The mathematical square root is a function, which maps a given domain
value to exactly one range value. That is why textbooks drum into
students' heads that, for example,
2^2 = (-2)^2
however
sqrt(2^2) = sqrt((-2)^2) = 2.
Functions do not map one domain value to one range value half the time
and another range value half the time.
If you change your original example to
f = -sqrt(x^2 + 2*x + 1)
f.full_simplify()
Sage does return
-x - 1
I'd say the two examples together show correct behavior. The Sage
user decides which value to assign to f, Sage and Maxima seem to
respect the standard mathematical definition of square root, and the
simplify returns the result that matches the user's assignment. That
may not be what the user wants, but it is correct.
What your user seems to be looking for seems more like an operation to
show all possible factorizations. I'm new to Sage and haven't
explored all the factorization capability, but if there is such an
operation and Sage only returns positive factors of a square, that
would be a problem.
On Mar 6, 1:17 pm, Michael Orlitzky <mich...@orlitzky.com> wrote:
> On 03/06/12 12:03, daniel.kho wrote:
>
>
>
>
>
>
>
>
>
> >> sage: f = sqrt(x^2 + 2*x + 1)
> >> sage: f.full_simplify()
> >> x + 1
>
> >> I think the user should have to try *really* hard to ask us for this
> >> simplification.
>
> >> Right now, simplify() just sends an expression to maxima and back. Full
> >> simplify does every simplification, including simplify_radical() which
> >> is to blame for the error above.
>
> > Wait a minute... isn't sqrt(x^2 + 2*x +1) = x + 1?
> > Sorry, my math could have turned rusty over the years and I'm *really*
> > new to sage, but I thought there was no error in the simplification?
>
> It is half the time. The other half it's -(x + 1). The Maxima 'domain'
> variable is supposed to control simplifications like this. From the docs[1]:
>
> Option variable: domain
>
> Default value: real
>
> When domain is set to complex, sqrt (x^2) will remain sqrt (x^2)
> instead of returning abs(x).
>
> In sage, we set this to 'complex', so the sqrt should be left alone.
> However, full_simplify() calls Maxima's radcan() function, which just
> chooses one of the square roots and sticks with it regardless of the
> 'domain' setting.
>
> This is very wrong over the reals, where we *should* get abs(x+1) rather
> than choosing +(x+1) or -(x+1) randomly.
>
> [1]http://maxima.sourceforge.net/docs/manual/en/maxima_9.html
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