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MathGroup Archive 2001

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RE: Showing intermediate steps in calculations

  • To: mathgroup at smc.vnet.net
  • Subject: [mg29054] RE: [mg29019] Showing intermediate steps in calculations
  • From: "David Park" <djmp at earthlink.net>
  • Date: Sat, 26 May 2001 21:53:45 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Maarten,

If your father is seriously interested in studying mathematics he would
definitely like Mathematica!

It is true that Mathematica in its "normal" use does a lot of automatic
simplification which often bypasses just the points that are at issue when
proving basic theorems and understanding basic steps. But Mathematica is a
tremendously versatile and general system for symbolic manipulation and
doesn't have to be used at all in the "normal" mode.

As an example, I have been studying an introductory real analysis book. It
starts out with the axioms which characterize a field. Then one is supposed
to prove a number of simple theorems. The Mathematica operators, Plus and
Times, already have all the attributes built in to them that automatically
implement what is to be proved. The solution is simple. I just used
CirclePlus and CircleTimes instead. They have no built in attributes, do no
automatic simplification, and they look fairly nice on the page. I was able
to write all the axioms as rules and then just apply them to expressions, or
parts of expressions, to prove the theorems.

One can also use Mathematica as a mathematical word processor. Sometimes it
is easier to just type out a proof, or parts of a proof, in text cells. Of
course, it is usually more educational to get Mathematica to perform the
steps.

Another technique is to put expressions in a HoldForm which prevents
evaluation. Then it is possible to do controlled evaluations or
transformations on the expression. Ted Ersek and I have written a package,
ExpressionManipulation, available at my web site below, which facilitates
such controlled evaluation. I think that Ted also has a similar package
available on MathSource. Allan Hayes has presented an EvaluateAt routine a
number of times on MathGroup, which is useful in such controlled
evaluations.

The tutorial for the ExpressionManipulation package has a number of
examples: Step-by-step evaluation of an expression; calculating a derivative
from the definition; step-by-step differentiation of a ratio; L'Hopital's
Rule with commentary; integration by substitution; simple fraction problems.

So yes, it is definitely possible to make Mathematica work the way you want,
and to show every step you wish to show - but it involves somewhat more
effort than just throwing a switch.

David Park
djmp at earthlink.net
http://home.earthlink.net/~djmp/

> From: Maarten D. de Jong [mailto:pleez.dont at rely.on.this.address]
To: mathgroup at smc.vnet.net
>
> I am writing this message on behalf of my father, who is having
> the following
> problem. He's working through various high-school and undergraduate books
> on algebra and calculus (out of fun and interest) and he meticulously
> writes down the answers to each problem he solves. He would like
> to do this
> on his computer, but after a bit of discussion, I discovered that any of
> the main symbolic math packages available do not solve his
> problem entirely.
> As far as I know, these programs take in the problem, and write out the
> answer, without showing intermediate steps and simplifications---and it is
> precisely this that my father wants to know about. Are there
> packages which
> show these steps? Or is it just a matter of turning on an option in the
> mainstream programs?
>
> Thanks for any help you can offer,
> Maarten
>



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