I do agree that a complex number has two square roots. Let me illustrate my
point of view on a simple example, on which actually I faced the problem.
I am using Sage to do differential geometry and in particular to manipulate
various charts on a manifold.
Let us consider the hyperbolic space H^2. The polar chart (R,phi)
associated with the Poincaré disk model of H^2 is related to the chart
(r,phi') associated with the hyperboloidal model by the following
transformations: phi'=phi and
r = 2*R/(1-R^2) <==> R = r / (1 + sqrt(1+r^2))
R in [0,1), r in [0, +infinity)
Now in Sage:
sage: R = var('R')
sage: assume(R>=0) ; assume(R<1)
sage: r = 2*R/(1-R^2)
sage: R == ( r / (1+sqrt(1+r^2)) ).simplify_full()
R == -1/R
which is false and is very annoying when manipulating fields defined on
H^2, when passing from one chart to the other.
A solution could be to define two functions:
- sqrt(x): the current one, which returns one of the two square roots of x,
with no control of which one
- sqrt_real(x), which assumes x is real and positive and always returns the
positive root of x, so that sqrt_real(x^2) = abs(x) would be guaranteed.
Eric.
Le lundi 17 juin 2013 17:49:07 UTC+2, rjf a écrit :
>
> Since your belief about what the correct behavior is, is at best
> arguable, and at worst, wrong, I would not expect it to
> be "corrected"
>
> Sqrt(anything) has TWO values.
> Choosing one should depend on the choice of square root branch, e.g.
> assume(sqrt(E))
>
> assuming something about E doesn't say much.
>
> e.g. even sqrt(9) is +-3.
> who cares about the "sign" of 9.
>
> In some engineering and physical circumstances, a branch of the sqrt
> can be intuited. But the math issue is not resolved by "assume(x>0)" etc
>
>
>
> On Saturday, June 15, 2013 8:31:02 AM UTC-7, Eric Gourgoulhon wrote:
>>
>> Hi,
>>
>> Here is some behavior of Sage which can be quite disturbing for a new
>> user and probably would be considered as a bug:
>>
>> sage: assume(x<0)
>> sage: sqrt(x^2).simplify_full()
>> x
>>
>> It appears that the instruction "assume(x<0)" is not sufficient to
>> enforce the desired behavior: x is still considered as a complex number and
>> sqrt(x^2) returns *a* square root of x^2, not necessarily the positive one.
>> To enforce x real, one should set:
>>
>> sage: maxima_calculus.eval('domain:real')
>> 'real'
>>
>> Then the output of simplify_full() is correct:
>>
>> sage: sqrt(x^2).simplify_full()
>> -x
>>
>> But the following problem remains:
>>
>> sage: sqrt((x-1)^2).simplify_full() # correct result:
>> -x + 1
>> sage: sqrt(x^2-2*x+1).simplify_full() # wrong result (and not consistent
>> with the above):
>> x - 1
>>
>> I know that the problem is due to Maxima function randcan and has been
>> much discussed, in particular here:
>> http://www.math.utexas.edu/pipermail/maxima/2011/026097.html
>> What is the latest status of this ? In particular, is there any
>> possibility to enforce the correct behavior by passing some options to
>> Maxima ? (I've noticed that maxima_calculus.eval('radexpand:true') does not
>> help).
>> Thank you for your help.
>>
>> Eric.
>>
>>
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