The study of patterns, structures, and abstract reasoning. Mathematics underpins science, engineering, computer science, and countless other fields.
Foundations
- Mathematical Proofs — Proof techniques, logic, and mathematical reasoning
- Discrete Mathematics — Logic, sets, combinatorics, graph theory
Core Analysis
- Calculus — Differentiation, integration, sequences and series
- Real Analysis — Rigorous foundations of calculus, limits, continuity
- Complex Analysis — Complex numbers, analytic functions, contour integration
Algebra
- Linear Algebra — Vectors, matrices, transformations, eigenvalues
- Abstract Algebra — Groups, rings, fields, Galois theory
Applied Mathematics
- Differential Equations — ODEs, PDEs, Fourier series
- Probability & Statistics — Probability theory, distributions, inference
Advanced Topics
- Number Theory — Primes, modular arithmetic, cryptographic applications
- Topology — Topological spaces, continuity, connectedness
Learning Path
A recommended progression through mathematical topics:
1. Prerequisites
- High school algebra and geometry
- Precalculus (functions, trigonometry)
2. Foundations
- Mathematical Proofs — Essential for all higher mathematics
- Discrete Mathematics — Builds logical thinking
3. Core
- Calculus — The language of change and accumulation
- Linear Algebra — Essential for applied mathematics and computer science
4. Intermediate
- Real Analysis — Rigorous understanding of calculus
- Abstract Algebra — Algebraic structures and symmetry
- Differential Equations — Modelling dynamic systems
5. Advanced
- Complex Analysis — Extends calculus to the complex plane
- Number Theory — Properties of integers, cryptography
- Topology — Abstract study of space and continuity
Popular Mathematics Books
Accessible books for exploring mathematical ideas:
- e: The Story of a Number by Eli Maor — History and significance of Euler’s number
- The Joy of X by Steven Strogatz — A guided tour of mathematics
- Fermat’s Enigma by Simon Singh — The story of Fermat’s Last Theorem
- A Mathematician’s Apology by G.H. Hardy — Reflections on mathematics as art
- Flatland by Edwin Abbott — A mathematical fiction exploring dimensions
- Proofs from THE BOOK by Aigner and Ziegler — Elegant proofs of fundamental theorems
Resources
Online Courses
- Khan Academy Mathematics — Free courses from arithmetic to calculus
- MIT OpenCourseWare Mathematics — University-level courses
- Brilliant — Interactive problem-solving
Visual Learning
- 3Blue1Brown — Beautiful visual explanations of mathematical concepts
Reference
- Paul’s Online Math Notes — Comprehensive calculus and differential equations notes
- Math Stack Exchange — Q&A community
Practice
- Project Euler — Mathematical and programming challenges