Difference Paradox
The Difference Paradox is one of those mathematical puzzles that makes you question your arithmetic. The setup is simple, the calculations seem correct, but the answer is clearly wrong. I…
Mathematics
Square Integrable Functions, Norms, Trial Method in Integral Equations
Building on integral equation fundamentals, this article covers square integrable functions, norms, and the trial method for solving integral equations. I explain L2 spaces, why they matter for integral equation…
Definitions and Types of Integral Equations
Integral equations look intimidating, but they're built from ideas you already know. I explain what integral equations are, their classification system (Fredholm vs Volterra, first vs second kind), and how…
Real Sequences: Definitions, Theorems, and Examples
Real sequences form the backbone of real analysis and advanced calculus. I've taught this topic to dozens of students, and the key is grasping what sequences represent: ordered lists of…
Irrational Numbers and The Proofs of their Irrationality
How do you prove a number is irrational? You can't check infinitely many decimal places. You need a proof. I present the classic proofs of irrationality for square root of…
The Area of a Disk
The area of a disk is pi times radius squared. You memorized this in middle school. But do you know why it's true? Most people don't. I show you the…
Triangle Inequality
The triangle inequality sounds obvious. No side of a triangle can be longer than the other two sides combined. But proving it rigorously is where things get interesting. I cover…
Find the Day of a Date using this Calendar Formula
You can figure out what day of the week any date falls on. No calendar needed. Just arithmetic. I'll teach you a genuine mathematical formula that works for any date…
Equations: A Basic Introduction
Equations are the reason we can solve problems in seconds that would take hours of guesswork. I cover the basics: what equations are, how they work, types of equations, and…
Numbers – The Basic Introduction
Numbers are the alphabet of mathematics. But here's what most people never learn: all the complex numbers you encounter, from imaginary numbers to quaternions, are built from a few simple…
Mathematical Logic – The basic introduction
Logic is the grammar of mathematics. You can have brilliant mathematical intuition, but without logical precision, you can't communicate your ideas or verify they're correct. I walk you through propositions,…
Fermat Numbers (Important Theorems & Examples)
Fermat was convinced he'd found an infinite source of primes. He was wrong. His formula n^2 + n + 41 produces primes for the first 40 values, then fails spectacularly.…