Degree Level Course
Mathematical Thinking
To introduce ideas of proofs and problem solving in mathematics and to help in the transition from learning basic mathematical methods to the learning of more advanced mathematical methods and ideas.
Course ID: BSMA2001
Course Credits: 4
Course Type: Elective
Pre-requisites: None
Course structure & Assessments
8 weeks of coursework, weekly online assignments, 2 in-person invigilated quizzes, 1 in-person invigilated end term exam. For details of standard course structure and assessments, visit Academics page.
| WEEK 1 | triangular numbers, Sigma Notation for Summation, Sequences, Peano's axioms for the natural numbers, Set Theory: the language of Mathematics, Hilbert Hotel, Bijections and Cardinality, The Natural Numbers, The Integers |
| WEEK 2 | A Trip to Cantorsville, Cantor's Diagonalization Argument, Towards the Real Numbers, Ordered Field, Completeness Axiom, The Least Upper Bound Property, Mathematical Logic and Statements |
| WEEK 3 | Currency Game, Divisibility, Greatest Common Divisor, The Euclidean Algorithm, Proof of the Euclidean Algorithm, Test for Divisibility |
| WEEK 4 | Fermat's Little Theorem, Fundamental Theorem of Arithmetic, Modular Arithmetic, Arithmetic with Congruences, Infinitude of Primes, Inclusion–Exclusion Principle |
| WEEK 5 | Proof of Inclusion-Exclusion Principle, Pigeonhole Principle, Sieve of Eratosthenes, More on the Infinitude of Primes, Gaps Between Primes |
| WEEK 6 | Binomial Coefficients, Binomial Theorem, Lattice Paths, Random Sampling, Permutations, Example of Inclusion-Exclusion Principle |
| WEEK 7 | Graph Theory - Introduction, Connectedness and Distance in Graphs, Adjacency Matrix, Graph Trees, Sperners Lemma, Graph Coloring |
| WEEK 8 | Limit of a Sequence, Properties of Limits, Sequential Continuity, Continuity of Trigonometric functions, Intermediate Value Theorem |
| WEEK 9 | Limit of Fibonacci series, Lamé formula, Sandwich Lemma, Shifting Lemma |
| WEEK 10 | Derivatives, Faà di Bruno's formula |
| WEEK 11 | Riemann Integrals, Mean Value Theorem |
| WEEK 12 | Uniform continuity, Fundamental theorem of calculus |
Prescribed Books
The following are the suggested books for the course:
“Mathematical Thinking: Problem-solving and Proofs”, John D’Angelo and Douglas West, Pearson, 2000 (2nd edition)
“Mathematical Proofs: A Transition to Advanced Mathematics”, G. Chartrand, A. D. Polimeni, P. Zhang, Pearson, 2012
About the Instructors
Amritanshu Prasad is a mathematician at The Institute of Mathematical Sciences, Chennai,
working in algebraic combinatorics and representation theory.
Sankaran Viswanath is a mathematician at The Institute of Mathematical Sciences, Chennai,
working in algebraic combinatorics and representation theory.