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 the rationals, filling in all the gaps. I walk through the construction step by step, making this foundational concept accessible.
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 topology, and more. Each link is verified and the books range from introductory to graduate level.
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 cases where it fails. If you’re studying real analysis or calculus, you’ll use this test constantly alongside the comparison and root tests.
The Jablonski diagram maps what happens to a molecule after it absorbs light. Fluorescence, phosphorescence, internal conversion, intersystem crossing — every pathway explained with diagrams, timescales, and real-world applications.
George Polya’s four-step method has guided mathematicians since 1945. Understand the problem, devise a plan, carry it out, look back. Here’s how it works in practice, from basic algebra to research-level mathematics, with worked examples and the heuristics that textbooks skip.
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 complete reference guide with downloadable PDF, covering the historical context, mathematical content, and enduring legacy of this remarkable text.
Poincare had the math. Einstein had the physics. The story of how two thinkers arrived at the same transformations from opposite directions, and why only one of them built a new theory of space and time.
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 does. Nobody can prove it. The Collatz Conjecture is simple enough for a child to understand and hard enough that no mathematician has cracked it.
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, examples, and practical applications. If you’re studying real analysis or probability theory, you’ll use these results constantly.
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 collisions, then react independently. I walk through the theory, the math, and the limitations that led to more refined models.
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 missing rupee go? I break down this elegant puzzle and reveal why our intuition about averages misleads us.
In 1904, Poincare asked: if every loop on a 3D shape can be shrunk to a point, must it be a sphere? It took 99 years. Perelman proved it using Ricci flow, then declined both the Fields Medal and the $1 million Millennium Prize.