Difference Paradox
Some of the most fascinating moments in mathematics come not from solving hard problems, but from stumbling into paradoxes. A mathematical paradox is a statement (or pair of statements) that…
Mathematics
Square Integrable Functions, Norms, Trial Method in Integral Equations
In this article, I will teach you about Square Integrable Functions, Norms and the solution of Integral Equations by the trial method. Don't forget to refer: Definitions and Types of…
Definitions and Types of Integral Equations
In this article, I will explain what Integral Equations are, how they are structured and what are certain types of Integral Equations. What is an Integral Equation? An integral equation…
Real Sequences: Definitions, Theorems, and Examples
Real sequences form the backbone of real analysis and much of advanced calculus. I've taught this topic to dozens of students over the years, and I've found that the key…
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…
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'm not talking about memorizing tricks or using apps. I mean a…
Equations: A Basic Introduction
Math teachers love to say "let x equal the unknown." But nobody explains why that simple trick is so powerful. Here's the thing: equations are the reason we can solve…
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…
Mathematical Logic – The basic introduction
If mathematics is a language, then logic is its grammar. You can have brilliant mathematical intuition, but without logical precision, you can't communicate your ideas or verify they're correct. Let…
Fermat Numbers (Important Theorems & Examples)
Fermat was convinced he'd found an infinite source of prime numbers. He was wrong. Here's the fascinating story of Fermat Numbers—and why one of history's greatest mathematicians got fooled by…