CONTINUED FRACTIONS WITH A CALCULATOR
Fred J. Schaal Lane Tech High School
Addison & Western
Chicago, IL 60613
To learn how to evaluate Continued Fractions by taking many reciprocals with
Paper, pencil and a hand calculator--preferable a freebie-from-the-bank-
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.
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