Raymond Toy wrote:
> <snip>
>> This is because Maxima simplifies:
>>
>> (%i7) sqrt(%i);
>> (%o7) (-1)^(1/4)
>>
>>
>
> Is there any reason why this shouldn't simplify to 1/sqrt(2)+%i/sqrt(2)?
>
Yes, there is at least one reason.
It can be argued that sqrt(2) is a shorthand for the positive number s
such that s^2 is 2.
From which it might seem that sqrt(E) is a shorthand for the positive
number s such that s^2 = E.
However the number offered here is not positive. So we can't use that rule.
In fact, the generalization of simplifying sqrt(x ^2) to x causes all
kinds of problems, even if you know
that x is assumed positive, because there are, of course, two square roots.
Going back to the original question, if we cannot choose the "positive"
s, can we choose a "principal" root? Sometimes, but you
do not have any assurance that this is the one of interest.
Here is another reason, or at least food for thought.
Mathematica 7.0 returns (-1)^(1/4).
evaluating it numerically however gives 0.707107 + 0.707107 I.
RJF