Euler's Equation: A sign from god?

Mar27-04, 12:13 AM

The first time I saw Euler's equation, it blew my mind.

$LaTeX Code: e^{i\\pi}+1 \\equal 0$

Here, we have three of the most important numbers in math, all related to each other in such a remarkably compact equation. Does anyone know what this means? I think you can prove this through Taylor Series, but that's not what I'm asking for. Is there some underlying, intuitive reason for why Euler's equation is true? Or is it just a big fat coincidence (hence sign from god)?

Edit: Why is there a stupid dash over my zero? This is why I hate Latex. Stupid, unexplained stuff always happens.

**ahrkron** · Mar27-04, 01:00 AM

Instead of "\equal", just use "="

$LaTeX Code: e^{i\\pi}+1 = 0$

I don't see any divine influence on the equation, but it is definitely a wonderful one. I think of it as a brief, yet elegant, summary of some of our most powerful and beautiful math (algebra, complex numbers and calculus).

In terms of the underlying reason, the proof is basically the reason you are looking for. Think of exponentiation in terms of the Taylor series (i.e., picture ("e^x" as a short notation for the series), and of complex numbers in terms of their intimate relation with rotations, which naturally bring pi to the arena. Instead of an unexplained coincidence, I see it as an inevitable result of the structure of math, lying in an unavoidable intersection of various branches of it.

**Bob3141592** · Mar27-04, 01:11 AM

Originally Posted by Jin314159

The first time I saw Euler's equation, it blew my mind.

$LaTeX Code: e^{i\\pi}+1 \\equal 0$

Here, we have three of the most important numbers in math, all related to each other in such a remarkably compact equation. Does anyone know what this means? I think you can prove this through Taylor Series, but that's not what I'm asking for. Is there some underlying, intuitive reason for why Euler's equation is true? Or is it just a big fat coincidence (hence sign from god)?

Edit: Why is there a stupid dash over my zero? This is why I hate Latex. Stupid, unexplained stuff always happens.

It's from the definition of

=

for complex z = x + iy. And yes, this definition does come about because if you use the Taylor Series expansion for

you can separate the resultant series into two which represent the trigonometric functions. However, such rearrangement of infinite series with changing signs is very dangerous, as this can change the resultant sum (as in conditionally convergent series). But Euler often played fast and loose with such unrigorous methods, and got away with it! In any case, having done that, setting x to 0 and y to $LaTeX Code: \\pi$ gives you $LaTeX Code: e^{i\\pi}$ = -1 which is a most ugly form. I much prefer it when the identity elements of addition and multiplication show in the equation.

But I'm not so sure it proves there's a God. Perhaps it proves that Euler was in league with the devil! :)

**philosophking** · Mar28-04, 03:39 PM

"Is there some underlying, intuitive reason for why Euler's equation is true? Or is it just a big fat coincidence (hence sign from god)?"

Since when did coincidences imply a sign from God?

**HallsofIvy** · Mar29-04, 07:47 AM

Are you saying that you consider any "coincidence" a sign from God? You are truly blessed!

selfAdjoint · Mar29-04, 10:37 AM

According to the vector interpretation of complex numbers $LaTeX Code: e^{i\\pi} = -1$ just says that if you turn through 180^o you will be facing the other way. Another way to look at it is that $LaTeX Code: e^{i\\theta}$ generates a circle in the complex plane, and of course $LaTeX Code: \\pi$ has its well-known relationship to a circle.

And why does $LaTeX Code: e^{i\\theta}$ generate a circle? Basically because if you multiply two complex numbers with magnitude and argument, the magnitude of the product will be the product of the magnitudes and the argument of the product will be the sum of th arguments, so the argument, or angle of a complex number behaves like a natural logarithm, and e is the basis of the natural logarithms.

And why does the product of complex numbers work that way? It comes out of the distributive law of multiplication and the representation of the complex numbers in components as $LaTeX Code: m cos \\theta + i sin \\theta$ which brings us back to Euler's series.

It hangs together because the plane has this rotation property. Whether you take that as a mystical sign or not is of course up to you.

**newton1** · Mar29-04, 11:30 PM

Originally Posted by philosophking

"Is there some underlying, intuitive reason for why Euler's equation is true? Or is it just a big fat coincidence (hence sign from god)?"

Since when did coincidences imply a sign from God?

you can use taylor expansion to proof this equation

**Janitor** · Mar29-04, 11:50 PM

John Baez has a lot of interesting stuff at his website. Complex numbers, and their "higher dimensional" analog, the quaternions, are pet topics of his. I took the liberty of pasting this from there:

There are very few dimensions in which the unit sphere is also a group. It happens only in dimensions 1, 2, and 4! In 1 dimensions the unit sphere is just two points, which we can think of as the unit real numbers, -1 and 1. In 2 dimensions we can think of the unit sphere as the unit complex numbers, exp(i theta). In 4 dimensions we can think of the unit sphere as the unit quaternions.

Only in these dimensions do we get polytopes that are also groups in a natural way. In 2 dimensions all the regular n-gons correspond to groups consisting of the unit complex numbers exp(2 pi i / n). In 4 dimensions things are more subtle and interesting. It's especially interesting because the group of unit quaternions, also known as SU(2), happens to be the `double cover' of the rotation group in 3 dimensions. Roughly speaking, this means that there is a nice function sending 2 elements of SU(2) to each rotation in 3 dimensions.

**ichiro_w** · Oct22-05, 06:10 PM

Richard Feynmann, in his lectures at Caltech (available in book form), referred to the underlying relation that produced the referenced equation as the "crown jewel of algebra", I think.

Oct22-05, 08:43 PM

Originally Posted by Jin314159

Or is it just a big fat coincidence (hence sign from god)?

Perhaps you should be a little less humble; it is an achievement of human intellect: we did it.

**Pengwuino** · Oct22-05, 08:58 PM

Originally Posted by Icebreaker

Perhaps you should be a little less humble; it is an achievement of human intellect: we did it.

since when did humans invent math? Did math not exist before us?

I'm still rather (extremely) confused as to how this relates to God. What's so important about this equation?

Oct22-05, 09:44 PM

Since, oh, about 4000 BC.

**thecolor11** · Oct23-05, 01:03 AM

Icebreaker, math existed before we discovered it, and it will exist long after we're gone.

**Pengwuino** · Oct23-05, 01:08 AM

Unless of course, you follow that "If I believe it's true, then it's true" line of thinking.

**fourier jr** · Oct23-05, 01:31 AM

i think euler himself thought that his equation was so perfect that it proved the existence of God.

**Tide** · Oct23-05, 01:51 AM

Originally Posted by thecolor11

Icebreaker, math existed before we discovered it, and it will exist long after we're gone.

Unless you are assuming there are/were other intelligent species in the Universe, how could that possibly be?