/math] all in one fell swoop. That is why many refer to this as the most important identity in all of mathematics.

 

 

the general form [math]e^{ix}=cosx +isinx[/math]

I would write that as [math]e^{ix}=\cos x + i\sin x[/math] Note the difference in appearance. This is apparently the target of the Feynman quote rather than Euler's identity.

 

 

Feynman is far from the only one, and far from the first, who has been enamored with Euler's identity ([math]e^{i\pi}+1=0[/math]) or Euler's formula ([math]e^{ix}=\cos x + i\sin x[/math]).

Edited February 1, 2012 by D H

[imatfaal]

Expressing this as [math]e^{i\pi}+1=0[/math] connects the two identity elements (0 and 1), the two most important transcendental numbers \pi and e), and [math]i[/math] all in one fell swoop. That is why many refer to this as the most important identity in all of mathematics.

 

 

I would write that as [math]e^{ix}=\cos x + i\sin x[/math] Note the difference in appearance. This is apparently the target of the Feynman quote rather than Euler's identity.

 

Feynman is far from the only one, and far from the first, who has been enamored with Euler's identity ([math]e^{i\pi}+1=0[/math]) or Euler's formula ([math]e^{ix}=\cos x + i\sin x[/math]).

Thanks DH that is a nice latex protip - I had experimented with trying spaces and bracket to make the difference between the variable and the sine/cosine and never been happy with the result.

[DrRocket]

for some reason I would show it with the one shifted to the other side [math] e^{i\pi}=-1[/math]. I know a group of physicists who play football with that as a slogan on their kit. And frankly I prefer euler's formula - the general form [math]e^{ix}=cosx +i \ sinx[/math]

 

The exponential function itself is, in the view of many, the most important function in mathematics.

 

The equation [math] e^{i\pi}=-1[/math] is rather remarkable, and is in fact the basis for a purely analytic definition of [math]\pi[/math]. I find that the simple fact that it includes the numbers [math] e, \ \pi , \ 0, \ 1[/math] to be not particularly important -- rather like a physicists version of numerology -- particularly given that is not known if either [math] e + \pi[/math] or [math]e \pi[/math] are rational (thought they cannot both be rational). However the relationship itself is profound. The equation [math]e^{ix}=cosx +i \ sinx[/math] is in fact the modern definition of the sine and cosine funtions.

 

What is done in a rigorous treatment is to start with the general formula [math] e ^z = \sum_{n=0}^\infty \frac {z^n}{n!}[/math] where [math]z[/math] is a complex variable and show that it is periodic. That period is defined to be [math] 2 i \pi[/math] and the sine and cosine functions are defined as above. This serves as a starting point for the study of analysis of one complex variable. (You can see a rather slick exposition of this approach in the first few pages of Rudin's Real and Complex Analysis.)

I would hope that the football team does not consist of a bunch of quantum physicists -- with the HUP they would never be quite sure where the ball was.

 

Another formula due to Euler seems to me to be equally remarkable. That is

 

V-E +F = 2 - 2g

 

which applies to trianulations of surfaces, where V is the number of vertices, E is the number of edges, F is the number of faces and g is the "genus" or number of holes in the surface. The expression on the left-hand side is called the Euler characteristic, and it completely classifies 2-dimensional manifolds up to homeomorphism (topological equivalence). Note that for an ordinary tetrahedrom (a toplogical sphere) that V-E+F = 4 - 6 +4 = 2 so that g=0 (no holes). Any 2-manifold with no holes (simply connected) is topologically equivalent to the 2-sphere. This is in fact one of the motivating thoughts behind the Poincare conjecture in dimension 3, the proof of which remained open for nearly a century until finally solved by Gregori Perelman using extremely powerful methods from analysis and geometry.

[D H]

(You can see a rather slick exposition of this approach in the first few pages of Rudin's Real and Complex Analysis.)

The very first sentence of which is "This is the most important function in mathematics". The "this" to which Rudin is referring is the exponential function.

[DrRocket]

The very first sentence of which is "This is the most important function in mathematics". The "this" to which Rudin is referring is the exponential function.

 

Which is what I said in the first sentence of my post. No disagreement here.

 

I, and I think most mathematicians, distinguish between a function, like the exponential function, which, colloquially, is a machine that takes in an argument and spits out a value, and a formula, which is a simple expression like the formula that defines the Euler characteristic which classifies two-manifolds.

 

Side note: Walter Rudin was an analyst. His feelings toward what is most important in mathematics reflect the natural bias of an analyst. I too am an analyst and share many of those same biases. However, mathematicians with other specialties (his wife Mary Ellen for instance) might find other functions to be more important to them.

Edited February 2, 2012 by DrRocket

[John Cuthber]

I think the spelling in the title is remarkable.

except, if it were, then it wouldn't be.

Oh damn! another paradox.

[DrRocket]

I think the spelling in the title is remarkable.

except, if it were, then it wouldn't be.

Oh damn! another paradox.

 

No mystery.

 

The OP is The french tourist.

[John Cuthber]

I know that, but I can't resist a joke about paradoxes (apart from the obvious one)

.

[The french tourist]

No mystery.

 

The OP is The french tourist.

 

Yes I'm sorry, I need to practice a bit more my English ! It's not easy to focus on both usual vocabulary and scientist vocabulary !

As for Euler's formulas, I also love them, especially [math]\cos(x)=\frac{e^{i\pi}+e^{-i\pi}}{2}[/math]

[DrRocket]

Yes I'm sorry, I need to practice a bit more my English ! It's not easy to focus on both usual vocabulary and scientist vocabulary !

As for Euler's formulas, I also love them, especially [math]\cos(x)=\frac{e^{i\pi}+e^{-i\pi}}{2}[/math]

 

Not a problem. Most mathematicians can read French. There is quite a bit of literature that requires it.

[The french tourist]

Not a problem. Most mathematicians can read French. There is quite a bit of literature that requires it.

 

By the way, I don't know where I was when I wrote down this but [math] \cos(x)=\frac{e^{ix}+e^{-ix}}{2} [/math] would be quite better. This is where it leads you to want to see [math]\pi[/math] anywhere...

Concerning scientist vocabulary, it is exotic between our two languages. For instance, literally translated from french to english, the field of complex numbers would be "the body of complex numbers" and the squeeze theorem would be "the policemen theorem".