620-150 Number, Shape and Choice

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Subject Description

Students completing this subject should comprehend

• some fundamental concepts in mathematics, both old and new;
• the value of well-chosen notation and terminology;
• the notion of a mathematical model;
• the concept of a logical, mathematical argument;

have developed the ability to

• use a simple mathematical model incorporating the ideas of number, shape, and choice;
• construct simple, logical arguments and draw conclusions;
• write a brief, clear account of a simple mathematical concept or a simple mathematical model.

and appreciate the beauty of mathematics and the usefulness of mathematics. This subject pursues basic themes in non-calculus based mathematics and selected applications, chosen from the following topics.

Number: The natural numbers, primes, divisors, congruences, applications to cryptography. Rational and irrational numbers. Countability. Powers and roots, algebraic equations, famous impossibilities, musical scales.Shape: Geometric similarity and scaling. Self-similarity and fractals. Classical and modern geometries and applications. Symmetries and tilings. Chaos: Calculator experiments, iteration, iterated mappings, geometrical notion of convergence, period-doubling and chaos. Choice: Competitive and cooperative games, linear programming, optimal decision making and voting systems. Probability at the casino: calculation of chances and expected winnings; variance as a measure of excitement of games; implications for the social consequences of gambling.

Assessment

Up to 24 pages of written assignments and a three-hour end-of semester written examination.