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Re: Re: More /.{I->-1} craziness
*To*: mathgroup at smc.vnet.net
*Subject*: [mg106573] Re: [mg106525] Re: More /.{I->-1} craziness
*From*: "David Park" <djmpark at comcast.net>
*Date*: Sat, 16 Jan 2010 06:14:58 -0500 (EST)
*References*: <hhf5s3$h4o$1@smc.vnet.net> <hhhmhl$o48$1@smc.vnet.net> <18423897.1263546085556.JavaMail.root@n11>
There is a perfectly simple explanation why it doesn't work for I. It's like
saying that because I can calculate a determinant for 2 x 2 and 3 x 3
matrices by adding the products on the diagonals I should be able to do the
same thing with any order matrix and students will be confused if they have
to do otherwise. No, they can just learn how to do it. It's part of their
education. And maybe they shouldn't be taught the limited method in the
first place.
Mathematica represents complex NUMBERS as atomic expressions, Complex[n1,n2]
where n1 and n2 are NUMBERS. It seems to me to be a pretty reasonable design
decision. It makes sense to think of complex numbers as entities in their
own right. Then they used I as a input shortcut, and displayed complex
numbers with \[ImaginaryI] to conform to standard notation. Maybe WRI should
not have allowed I for input or ImaginaryI on display. If everyone had to
type Complex[a,b], and saw it on output, they might not have to learn the
difference between input, display and internal representation. If WRI did
that, I'm sure there would be many more complaints and you would be near the
head of the line.
The other choice would be to do away with the Complex representation
altogether. Then complex NUMBERS would be represented as Plus expressions
and would no longer be number entities in their own right. I don't know what
all the implications of that would be in the Mathematica handling of complex
expressions, but I don't think it would be good. Mathematica would have to
continually parse expressions to recognize complex numbers. For example, how
would Mathematica know to simplify (3+4I)/(1-5I)? It should be able to
simplify it just as easily as 4/2.
When students get to the point where they are treating complex numbers and
functions they can just learn the difference between input and internal
representation. If they can't understand that, I don't see how they can
understand much about complex algebra. And they should long ago have learned
that it is worth while having convenient input forms and output displays.
So don't teach them I -> -I. Teach them Complex[a_,b_] -> Complex[a,-b], or
Conjugate and ComplexExpand. Teach them about complex NUMBERS. It's not that
difficult.
David Park
djmpark at comcast.net
http://home.comcast.net/~djmpark/
From: AES [mailto:siegman at stanford.edu]
But a single character on the lhs? It works correctly AFAIK for every
other single-character in the alphabet. Why shouldn't one expect it to
work for I?
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