CONTINUED FRACTIONS WITH A CALCULATOR

Fred J. Schaal                 Lane Tech High School  
                               Addison & Western      
                               Chicago, IL 60613
                               1-312-880-8100
 
Objectives: 
     To learn how to evaluate Continued Fractions by taking many reciprocals with 
a calculator. 

Materials: 
     Paper, pencil and a hand calculator--preferable a freebie-from-the-bank-
four-banger type.  

Strategies: 
     Use the definition of a Continued Fraction to generate various examples.  Try 
to predict the value of the current fraction in the light of previous work.  Keep 
it very experimental.  Should interest wane, give several examples of rational 
numbers expressed as Continued Fractions.  Should even this become boring, give 
the expression for tan z as a Continued Fraction and try it out for several values 
of z in radian measure.  (I doubt that either of these alternatives will be 
necessary in a single class of 40 minutes.) 

A FEW NOTES:
    I initially encountered Continued Fractions (henceforth C.F.) as problems from 
the contests that our Lane-Tech-math teams enter.  I began to play around with 
them just to see what would happen.  They are fun this way.This is the 
phenomenological way I want to present them in this mini-teach.  
    This mini-teach on C.F. has driven me to the library for a look at several 
math dictionaries.  Most of the 7 or 8 that I found discuss Continued Fractions.  
Some tell you more than you would ever want to know.  
    Barnes and Noble's Dictionary of Mathematics--1972, Millington and Millinton--
defines a continued fraction as "an integer and a fraction, the denominator of 
which is also an integer and a fraction, etc."  This definition is the springboard 
for my lesson. 
    Simon and Schuster's The Universal Encyclopedia of Mathematics--1969--states 
that:"Every rational number a/b (a, b positive integers) can be developed as a 
continued fraction."  This idea is beyond  the original scope of my mini-teach, 
but if I get stuck I shall try several examples. Recall my strategies above. 
    Also from the above source I found these lines: "Irrational numbers can be 
developed as infinite continued fractions.  Conversely every infinite continued 
fraction in an irrational number."  I shall not get to these ideas in this 
introductory mini-teach, but they could be dealt with in a future class, should 
sufficient interest arise. 
    In MIT Press's Encyclopedic Dictionary of Mathematics, written by the 
Mathematics Society of Japan, I found an expansion of tan z as a continued 
fraction with powers of z in the denominators.  A trig-conscious class could try 
several values of z (in radians) just to see if it works.