Return to Mathematics IndexCONTINUED FRACTIONS WITH A CALCULATORFred J. Schaal Lane Tech High School Addison & Western Chicago, IL 60613 1-312-880-8100Objectives: 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 thephenomenological 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'sDictionary 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."Thisdefinition is the springboardfor my lesson.Simon and Schuster'sThe 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'sEncyclopedic 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.