Message-Id: <l03102800b0f482949641@[209.142.17.34]>
Date: Wed, 28 Jan 1998 08:22:47 -0700
To: pinhole@exploratorium.edu
From: Ron Wong <ronwong@inreach.com>
Subject: re math questions
Geoff asked:
>Right now in my algebra 2 class, we are starting to study rational and
>irrational functions. Does anyone know of any particularly interesting
>places that these sorts of functions show up in nature? (I know I could
>look through a physics book and find that they show up in gravitational
>attraction, magnetism, and so forth. But does anyone have any particularly
>salient examples that really could grab students...
I believe that "rational functions" are defined as follows: If m and n are
both polynomials, then m/n is rational function. This function involves a
ratio and for this reason is called a rational function (a footnote: if m
and n are polynomials in x, then usually m/n is not a polynomial in x).
As a consequence, you will find a rational function anytime you find a
ratio or proportion in nature - a fairly common phenomena.
I've never heard of "irrational functions" before - irrational *numbers*,
yes; irrational *functions*, no. As far as the former is concerned, PI is
the most ubiquitous example. Whether that fact or the appearance of any
other irrational number would necessarily "grab" a student is open to
question.
>Also, what is the purpose of studying long-division of polynomials?
It probably serves the same purpose as studying the multiplication of
polynomials.
Any quantity can be represented as a polynomial in x where x represents the
base of a particular number system (eight oranges would be 1000 oranges in
base 2 -> 1*x^3 + 0*x^2 + 0*x + 0 where x = 2, 22 oranges in base 3 -> 2*x
+ 2 where x = 3, and 8 oranges in base ten for instance). When you multiply
or divide two numbers you are actually multiplying or dividing two
polynomials in x. So, learning how to multiply and divide polynomials is in
fact learning what multiplication and division is in the most general sense
of the word.
Try dividing eight oranges in base 3 by four in base 3 and see what
happens. Do the same in base 2. Try other quantities/bases. If you do this
using polynomial expressions (the hard way, as some students would say),
then notice that you don't have to substitute the values of the base for x
in the initial expression. Just solve for the answer in terms of x and then
do the substitution (if necessary). The same is true for multiplication.
It's the basis for various "speed" methods of basic arithmetic (the
Trachtenberg method being one of the more popular ones).
ron