Interested in joining our January 2025 batch? Applications opens on September 30, 2024.

Interested in joining our January 2025 batch? Applications opens on September 30, 2024.

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.

by Prof. Amritanshu Prasad , Prof. Sankaran Viswanath

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
+ Show all weeks

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

Prof. Amritanshu Prasad
Professor, The Institute of Mathematical Science, Chennai

Amritanshu Prasad is a mathematician at The Institute of Mathematical Sciences, Chennai,

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working in algebraic combinatorics and representation theory.

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Prof. Sankaran Viswanath
Professor, The Institute of Mathematical Science, Chennai

Sankaran Viswanath is a mathematician at The Institute of Mathematical Sciences, Chennai,

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working in algebraic combinatorics and representation theory.

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