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April 21st, 2015, 11:11 AM   #11
Tau
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Yet, I could just as well label i as the error number, is it imaginary or in reality an error? We can imagine both real things as well as errors. This course of thought leads me to the idea of quantifying errors.
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April 21st, 2015, 11:20 AM   #12
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Do you agree with De Morgan's observation that they arise only whenever we deal with an error ("inconsistency or absurdity")?
No. We've learned vastly more about them in the time since De Morgan suggested that. (And frankly, he was a bit behind the times in that respect -- that view was typical 300 years ago but already old-fashioned by De Morgan's day.)
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April 21st, 2015, 11:24 AM   #13
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If De Morgan was wrong then how does the mother of complex numbers, the equation x^2 = -1, reconciles with the property that -1 times -1 = 1?
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April 21st, 2015, 12:59 PM   #14
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If De Morgan was wrong then how does the mother of complex numbers, the equation x^2 = -1, reconciles with the property that -1 times -1 = 1?
What is there to reconcile?
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April 21st, 2015, 07:12 PM   #15
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If De Morgan was wrong then how does the mother of complex numbers, the equation x^2 = -1, reconciles with the property that -1 times -1 = 1?
Well, if x^2 = y, and y^2 = z, then x is a fourth root of z. In this case i is a fourth root of unity. This shouldn't be surprising, though, given the Fundamental Theorem of Algebra.
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