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…
Study Notes
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…
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…
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…
Wein’s Formula & Wein’s Laws
Wien's displacement law and Wien's distribution law are fundamental to understanding blackbody radiation. This guide covers both formulas, their derivations, and how they predict the relationship between temperature and peak…
Solving Ramanujan’s Puzzling Problem
Ramanujan's nested radical problem looks impossible at first glance. Infinitely many nested square roots, each multiplied by increasing integers. How do you evaluate that? I walk through the solution step…