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Wizard
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Wizard
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October 9th, 2013 at 4:15:10 PM permalink
This is a follow up to the wonderful thread What does 0^0 equal?. This question posed in the title was brought up in that thread here, but I think it is worthy of a thread on its own.

To address the first three choices, "-1, no ifs ands or buts" means it is a firm -1, and the laws of mathematics can prove it beyond any reasonable doubt. "-1, by convention" means that it is assigned a value of -1, so that all cases of other general formulas work, but the value is more given in the interests of consistency, rather than something that can be proven. "Indeterminate form" means that the answer doesn't exist, like what some would say 0^0 is.
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thecesspit
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October 9th, 2013 at 4:25:07 PM permalink
If your an electronic engineer, i is always written as j :)
"Then you can admire the real gambler, who has neither eaten, slept, thought nor lived, he has so smarted under the scourge of his martingale, so suffered on the rack of his desire for a coup at trente-et-quarante" - Honore de Balzac, 1829
teliot
teliot
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October 9th, 2013 at 4:29:00 PM permalink
Quote: Wizard

This is a follow up to the wonderful thread What does 0^0 equal?. This question posed in the title was brought up in that thread here, but I think it is worthy of a thread on its own.

To address the first three choices, "-1, no ifs ands or buts" means it is a firm -1, and the laws of mathematics can prove it beyond any reasonable doubt. "-1, by convention" means that it is assigned a value of -1, so that all cases of general other formulas work, but the value is more given in the interests of consistency, rather than something that can be proven. "Indeterminate form" means that the answer doesn't exist, like what some would say 0^0 is.

While it is a convention that 0^0 = 1, since an empty-product is not defined, it is a fact that e^(i*pi) = -1, and the proof is given in about the fourth week of any undergraduate course on Complex Variables.
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rdw4potus
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October 9th, 2013 at 4:37:58 PM permalink
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Wizard
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Wizard
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October 9th, 2013 at 4:38:29 PM permalink
Quote: teliot

... and the proof is given in about the fourth week of any undergraduate course on Complex Variables.



Is that proof based on Euler's Equation?
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thecesspit
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October 9th, 2013 at 4:57:47 PM permalink
I see Babs noted that -1 is an imaginary number in a previous thread. It's not. -1 is real and negative. The real and imaginary parts of a complex number (e.g. 3 + 4i) can be used to do all sorts of useful sums, we used for modelling frequency response of a circuit. I forget exactly how, but capacitors and inductors could be modelled in terms of imaginary numbers, resistors as real numbers and the response of a circuit to various inputs could be worked out.

Edit : http://scipp.ucsc.edu/~johnson/phys160/ComplexNumbers.pdf looks very familiar to me...
"Then you can admire the real gambler, who has neither eaten, slept, thought nor lived, he has so smarted under the scourge of his martingale, so suffered on the rack of his desire for a coup at trente-et-quarante" - Honore de Balzac, 1829
teliot
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October 9th, 2013 at 4:58:59 PM permalink
Quote: Wizard

Is that proof based on Euler's Equation?

You're not going to ask for a proof of Euler's are you? Hint. Taylor series.
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dwheatley
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October 9th, 2013 at 5:31:32 PM permalink
I was taught the equation in high-school but saw the proof in a 1st year Calculus course. A Taylor series expansion does prove the result.
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beachbumbabs
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October 9th, 2013 at 5:50:05 PM permalink
Quote: thecesspit

I see Babs noted that -1 is an imaginary number in a previous thread. It's not. -1 is real and negative. The real and imaginary parts of a complex number (e.g. 3 + 4i) can be used to do all sorts of useful sums, we used for modelling frequency response of a circuit. I forget exactly how, but capacitors and inductors could be modelled in terms of imaginary numbers, resistors as real numbers and the response of a circuit to various inputs could be worked out.

Edit : http://scipp.ucsc.edu/~johnson/phys160/ComplexNumbers.pdf looks very familiar to me...



cess,

Thanks for the correction; I asked for it aand appreciate the opportunity to learn from it.
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Doc
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October 9th, 2013 at 7:00:52 PM permalink
Quote: teliot

You're not going to ask for a proof of Euler's are you? Hint. Taylor series.


Note: This post is an admission of the weakness of either my math skills or my memory.

Earlier today, when this question was posed in the original thread, I was thinking that a Taylor Series expansion might offer the proof that the Wizard was looking for then. Since I am lazy and like to expand about a=0, I tried to look up the proper spelling of the name for the Maclaurin Series. I was reckless and looked at Wikipedia, where I came across the comment that, "A function may not be equal to its Taylor series, even if its Taylor series converges at every point." They even give an example of a function for which the Taylor Series converges readily and does not represent the function.

If I had heard that when studying the calculus, the notion completely evaporated from my memory, as did so many other things I "studied" in school.

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