Dedekind’s Theory of Real Numbers
Most math students use real numbers without ever understanding how they're constructed. Dedekind solved this with his theory of cuts. He showed how to build the real numbers rigorously from…
20+ Free Algebra Books for Students & Teachers (PDF Ebooks)
I've compiled 20+ free algebra and topology ebooks. Not sketchy PDFs from torrent sites. Real textbooks that professors and universities have made freely available. Covers abstract algebra, linear algebra, algebraic…
D’Alembert’s Ratio Test of Convergence of Series
The D'Alembert Ratio Test is one of the most reliable tools for determining whether an infinite series converges or diverges. I cover the theorem statement, proof, worked examples, and the…
Jablonski Diagram – Consequences of Light Absorption
When a molecule absorbs light, the energy has to go somewhere. It can trigger a chemical reaction, release heat, or emit light as fluorescence or phosphorescence. The Jablonski diagram maps…
Essential Steps of Problem Solving in Mathematical Sciences
Problem solving in mathematics isn't about memorizing formulas. It's about knowing what to look for. I break down the essential steps every math student needs: understanding the problem, devising a…
Bhaskara II’s Lilavati: Complete Reference Guide (Download PDF)
Lilavati is a 12th-century mathematics treatise that reads like poetry. Written by Bhaskara II, it wraps arithmetic, algebra, and geometry inside riddles about bees, snakes, and peacocks. I've created a…
Un-Popular circumstances connected with Most Popular, Theory of Relativity
Einstein gets all the credit for relativity, but Poincare was circling the same ideas years earlier. He was trying to salvage classical physics with modifications when the Michelson-Morley experiment shattered…
The Collatz Conjecture: Unsolved but Useless
Pick any positive integer. If it's even, halve it. If it's odd, triple it and add one. Repeat. You always reach 1. At least, that's what every number ever tested…
Dirichlet’s Theorem and Liouville’s Extension of Dirichlet’s Theorem
The Dirichlet theorem connects the Gamma and Beta functions to evaluate complex multiple integrals. Liouville extended it further, making it even more powerful. I cover both theorems with complete proofs,…
The Lindemann Theory of Unimolecular Reactions
Unimolecular reactions seem impossible. A single molecule just reacts on its own? Where does the energy come from? Lindemann solved this puzzle by proposing that molecules first get energized through…
Mystery of Missing Money – 1 Rupee by Shakuntala Devi
Shakuntala Devi's missing money puzzle is a classic in mathematical logic. Two women sell marbles at different rates. When they combine forces, the math doesn't add up. Where did the…
Understanding Poincare Conjecture
In 1904, Poincare asked a deceptively simple question: if a three-dimensional shape is simply connected, must it be a sphere? This question stumped mathematicians for nearly a century until Grigori…