Any strictly periodic phenomena, actually.
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8 years ago
OT: mathematical conundrum
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8 years ago
The question is imprecise. i is not *the* square root of -1, it is *a* square root of -1.
Cheers
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8 years ago
and the others are?
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8 years ago
And the others are *what*, precisely?
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8 years ago
Can you be a little more specific on the very definition of "i"?
I'm left wondering what other square roots of -1 there are?
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8 years ago
The other is -i
Cheers
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8 years ago
Can you explain why (-i)^2 != 1 ?
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8 years ago
Yes, but not as well as others, so...
formatting link
Section 1, 'Definition'.
Cheers
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8 years ago
Ah, OK. I assumed, wrongly, that negative numbers would stay negative numbers. And it's a long time since I took A-level maths.
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8 years ago
Should've been 'imaginary numbers', not negative.
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8 years ago
By the definition of ^2,
(-i)^2 == (-i)*(-i) (Definition of squaring) == (-1*i)*(-1*i) Expanding -i to -1*i == -1*i*-1*i * is associative == -1*-1*i*i Relying on the fact that the middle * is commutative == (-1*-1)*(i*i) For clarity == (1)*(i*i) -1*-1 is 1 == (i*i) ... and we can drop the 1 == -1 By definition of i != 1
QED.
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8 years ago
(-1)^2 = (-i) x (-i) = (-1 x i) x (-1 x i) = (-1 x -1) x (i x i) = (1) x (i^2) = 1 x -1 = -1
as ene skoolboy kno.
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8 years ago
Or in short, the 2 -ves, squared, become +1 leaving i2 with is -1.
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8 years ago
Interesting link, I've used i quite a bit in programming but never realized the functional equivalency -i and i. This quote from that article is pretty cool. "Attempting to apply the calculation rules of the principal (real) square root function to manipulate the principal branch of the complex square root function will produce false results:"
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8 years ago
Like broken electoral promises you mean?
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8 years ago
The number of roots is the same as the highest power in any polynomial (fundamental theorem of algebra) so there are no others.
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8 years ago
I was half believing you were correct on the basis of phase, where i was
90degs and -i was 270degs. Doubling of angle would lead to -1 in both cases. Its a long time since I dealt with anything so esoteric.- Vote on answer
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8 years ago
Well they certainly are in some imaginary dimension, yes.:-0)
It still amazes me that people vote for what they are *promised* (Love peace on earth, goodwill to all men, social justice and tolerance, intentional harmony) instead of what they must know they will get (societal destruction, mass immigration, hatred, bigotry, loss of earnings, loss of freedom, serfdom under a European superstate) etc.
Hey, If I promise unlimited national health heroin, people can vote for me, be happy AND get what they were promised!
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8 years ago
I guess that would be ignoring roots consisting of complex numbers.
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8 years ago
No. You've already had the example of -1 having two square roots, both complex.
The circle showing the N Nth roots of 1 is a standard thing when introducing complex numbers - 2 square roots (1,-1), 3 cube (1, -0.5 += .866 i), 4 fourth (1,i,-1,-i) etc. (stops before having to work out what sin and cos 108 degrees is...)