Re: Some bugs in Mathematica
- To: mathgroup at smc.vnet.net
- Subject: [mg59617] Re: Some bugs in Mathematica
- From: "Alex" <akhmel at hotmail.com>
- Date: Sun, 14 Aug 2005 04:38:05 -0400 (EDT)
- References: <ddk7th$13o$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
I accept your challenge to improve algebra of complex numbers. Here is
my first installment. I claim that
\!\(\@\(1 - z1/z2\)\/\@\(z2 - z1\)\)
can be simplified to \!\(1/\@z2\), if Re[z2] > Re[z1]. Otherwise, the
result is Sign[Re[z1]Im[z2]-Im[z1]Re[z2]] Sign[Im[z2]-Im[z1]] /
Sqrt[z2].
I challenge Kozlowski and anybody else to give me a numerical example
where my formulas are incorrect.
Alex
Andrzej Kozlowski wrote:
> As for the first sentence above I think I should leave it to others
> to make the judgement whether it applies to my posting more than to
> everything posted by Mr. Khmelnitsky.
>
> The statement that the square root of a complex number has two
> brunches is well... not very deep: we could equally say that the cube
> root has three branches, the fourth root has four etc... For example,
> Mathematica gives
>
>
>
> Solve[x^5 == a, x]
>
> {{x -> a^(1/5)}, {x -> (-(-1)^(1/5))*a^(1/5)},
> {x -> (-1)^(2/5)*a^(1/5)},
> {x -> (-(-1)^(3/5))*a^(1/5)},
> {x -> (-1)^(4/5)*a^(1/5)}}
>
> so we could also say that we could cancel 5-th roots since the result
> is determined "up to one of the 5-th roots of 1". Of course, the real
> question is how Mathematica should have expressed this in a way that
> would have been more useful and clearer than leaving the expression
> un-cancelled. In particular, this expression would have to be such
> that it could easily serve as input for subsequent algebraic
> operations. The information about the possible "error" would have to
> be carried along to the next operation. If there were a large number
> of such cancellations in an expression, and a number of consecutive
> operations had to be perfumed on it, all of this information would
> have to be combined and carried over successive steps. ALl of this
> would involve mounting computational effort. In principle such an
> approach is possible and I am sure it has been considered, but the
> fact is that no CAS system known to me does anything of this but
> instead such expressions are left unchanged until the user provides
> additional information in the form of assumptions is the approach
> adopted by all the CAS know to me. If Mr. Khmelnitsky or any one
> knows of better way I am sure he can make a career in the
> computational algebra world, where there are quite many pretty clever
> people who for at last 3 decades have been thinking about these
> matters. However, before Mr. Khmelnitsky decides to embark on this
> project I suggest he learns about the InputForm of expressions
> because very few people, and certainly not me, will ever bother to
> try to decode the sort of input that the originator of this thread
> has been providing (particularly that almost all of turned out to
> have been incorrect).
>
> Andrzej Kozlowski
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