sqrt question

04282017, 06:09 PM
Post: #22




RE: sqrt question
(04092017 03:59 AM)Claudio L. Wrote:(04072017 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 nonnegative. 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 nonnegative 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 onetoone (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  04062017, 03:25 PM
RE: sqrt question  pier4r  04062017, 04:01 PM
RE: sqrt question  Namir  04062017, 04:02 PM
RE: sqrt question  KeithB  04062017, 04:51 PM
RE: sqrt question  Han  04062017, 05:46 PM
RE: sqrt question  pier4r  04062017, 05:15 PM
RE: sqrt question  KeithB  04062017, 06:03 PM
RE: sqrt question  Han  04062017, 06:18 PM
RE: sqrt question  Claudio L.  04072017, 01:23 PM
RE: sqrt question  Han  04072017, 04:48 PM
RE: sqrt question  Claudio L.  04072017, 09:15 PM
RE: sqrt question  Han  04072017, 10:54 PM
RE: sqrt question  Claudio L.  04092017, 03:59 AM
RE: sqrt question  David Hayden  04242017, 09:36 PM
RE: sqrt question  Claudio L.  04262017, 03:08 AM
RE: sqrt question  Han  04282017 06:09 PM
RE: sqrt question  nsg  04072017, 11:34 PM
RE: sqrt question  Vtile  04092017, 10:41 AM
RE: sqrt question  nsg  04092017, 05:26 PM
RE: sqrt question  Vtile  04092017, 11:07 PM
RE: sqrt question  nsg  04102017, 01:44 AM
RE: sqrt question  Vtile  04252017, 11:38 PM

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