MIT Massachusetts Institute of Technology

Rational versus Irrational Numbers

Professor Nabil Nassif 
Mathematics Department
American University of Beirut

Sophie Moufawad
PhD Student
Institut National de Recherche en
Informatique et Automatique (INRIA)
Paris, France

The scope of this video lesson consists in studying the sets of Rational and Irrational numbers. The main objectives for the students are: 1) Learning how to distinguish between a rational number and an irrational number; 2) Learning about the existence of 2 infinities: the “countable infinity” for sets of Rational numbers and the “uncountable infinity” for the set of irrational numbers; and 3) Learning to use straightforward and advanced tools from Mathematics to reach goals (1) and (2). Such tools include: Methods of proofs, Euclidean Division and the Pigeonhole Principle, at the basic level, in adition to Cantor’s proof of uncountability at an upper level. The students should be aware of some definitions in Discrete Mathematics such as: sets, unions, intersections; logical definitions of “implies that” (“if then”) and equivalences (“if and only if”); some knowledge of elementary probability definition; and one-to-one mappings (bijection). This video lesson spans about 32 minutes and provides 7 exercises for students to work out in groups and in consultation with their classroom teacher. The exercises exclude the main question of the lesson, which should trigger the students’ interest in the lesson. The entire duration of the video demonstration and exercises should take about 50 minutes or equivalently one classroom session. To complete the lesson, a calculator is needed. The calculator’s functionalities should include the square root function and the number Pi. (Note that, nowadays, smartphone calculators include those functionalities.) 

Nabil Nassif has a Diplôme d'Ingénieur from the Ecole Centrale de Paris, followed by a Master's and a PhD in Applied Mathematics from Harvard University. He is Professor in the Mathematics Department at the American University of Beirut where he teaches and conducts research in Applied Mathematics, Numerical Analysis and Scientific Computing. Professor Nassif has held several visiting positions in France, Switzerland, USA and Sweden.

Sophie Moufawad has a BS in Mathematics, a Teaching Diploma and a Masters degree in Computational Science from the American University of Beirut. She is now completing her Doctorate in Computer Science at the Institut National de Recherche en Informatique et Automatique (INRIA), Paris, France. Her research interests lie in Applied Linear Algebra, Parallel Programming and High Performance Computing.

This site, sponsored by Fact Monster, provides a discussion about Rational and Irrational Numbers that is at an elementary level.

This site, sponsored by Math Is Fun, presents a fun discussion on an elementary level that includes the history of irrational numbers and “famous” irrational numbers.

This video animation on YouTube defines and compares rational and irrational numbers, again at an elementary level.

This YouTube video animation discusses, on an intermediary level, how we know that all numbers are not rational.

This site, sponsored by Wolfram Math World, presents a comprehensive and more advanced discussion of irrational numbers.

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