Objectives
Continued fractions terminate for rational numbers and become infinite for irrational. From the simplest of expressions come some of the most common irrational numbers. Students are also subtly introduced to phi (golden ratio), but the fanfare comes later. The purpose of this activity is for students to experience some of the struggles ancient mathematicians would have grapled with as they try and understand the concept of infinity.
Vocabulary
- Pythagoras
- Rational and Irrational
- Finite and Infinite
- Continued Fraction
- Infinite Fraction
- Rational and Irrational
About the Lesson
The Pythagorean Circle struggled with the notion of irrational numbers, the sort that their theorem generated. Evidence exists that continued fractions were being explored. Remember that rational numbers can be expressed as the ratio between two whole numbers. Can these numbers be expressed as continued fractions? The answer lies in this stage of the investigation that also includes some cryptic clues and a beautiful fraction that expresses the best known irrational number: pi.