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Topic: 1=0? (Math skills needed) (Read 2024 times) |
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kellys
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Re: 1=0? (Math skills needed)
« Reply #25 on: Mar 11th, 2004, 5:32pm » |
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So the "nth root" problem isn't particular to the complex numbers, since the nth root is a natural definition in *any* field (or group). I think the thing to tell early students of mathematics is that the = sign is constantly abused, and its meaning will often change. For example, I would write,
a = [sqrt]i
To mean, "a is a square root of i," even though what would be more appropriate is, a [in] { z | z2=i }.
Of course, another way to make the nth root a function in the complex number is to use z1/2 = exp(log(z) / 2), since by log we generally mean the function defined on the primary branch. But that might not help.
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Icarus
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Re: 1=0? (Math skills needed)
« Reply #26 on: Mar 13th, 2004, 8:27am » |
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No, it definitely would not help, since by log(z) of complex numbers we often do not mean the primary branch!
You could define all exponentials that way, by specifying the primary branch, but to do so means that you have to give up all the nice laws of handling exponents, since they would no longer hold in some cases, as godskook's example shows.
We generally find it preferable to keep the laws, but with the understanding that they hold only "by abuse of notation", in that the two expressions on either side of the equal signs may actually be conjugal roots rather than the same one.
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When their digits are reversed? " - Anonymous
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Nigel_Parsons
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Re: 1=0? (Math skills needed)
« Reply #27 on: Apr 10th, 2004, 3:43pm » |
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This may appear elsewhere, but:
if A=B
Then A*A=A*B (multiplying both by A)
Then A*A - B*B = A*B - B*B (subtract B^2 from both sides)
Then (A+B)*(A-B) =B*(A-B) [left side difference of 2 sqrs, right side simplification!]
Then (A+B) = B (divide both sides by (A-B)
Then (B+B) =B (substitution, A=B line 1)
thus 2B=B thence 2=1
Subtract 1 from each side & 1=0
QED! 
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Benoit_Mandelbrot
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Re: 1=0? (Math skills needed)
« Reply #28 on: Apr 13th, 2004, 6:06am » |
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Yes, but from the original equation A=B, you can find that A-B=0. You divided by zero.
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| « Last Edit: Apr 13th, 2004, 8:57am by Benoit_Mandelbrot » |
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Because of modulo, different bases, and significant digits, all numbers equal each other!
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towr
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Re: 1=0? (Math skills needed)
« Reply #29 on: Apr 13th, 2004, 6:26am » |
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that and 2B=B gives (B=0 [vee] 2=1) and since 2=1 [equiv] [perp] this simplifies to just B=0
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Wikipedia, Google, Mathworld, Integer sequence DB
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