Quantitative Reasoning and Mathematical Thinking

This course will introduce students to the basics of mathematical and logical reasoning, algorithmic thinking, and data‑driven analysis. The course will use examples from diverse fields to highlight the importance of these in a problem‑solving approach.

An open mind. No fear of Mathematics. This course will not assume prior background in high school level mathematics or computing.

1. God gave us numbers, and human thought created algorithms

   Sets, sizes and natural numbers. Early math is constructive and algorithmic: addition, subtraction, multiplication (repeated addition), the division theorem and fractions. Illustration of the basic algorithms in different models — ruler and compass, abacus, marbles, pencil and paper. (Lecture notes updated on September 6)

2. Abstraction turns problems and concepts into principles

   Functions and relations, counting, one‑one and onto functions, finite and countably infinite sets, equivalence classes and partitions. Modular arithmetic: the clock, the magic square, one‑time pad and perfect secrecy, etc. (Lecture notes updated on September 6)

3. We need precision in thought and action to win arguments

   Basic logic, truth tables, De Morgan’s laws, truth table of implication. Proofs: explicit construction, counter‑example, exhaustion of cases, contradiction, contrapositive. (Lecture notes updated on September 21)

4. Different types of geometry

   Euclidean and affine geometries. What about the (pin‑hole) camera?

5. From geometry to algebra

   Polynomials and the place‑value number system. Problem solving with systems of equations. The magic of the geometric series. Compound interest. How many years will it take for my money to double? What is the value of my home loan? Function plots — linear, quadratic, exponential, logarithmic. Curve fitting and function estimation from data. Leads to machine learning?

6. The prime thing

   Primes, GCD and LCM, there are infinitely many primes, the fundamental theorem of arithmetic. Primes and the intrigue of crypto.

7. Counting can be tricky

   Basics of combinatorics: rule of sum and rule of products, the principle of inclusion and exclusion, permutations and combinations. Pigeonhole arguments. Problem solving with graphs.

8. The irrational persecution of Hippasus

   If we have numbers and division then we must have rationals. If we have rationals and ruler‑and‑compass, then we must have irrationals. Sequences of rationals may also lead to irrationals.

9. Even infinity admits a hierarchy we must confront

   Real numbers: why do we need them? Reals are uncountable. Power sets.

10. Algorithmic thinking – the step‑by‑step thing

    Principle of mathematical induction, recursion and algorithms, step‑by‑step. Recursive model of computation. Algorithms in everyday life.

11. To feel no fear is a luxury of the unwise

    One is algorithmically illiterate till one learns to fear the exponential and non‑linearity. Time and space complexity of algorithms. Examples of intractable problems. Limits of computation.

12. Playing the dice

    Sample spaces and events. Probability distribution. Mean, median and mode. How to draw a representative sample? Conditional probability and Bayes’ theorem. Examples.

13. Algorithmic reasoning

    Propositional and first‑order logic. Soundness and completeness. Examples of theorem proving using resolution.

Emphasis on examples and uses for each of the above.

• All students are expected to follow a high ethical standard.
• Collaborations and discussions are encouraged. However, all students are required to write up all solutions and submitted work entirely on their own. Any collaboration, or help taken, must be declared. All submissions must include a declaration of originality.
• Students are encouraged to refer to books, papers, internet resources and tools like ChatGPT. They may even consult other individuals. However, the source must be clearly cited if any part of the solution (or even an idea) is taken from such a source.
• Failure to declare any help taken will be interpreted as academic misconduct and result in a F grade in the course.

The course will require 100% attendance. It will be hard to catch up if there are too many missed classes. Class participation will count towards grading. There will be no make-up provisions for missed quizzes and class participation, for whatever reason. Best of n-1 quizzes out of n will count toward the final grade. Make-up tests may be allowed for the midterm and final exams (only due to illness) on production of a medical certificate clearly stating that the student was not in a position to take the test. A medical prescription will not be sufficient.

• Quizzes: 30%
• Assignments and project: 40%
• Midterm examination: 15%
• Final examination: 15%