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Nigel (UK) · 10-27-2018, 10:04 AM

(10-26-2018 06:33 AM)sasa Wrote: Albert,

That is exactly the point.

Since here \(\frac{1+i}{\sqrt{2}}\) is just a "principal value", equally good/bad could be \(-\frac{1+i}{\sqrt{2}}\). Furthermore, logical question here is how to choose "principal value"? The answer is: arbitrary.

That is the reason to keep it as is, since evaluating it in an expression with "principal value", we may get result we hardly expect. It is probably not likely to happens in most of practical uses, but it is theoretically possible.

I now understand your point, but I do not agree that the Prime's current behaviour is wrong. My reasons:

People normally assume that \(\sqrt{z}\) is single-valued. As evidence for this assertion, I point to the universal appearance of \(\pm\) in front of the square root in the formula for the roots of a quadratic equation.
Sometimes people genuinely need to know the value of the square root of a complex number. If I'm looking at a damped simple harmonic oscillator, the real and imaginary parts of \(\omega\) have separate meanings and I would like to see them separately.
As an educational point, it is good to show students that the square root (or any other power) of a complex number is again a complex number. Leaving \(\sqrt{i}\) unevaluated when an exact representation is possible would make a mystery where there is none.
Suppose that instead of remaining unevaluated, \(\sqrt{z}\) did return a list of two values. This list would double in size each time another square root was taken. This would quickly become impractical.
Another suggestion you made in an earlier post is to re-write \(\sqrt{i}\) as \((-1)^{1/4}\), which is what Wolfram Alpha does. This works for Wolfram Alpha because it regards both the square root and the power as functions, each returning a single value. It wouldn't work here because if you wish the square root to have two values, then you would have to allow that \((-1)^{1/4}\) has four values, so one would not be an acceptable re-writing of the other.
Finally - and perhaps most importantly - I just do not see the problem with the current situation. Students learning about complex numbers already know that although \(x^2=4\) has two solutions, \(\sqrt4=+2\). This matches perfectly with what the Prime does with complex (and real) numbers. Students do not need to know about Riemann surfaces to be able to answer exam questions such as: Write \(\sqrt{i}\) in the form \(a+bi,\quad a,b\in R\).

If this comes across as grumpy and aggressive, I apologise: that certainly isn't my intention.

Best wishes!

Nigel (UK)