[2011-May-18] Mathness of 1

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Astrogirl
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[2011-May-18] Mathness of 1

Post by Astrogirl »

http://www.smbc-comics.com/index.php?db=comics&id=2249

Is this last thing really 1? It surely somehow draws from e^(i*pi) being -1, but how?

I can't even read properly where is the base and where is the exponent. Either it says e^(i*pi/2) * i^i or it says e^(i*pi/2 * i^i).

i^i has infinitely many results: e^(-Pi/2 - 2*Pi*n) with n an integer. This is where I get lost.
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Re: Mathness of 1 [2011-May-18]

Post by zomgmouse »

(e^[pi*i]) +2 = 1
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Astrogirl
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Re: Mathness of 1 [2011-May-18]

Post by Astrogirl »

Is this ... an answer to my question?
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Re: Mathness of 1 [2011-May-18]

Post by Felstaff »

HE MATHED AT YOU DIDN'T HE?!?
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Re: Mathness of 1 [2011-May-18]

Post by K^2 »

Astrogirl wrote:Is this last thing really 1? It surely somehow draws from e^(i*pi) being -1, but how?
e^(pi/2) * i^i = e^(pi/2) * e^(i*ln(i)) = e^(pi/2) * e^(i * i * pi/2) = e^(pi/2 - pi/2) = 1.

And yes, there are multiple roots for e^x = i, but only i*pi/2 is used here. I can't quite think of a good argument for that just now.

shining2k1

Re: Mathness of 1 [2011-May-18]

Post by shining2k1 »

Hey Astrogirl,

you smuggled an extra i into your first expression (force of habit, I guess). With e^(pi/2) *i^i instead of e^(i*pi/2) *i^i, the whole thing resolves to
e^(Pi/2-Pi/2 - 2*Pi*n)=e^(-2Pi*n)=1 for all whole numbers n

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Astrogirl
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Re: Mathness of 1 [2011-May-18]

Post by Astrogirl »

Ah, that suddenly makes a lot of sense. Indeed an extra i crept in on me.
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Re: Mathness of 1 [2011-May-18]

Post by K^2 »

shining2k1 wrote:e^(Pi/2-Pi/2 - 2*Pi*n)=e^(-2Pi*n)=1 for all whole numbers n
e^(-2i*Pi*n) = 1 for all integer n, but what you actually get is only 1 for n=0. That doesn't go away on its own, and has to be part of definition of imaginary exponents.

orionsbelt

Re: Mathness of 1 [2011-May-18]

Post by orionsbelt »

No one is going to point out that the caption says "Mathematicians are no longer allowed to sporting events"?
Sorry guys. I was a math major but I was also a writing minor. Awkward wording pops out to me. At me. Whatever you prefer.

JS

Re: Mathness of 1 [2011-May-18]

Post by JS »

Oh, no, Zach; do you know what kind of can of worms you've opened? http://tinyurl.com/3tnvqh5

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Re: Mathness of 1 [2011-May-18]

Post by Kaharz »

In the pop up he did note that that 0^0=1 depends on your definition. Always read the footnotes...
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Mr. Pedantic

Re: Mathness of 1 [2011-May-18]

Post by Mr. Pedantic »

It is multi-valued.
e^(Pi/2) i^i = e^(Pi/2) e^(i ln i) = e^(Pi/2) e^(i (ln |i| + i arg(i))) = e^(Pi/2) e^(- Pi/2 - 2 n Pi) = e^(- 2 n Pi)
for any integer n. There is no 'i' term in the exponent, so this is not a single value. For n=0, we get 1 (and indeed, this would be the value associated with the principal branch of the complex logarithm), but it's just as reasonable to claim that this sign says "We're #e^(16 Pi)", (approximately 6.7 x 10^21)

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Re: Mathness of 1 [2011-May-18]

Post by Oldrac the Chitinous »

You know, if you just glance at the comic quickly, it looks kind of like the last sign says "We're Hentai."

Or maybe that's just me.
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Re: Mathness of 1 [2011-May-18]

Post by Edminster »

Oldrac the Chitinous wrote:You know, if you just glance at the comic quickly, it looks kind of like the last sign says "We're Hentai."

Or maybe that's just me.
You being hentai does bad things to my peace of mind.
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Re: Mathness of 1 [2011-May-18]

Post by Tom Swift »

Although it's a reasonable gripe that i^i has infinitely many possible values, I'm more concerned that the guy in the middle thinks 0^0 is 1. I think that maybe the mathematicians had a buddy from the physics department come along.

The value of 0^0 is "undefined". It's a classic trick question about double limits (limits of limits). It isn't 1: think about the limit of the limit of f(u,v) = u^v as both u and v go to 0. If you take the limit as u goes to 0 first (with v not 0) then that limit is 0. Do it the other way around and take the limit as v goes to 0 first and you get 1. So you can approach 0^0 two different ways and get two different answers for what it should be. If 0^0 really had a value, you couldn't get two different limits for it.

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