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From: Christoph Lhotka on 9 Mar 2010 06:17
hi,

the concept of testing the algebraic equality/inequality of algebraic
expressions is indeed one of the most complex issues to do (it is easy for
you, because you already did it - and see it) but not easy from an
algorthmic point of view!

I would suggest 3 things:

1) try to find a normal form for the expressions, so they can be compared
easily:

PowerExpand[r^2 Sqrt[(r^3 + r + 2)/r] - Sqrt[r^3 (r^3 + r + 2)]]
0

2) use Simplify with all its power using Assumptions (are the any
conditions on the subexpressions, are they real/positive/etc...).

e.g.
Simplify[r^2 Sqrt[(r^3 + r + 2)/r] - Sqrt[r^3 (r^3 + r + 2)],r>0]

will work, but r<0, etc unfortunatly not.

3) it is easy to implement the identites you (really) know into
Mathematica by using rules.

whatIknow=
{
r_^2 Sqrt[(r_^3 + r_ + 2)/r_] - Sqrt[r_^3 (r_^3 + r_ + 2)]->0,
r_^2 Sqrt[(r_^3 + r_ + 2)/r_] -> - Sqrt[r^3 (r^3 + r + 2)],
....
};

This looks laborous, but after a while you while have a list of things,
you know and Mathematica can immediatly apply to your expressions:

r^2 Sqrt[(r^3 + r + 2)/r] - Sqrt[r^3 (r^3 + r + 2)]/.whatIknow
0

or combine 1), 2) and 3) !

Christoph


On Mo, 8.03.2010, 12:09, mmdanziger wrote:
> This isn't the first time that I've encountered something like this in
> Mathematica but in my calculations I got a term like this:
>
> r^2 Sqrt[(r^3 + r + 2)/r] - Sqrt[r^3 (r^3 + r + 2)]
>
> Which is obviously identically zero. For some reason Simplify or even
> FullSimplify can't figure this out. Once you get dependent on
> Mathematica these things are pretty disturbing...you forget about your
> own knowledge because the program tells you that things are
> different. Then you sit there like an idiot checking an algebraic
> identity that any beginning precalc student should be able to solve no
> problem.
>
> Is there any way to get Mathematica to "wake up" to these things? It
> has such a powerful algebraic engine for most things, why can't it see
> something simple like the above? Do you really have to manually
> override and tell the program when things should be zero?
>
> For the time being I'll just sift through and test things by hand but
> I can't believe that there isn't a better way.
>
> Best,
> md
>


From: DrMajorBob on 9 Mar 2010 06:20
Your expression is NOT identically zero, since, for instance:

r^2 Sqrt[(r^3 + r + 2)/r] - Sqrt[r^3 (r^3 + r + 2)] /. r -> I

-2 + 2 I

When r is Real, on the other hand:

r^2 Sqrt[(r^3 + r + 2)/r] - Sqrt[r^3 (r^3 + r + 2)] // PowerExpand

0

Bobby

On Mon, 08 Mar 2010 05:09:56 -0600, mmdanziger <mmdanziger(a)gmail.com>
wrote:

> This isn't the first time that I've encountered something like this in
> Mathematica but in my calculations I got a term like this:
>
> r^2 Sqrt[(r^3 + r + 2)/r] - Sqrt[r^3 (r^3 + r + 2)]
>
> Which is obviously identically zero. For some reason Simplify or even
> FullSimplify can't figure this out. Once you get dependent on
> Mathematica these things are pretty disturbing...you forget about your
> own knowledge because the program tells you that things are
> different. Then you sit there like an idiot checking an algebraic
> identity that any beginning precalc student should be able to solve no
> problem.
>
> Is there any way to get Mathematica to "wake up" to these things? It
> has such a powerful algebraic engine for most things, why can't it see
> something simple like the above? Do you really have to manually
> override and tell the program when things should be zero?
>
> For the time being I'll just sift through and test things by hand but
> I can't believe that there isn't a better way.
>
> Best,
> md
>


--
DrMajorBob(a)yahoo.com

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