20 Best Calculus Books [Best Sellers Included]
Fresh into Calculus? Are you looking for the best calculus books that you should start with? Well, when I was your age, I had tried several books from the library,…
Calculus
Irrational Numbers and The Proofs of their Irrationality
How do you prove a number is irrational? You can’t just look at a number and declare it irrational. You can’t check infinitely many decimal places. You need a proof.…
The Area of a Disk
The area of a disk with radius \( R \) is \( \pi R^2 \). You probably memorized this in middle school. But do you know why it’s true? Most…
Triangle Inequality
The triangle inequality is one of those results that sounds obvious until you try to prove it rigorously. Here’s the geometric intuition: no side of a triangle can be longer…
Numbers – The Basic Introduction
Numbers are the alphabet of mathematics. Logic is the grammar. And everything we build in math, from simple arithmetic to quantum field theory, comes from arranging these symbols according to…
Everywhere Continuous Non-differentiable Function
Weierstrass introduced a shocking idea in 1872. There exist functions that are continuous everywhere but differentiable nowhere. Not just at a few points. At no points whatsoever. This was a…
Free Online Calculus Text Books
I’ve spent years collecting free math resources, including a variety of calculus text books. Not the sketchy PDFs from torrent sites. Not the scanned copies with missing pages. Real Calculus…
D’Alembert’s Ratio Test of Convergence of Series
The D’Alembert Ratio Test is one of the most useful tools for determining whether an infinite series converges or diverges. If you’re studying real analysis or calculus, you’ll use this…
Dirichlet’s Theorem and Liouville’s Extension of Dirichlet’s Theorem
The Gamma function is one of the most important special functions in mathematics. It extends the factorial to non-integer values. The Beta function is its close relative, and together they…
Derivative of x squared is 2x or x ? Where is the fallacy?
Derivative of x squared As we know, the derivative of x squared, i.e., differentiation of $ x^2$ , with respect to $ x$, is $ 2x$. i.e., $ \dfrac{d}{dx} x^2…
Solving Ramanujan’s Puzzling Problem
Here’s a problem that looks impossible at first glance. You’ve got infinitely many nested square roots, each level multiplied by an increasing integer. How do you even begin to evaluate…