Re: x=@n3*PI

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Re: x=@n3*PI

Dick Smith wrote:
> ] >
> ] > Yes, but to what purpose are we put through this game?  Why is it there?

and Glenn Fisher responded:
> ] It is there because the solution of the problem has many values all
> ] of the form returned with the arbitrary integer.

And Smith then asked:
> Yes, I'm aware of that, but my point was that I didn't understand why the 'n'
> has to be qualified as it is.  E.g., taking the example from the subject line
> of this thread, surely x=@n3*PI can be written just as accurately as x=n*PI?
>
> The '@' and the 3 (in this case) are redundant and just clutter up the
> answer in my opinion.

Except that "n" is an ordinary TI-89/92 variable, and might have been
assigned some value.  Suppose you typed
   solve(sin(x)=0.4,x)
[I know that this equation would be better handled with the arcsine
function; it's just used as an example.]  If the 89/92 behaved the way you
suggest (with just "n"), it would report
   x=2 n pi + 2.730 ...  or  x=2 n pi + 0.4115 ...
which is just fine --- UNLESS you had previously set n equal to, say, 14, in
which case you would see
   x=90.69 ...  or  x=88.37...

So, at least the "@" (for "arbitrary") is useful to distinguish this from
"regular" variables.  What about the number after "@n"?

Fisher wrote:
> ] The number attached to the @n is to differentiate it from other
> ] arbitrary integers in the same session or problem.  It may take
> ] two or more arbitrary integers to represent all solutions to
> ] the problem.

Smith responded:
> I don't see this (sorry); if n is an arbitrary integer (0, 1, 2, 3, etc.)
> then how is @n25 any different?  I mean, @n25 can take values 0, 1, 2, 3,
> etc. in just the same way.  We don't need @n25, we only need n, IMHO.

There IS a difference -- one that could easily escape students who are just
learning to solve these equations, and even some who are used to such
things.  If you are trying to solve a system of equations for ordered pairs
(x,y), and you see the solutions
   x = 2 n pi + 0.1
and
   y = 2 n pi - 0.3
it would be very easy to think that the solutions to the system are
   (x,y) = (2 n pi + 0.1, 2 n pi - 0.3)
for an arbitrary integer n.  However, the correct interpretation would be
that the solutions are
   (x,y) = (2 n pi + 0.1, 2 m pi - 0.3)
for two (potentially different) arbitrary integers m and n.  The numbers
after "@n" make it evident that these integers can be different.

Such situations may not arise often, but it seems to me that the minor
nuisance of dealing with the "@n3" scheme is worth it for the benefit of
that scheme in such cases.

--
Darryl K. Nester                E-mail: mailto:nesterd@bluffton.edu
Assoc. Prof. of Mathematics        WWW: http://www.bluffton.edu/~nesterd
Bluffton College                 Phone: 419-358-3483
Bluffton, OH  45817-1704           Fax: 419-358-3232

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