Euler's Equation: A sign from god?

[Jin314159]

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

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

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]

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

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

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]

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

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

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 $e^z$ = $e^x(Cos(y) - i Sin(y))$ for complex z = x + iy. And yes, this definition does come about because if you use the Taylor Series expansion for $e^z$ 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 $\pi$ gives you $e^{i\pi}$ = -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]

Hm?

"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]

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

 

[selfAdjoint]

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

And why does $$e^{i\theta}$$ 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 $$m cos \theta + i sin \theta$$ 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]

"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]

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]

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.

 

[Icebreaker]

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]

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?

 

[Icebreaker]

Since, oh, about 4000 BC.

 

[thecolor11]

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

 

[Pengwuino]

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

 

[fourier jr]

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

 

[Tide]

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?

 

[Pengwuino]

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

1 + 1 = 2 existed before humans did. We didn't invent math, we discovered it.

 

[Tide]

1 + 1 = 2 existed before humans did. We didn't invent math, we discovered it.

I don't think so! Number is a concept and you cannot have concepts without mind. There may have been a "number" of things before the mind existed but the numbers require realization in order to exist.

 

[robert Ihnot]

Euler Formula: http://mathworld.wolfram.com/EulerFormula.html

Gauss is reported to have commented that if this formula was not immediately obvious, the reader would never be a first-class mathematician (Derbyshire 2004, p. 202).

I doubt most of us are in that class, but, according to Gauss, so much for mystification of the formula. (It might be added that Gauss was a hard master who cared little for his students, and did absolutely nothing to help Galois or Able.)

 

[robert Ihnot]

Tide: I don't think so! Number is a concept and you cannot have concepts without mind. There may have been a "number" of things before the mind existed but the numbers require realization in order to exist.

Does a tree falling in the forest make a sound if there is no one to hear it?

 

[Pengwuino]

Does a tree falling in the forest make a sound if there is no one to hear it?

I think the argument here is that does the forest even exist if no one is around to hear it.

 

[robert Ihnot]

Pengwuino: I think the argument here is that does the forest even exist if no one is around to hear it.

Correct, if you want to put the argument that way.

 

[Tide]

Tide: I don't think so! Number is a concept and you cannot have concepts without mind. There may have been a "number" of things before the mind existed but the numbers require realization in order to exist.
Does a tree falling in the forest make a sound if there is no one to hear it?

How, exactly, do you know a tree is falling in the forest?

 

[Icebreaker]

Saying mathematics is discovered and that it somehow trascends us only serves to add grandeur to where it's not really needed.

 

[George Jones]

Number is a concept and you cannot have concepts without mind.

This is not a fact, it is a personal philosophy of mathematics held by some. Others believe that mathematics exists independently of the human mind. There is no way to "prove" that one philosophy is more true than the other.

As I have said before, I personally am a whole-hearted enough Platonist that I like:

"... and there is no sort of agreement about the nature of mathematical reality among either mathematicians or philosophers. Some hold that it is 'mental' and that in some sense we construct it, others that it is outside and independent of us ... I believe that mathematical reality lies outside of us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations', are simply our notes of our observations."

G. H. Hardy

Regards,
George

 

[Karlsen]

Euler Formula: http://mathworld.wolfram.com/EulerFormula.html
Gauss is reported to have commented that if this formula was not immediately obvious, the reader would never be a first-class mathematician (Derbyshire 2004, p. 202).
I doubt most of us are in that class, but, according to Gauss, so much for mystification of the formula. (It might be added that Gauss was a hard master who cared little for his students, and did absolutely nothing to help Galois or Able.)

The name is Abel.

Anyway, a simple proof for e^(pi*i) + 1 = 0 can be obtained with only some basic knowledge of math.
Look at the definitions of sin(x) and cos(x):

cos(x) = (e^(i*x) + e^-(i*x))/2
sin(x) = (e^(i*x) - e^-(i*x))/(2i)

cos(x) + i*sin(x) = (e^(i*x) + e^-(i*x))/2 + i*(e^(i*x) - e^-(i*x))/(2i) = (2*e^(i*x))/2 = e^(pi*x), so we have
e^(i*x) = cos(x) + i*sin(x), if we plug in x = pi, we get
e^(i*pi) = cos(pi) + i*sin(pi) = -1 + i*0 = -1, hence
e^(i*pi) + 1 = 0

¡b00m!

 

[hypermorphism]

The name is Abel.
Anyway, a simple proof for e^(pi*i) + 1 = 0 can be obtained with only some basic knowledge of math.
Look at the definitions of sin(x) and cos(x):
cos(x) = (e^(i*x) + e^-(i*x))/2
sin(x) = (e^(i*x) - e^-(i*x))/(2i)
...

That's a pretty post-hoc proof! These expressions for sine and cosine are usually only understood by deriving them from Euler's formula or their Taylor expansions.

 

[Karlsen]

Well, yeah, but it's still how sin(x) is defined. (For complex x)

 

[Tide]

This is not a fact, it is a personal philosophy of mathematics held by some. Others believe that mathematics exists independently of the human mind. There is no way to "prove" that one philosophy is more true than the other.

It is not a fact that number is a concept?

da Vinci often said that his scultpures already existed inside his marble slabs and that all he had to do was to remove the unneeded material. He was being too modest. He had to conceptualize the creation of the form in order to make it a reality.

 

[Gokul43201]

I guess computers always existed too ! It just took someone to put the necessary pieces together...after all, there is a non-zero probability that a Thinkpad could have been created in a dust storm.

 

[inquire4more]

I guess computers always existed too ! It just took someone to put the necessary pieces together...after all, there is a non-zero probability that a Thinkpad could have been created in a dust storm.

A computer is an entity, not an idea or truth. The truth of the statement 1+1=2 holds, ireespective. One object added to another object give us two such objects. We may have given names to the number, but we did not invent number, or truth.

 

[eNathan]

What a coincidence, I was just reading up on Euler's equation, and I check out physicsforums (considering starting a new thread), and there it is already.

Math is both an invention and a discovery. We invented our mathematical system based on the consistency logic which we observe in the universe. This consistency and logic existed before humans did, but we invented our systems of math to define it, if you will. But that's just my theory =)

By the way, Jin, I think what people here have tried to express is that the reason why the constants i and pi work soo beautifully in that equation is because they are already related. They are not just totally random, un-related constants. But I was thinking the same thing too when I saw the equation. That is like saying "why does e=mc^2, the equation is too perfect" -- I already asked this question, and as it turns out, it's merely because the units of joules were designed to work perfectly like this.

 

[Tide]

A computer is an entity, not an idea or truth. The truth of the statement 1+1=2 holds, ireespective. One object added to another object give us two such objects. We may have given names to the number, but we did not invent number, or truth.

You're saying that something physical is less real than an idea or truth both of which you further imply exist without mind. Did the corner that you just boxed yourself into always exist or did we just now discover it?

 

[inquire4more]

You're saying that something physical is less real than an idea or truth both of which you further imply exist without mind. Did the corner that you just boxed yourself into always exist or did we just now discover it?

If that corner I'm in is an idea, then it was always there...:tongue2:
In all seriousness, I do not imply in any way that physical entities are any less real. I only state that some things, though not all, require a construction for their existence as such. A computer requires a construction of its form from the material parts to exist, but the concept of number, or perhaps another such thing, reuires no such construction. Three computers is exactly three computers, whether we define a "three" or not. Take that, intuitionist!

 

[matt grime]

But an assmeblage of parts into a functioning computer is a computer irrespective of whether we define that or not.

You are commiting the cardinal sin of assuming that our western ideas of mathematics are absolutely correct, the ideal state of affairs, and just waiting for us to discover them. Many peoples have managed prefectly adequately without inventing the classfication of sets according to cardinality.

There is no right or wrong interpretation on that one, however mathematics is more than just saying '3 oranges' it is disassociating the 'threeness' from the 'orangeness' of the objects that is important, the idea that somethings can be discretized and that this is useful and independent of the objects is the achievement of mathematics.
I like the Blackadder thing:

Blackadder: If I have two beans, and I add two more beans, what do I have?

Baldrick: A very small casserole.And then there is the fact that mathematics is of course far more complicated than simply the natural numbers (though they are complicated enough - you might want to ponder what the natural numbers really are, and learn about the Peano Axioms; there are statements that are true about the natural numbers that do not follow from the axiomatic description of them, I think google and wolfram can help you there).