Funny math cartoon

[aeest400]

cartoon (http://xkcd.com/c179.html)

Found this pretty funny. Can anyone explain in semi "common sense" terms why the second relationship holds. I'm sure I can find a proof, but I'd like some shorthand explanation (or some simpple algrbra) showing why e^pi(sqrt-1)=-1

[surftheiop]

http://en.wikipedia.org/wiki/Euler's_identity

[LuckOfTheDraw]

That's actually pretty funny.

However, I'm a little confused. I never really studied much of imaginary numbers, a little in 10th grade, but that's it. Anyways, while I know i = sqrt(-1) by definition, isn't i in the identity e^(pi*i) + 1 = 0 more like a unit, simply defining the direction on the cartesian plane, rather than actually saying that e^(pi*sqrt(-1)) + 1 = 0? Sorry if I'm being a retard here and they simply mean the same thing and my puny intellect is unable to comprehend it.

[gull]

e^(i*x) = cos(x) + sin(x)

So when x = pi, e^(i*x) = 1.

This wikipedia entry shows a number of different proofs. The taylor series is the one that I've seen before.
http://en.wikipedia.org/wiki/Euler%27s_formula#Proofs

One way to think of e^(x*i) is by using the complex plane.
http://en.wikipedia.org/wiki/Complex_plane

[PairTheBoard]

I've always felt a lot like this quote from the first Wikipedia link:

[ QUOTE ]
After proving the identity in a lecture, Benjamin Peirce, a noted nineteenth century mathematician and Harvard professor, said, "It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth."

[/ QUOTE ]

I understand the proofs. I understand that if you want to define e^ix in such a way that the Taylor series for e^x still holds then Euler's Formula must hold. I understand that if you want to extend the function e^x on the Reals to the Complex numbers in a reasonable way, then Euler's Formula must hold. I understand that if you want to define e^ix in such a way that it differentiates similiarly to e^x then Euler's Formula must hold. But I've never been able to make sense of e^ix as some kind of exponentiation.

PairTheBoard

[ApeAttack]

[ QUOTE ]
e^(i*x) = cos(x) + sin(x)

So when x = pi, e^(i*x) = 1.

This wikipedia entry shows a number of different proofs. The taylor series is the one that I've seen before.
http://en.wikipedia.org/wiki/Euler%27s_formula#Proofs

One way to think of e^(x*i) is by using the complex plane.
http://en.wikipedia.org/wiki/Complex_plane

[/ QUOTE ]

The Taylor Series is how I learned it as well.

[holmansf]

[ QUOTE ]
I've always felt a lot like this quote from the first Wikipedia link:


Quote:
--------------------------------------------------------------------------------

After proving the identity in a lecture, Benjamin Peirce, a noted nineteenth century mathematician and Harvard professor, said, "It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth."


--------------------------------------------------------------------------------



I understand the proofs. I understand that if you want to define e^ix in such a way that the Taylor series for e^x still holds then Euler's Formula must hold. I understand that if you want to extend the function e^x on the Reals to the Complex numbers in a reasonable way, then Euler's Formula must hold. I understand that if you want to define e^ix in such a way that it differentiates similiarly to e^x then Euler's Formula must hold. But I've never been able to make sense of e^ix as some kind of exponentiation.

PairTheBoard

[/ QUOTE ]

Actually I think Euler's identity massively clarifies the nature of complex numbers. It makes it clear that multiplication by a complex number is a dilation and rotation of the plane. Dilation by the modulus of the complex number and rotation by the argument.

[flipdeadshot22]

[ QUOTE ]
[ QUOTE ]
I've always felt a lot like this quote from the first Wikipedia link:


Quote:
--------------------------------------------------------------------------------

After proving the identity in a lecture, Benjamin Peirce, a noted nineteenth century mathematician and Harvard professor, said, "It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth."


--------------------------------------------------------------------------------



I understand the proofs. I understand that if you want to define e^ix in such a way that the Taylor series for e^x still holds then Euler's Formula must hold. I understand that if you want to extend the function e^x on the Reals to the Complex numbers in a reasonable way, then Euler's Formula must hold. I understand that if you want to define e^ix in such a way that it differentiates similiarly to e^x then Euler's Formula must hold. But I've never been able to make sense of e^ix as some kind of exponentiation.

PairTheBoard

[/ QUOTE ]

Actually I think Euler's identity massively clarifies the nature of complex numbers. It makes it clear that multiplication by a complex number is a dilation and rotation of the plane. Dilation by the modulus of the complex number and rotation by the argument.

[/ QUOTE ]

I'm sure that PTB understands quite well the geometric interpretation of a complex number, but is more baffled by the "meaning" of complex exponentiation. I define meaning in the way that the addition, multiplication, real valued exponentiation, etc. has meaning, or some sort of isomorphic connection to reality (two apples plus two apples equals four apples, multiply 2 apples by 2, get 4 apples, take two sets of two apples, multiply, get 4, etc.) But what the hell would be an analogous "real world" interpretation of complex exponentiation?

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]
I've always felt a lot like this quote from the first Wikipedia link:


Quote:
--------------------------------------------------------------------------------

After proving the identity in a lecture, Benjamin Peirce, a noted nineteenth century mathematician and Harvard professor, said, "It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth."


--------------------------------------------------------------------------------



I understand the proofs. I understand that if you want to define e^ix in such a way that the Taylor series for e^x still holds then Euler's Formula must hold. I understand that if you want to extend the function e^x on the Reals to the Complex numbers in a reasonable way, then Euler's Formula must hold. I understand that if you want to define e^ix in such a way that it differentiates similiarly to e^x then Euler's Formula must hold. But I've never been able to make sense of e^ix as some kind of exponentiation.

PairTheBoard

[/ QUOTE ]

Actually I think Euler's identity massively clarifies the nature of complex numbers. It makes it clear that multiplication by a complex number is a dilation and rotation of the plane. Dilation by the modulus of the complex number and rotation by the argument.

[/ QUOTE ]

That allows you to make sense of a complex number, say z, raised to a real power. If |z|=1 then you can see how z^n rotates around on the unit circle of the complex plane for intergers n. But I don't see how it makes sense out of an expression like z^i if you are trying to see it as exponentiation in the same way we understand exponentiation normally. That is, repeated multiplication of z by itself for z^n, or the reversal of that process for z^(1/n), or a combination of the two for z^q where q is rational, or aproximations by z^q for z^r where r is real. That's how I think of exponentiation. I can't see how z^i or e^i can be viewed as any kind of such exponentiation that makes sense.

PairTheBoard

[Ampelmann]

[ QUOTE ]
e^(i*x) = cos(x) + i sin(x)

[/ QUOTE ]
FYP

Proof e.g. by comparison of the Taylor series expansions of both sides. Pretty straightforward.