oldk1331 wrote:
>
> Is this the same as not simplifying "sqrt(6)-sqrt(2)*sqrt(3)" to 0?
Somewhat related but not the same. One thing is that internaly
'((2)^(1/4))^2' and '(2)^(1/2)' are different. This is closely
related to the fact that internally 'sqrt(6)' is different than
'sqrt(2)*sqrt(3)'. In both cases internal representations are
different for mathematical reasons: we can not assert equality
without extra assumptions/information.
However, the main issue is that we should not introduce things
like "sqrt(6)-sqrt(2)*sqrt(3)" or "((2)^(1/4))^2 - (2)^(1/2)"
during default simplifications. And our square root is
creating such things.
To put this differently, much of Expression design is build
on concept of normal representation: we should be able to
recognize 0 just looking at representation. Users can give
us nasty inputs like "sqrt(6)-sqrt(2)*sqrt(3)" for which
we can not build normal representation, but we can resonably
declare such inputs as user errors: FriCAS is not designed
to give unswers to ill-posed questions. However, given
well-posed problem FriCAS should never transform it to
ill-posed one and this is essence of this bug:
"(((2)^(1/2))^(1/2))^2 - (2)^(1/2)" is well-posed and
should give 0.
Actually, I discoverd this bug looking at
integrate(1/(sqrt(10) - x^2)^(1/2), x)
(reported in sci.math.symbolic by Martin "clicliclick")
and why it falls into infinite loop. In core integrator
we need fields and square root broke field assumption
by introducing unresolvable redundancy. The integral
above ilustrates also other problems with handling of
square roots, but the above was the first one...
--
Waldek Hebisch
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