20 Best Calculus Books [Best Sellers Included]
Calculus textbooks can make or break your understanding of the subject. I struggled with calculus the first time around because I had the wrong book. After years of studying and…
Calculus
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…
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…
Everywhere Continuous Non-differentiable Function
Weierstrass shocked the mathematical world in 1872 by constructing a function that's continuous everywhere but differentiable nowhere. Not at a few points. At no points whatsoever. I walk through the…
Free Online Calculus Text Books
I've spent years collecting free calculus textbooks. Not scanned copies with missing pages. Real textbooks that professors and universities have made freely available online. This curated list covers single-variable calculus,…
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…
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,…
Derivative of x squared is 2x or x ? Where is the fallacy?
Here's a mathematical fallacy that trips up even sharp students. Can you prove that the derivative of x squared is x instead of 2x? I'll show you the flawed proof,…
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…