sqrt question
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04-28-2017, 06:09 PM
Post: #22
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RE: sqrt question
(04-09-2017 03:59 AM)Claudio L. Wrote:(04-07-2017 10:54 PM)Han Wrote: Is this due to the sqrt() function, though? This seems like a consequence of assuming factorization properties of 1 and -1 that may not still hold true for complex numbers. It is not clear to me what you mean by the factorization holds true for complex numbers. I agree that \( \sqrt{ab} = \sqrt{a} \sqrt{b} \) provided that \( a \), \( b \), and \(ab \) are non-negative. However, I question whether the definition of \( \sqrt{x} \) has been implicitly changed when you allow \( a \) and \(b \) to be negative. For complex numbers, which can be represented as \( re^{i\theta} \), (where \( r \) is a non-negative real number and \(-\pi < \theta \le \pi \) ), we have the "principal root" \[ \sqrt{z} = \sqrt{re^{i\theta}} = \sqrt{r} e^{i\theta/2} \] The reason your example produces two outcomes is because you did not define the square root function (over the complex plane) to be one-to-one (unless you are restricting \(\theta \) to be strictly positive and less than or equal to \( 2\pi \) ). My point here is that it is mathematically possible to define the square root function for a complex number without obtaining ambiguous results. Graph 3D | QPI | SolveSys |
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Messages In This Thread |
sqrt question - KeithB - 04-06-2017, 03:25 PM
RE: sqrt question - pier4r - 04-06-2017, 04:01 PM
RE: sqrt question - Namir - 04-06-2017, 04:02 PM
RE: sqrt question - KeithB - 04-06-2017, 04:51 PM
RE: sqrt question - Han - 04-06-2017, 05:46 PM
RE: sqrt question - pier4r - 04-06-2017, 05:15 PM
RE: sqrt question - KeithB - 04-06-2017, 06:03 PM
RE: sqrt question - Han - 04-06-2017, 06:18 PM
RE: sqrt question - Claudio L. - 04-07-2017, 01:23 PM
RE: sqrt question - Han - 04-07-2017, 04:48 PM
RE: sqrt question - Claudio L. - 04-07-2017, 09:15 PM
RE: sqrt question - Han - 04-07-2017, 10:54 PM
RE: sqrt question - Claudio L. - 04-09-2017, 03:59 AM
RE: sqrt question - David Hayden - 04-24-2017, 09:36 PM
RE: sqrt question - Claudio L. - 04-26-2017, 03:08 AM
RE: sqrt question - Han - 04-28-2017 06:09 PM
RE: sqrt question - nsg - 04-07-2017, 11:34 PM
RE: sqrt question - Vtile - 04-09-2017, 10:41 AM
RE: sqrt question - nsg - 04-09-2017, 05:26 PM
RE: sqrt question - Vtile - 04-09-2017, 11:07 PM
RE: sqrt question - nsg - 04-10-2017, 01:44 AM
RE: sqrt question - Vtile - 04-25-2017, 11:38 PM
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