π (pi), e (Euler's number), φ (the golden ratio)... You've heard the names, but exactly what kind of numbers are they? Let's look at where these numbers come from, why their decimals go on forever, and what makes some of them especially mysterious "transcendental numbers."
Every number splits broadly into rational numbers (numbers that can be written exactly as a fraction) and irrational numbers (numbers that can't). Irrational numbers, though, come in two kinds — an algebraic irrational can be a solution to an equation with integer coefficients, while a transcendental number can never be a solution to any such equation.
For example, √2 is irrational, but it's a solution to x²−2=0, so it's an algebraic number. π and e, on the other hand, have been proven to never be a solution to any equation you build, no matter what — which is why they're called "transcendental."
Divide a circle's circumference by its diameter, and you always get the exact same value, no matter the circle's size. That value is π ≈ 3.14159265... Measure the diameter around the circumference, and you'll see it fits about 3 times, plus a bit more.
Imagine a bank that pays 100% interest per year. Pay it out just once, and your principal doubles — but split the interest into smaller, more frequent payments, and the final amount keeps growing. Yet no matter how finely you split it, the amount never crosses a certain value — that limit is e ≈ 2.71828182...
Keep computing the ratio of two neighboring numbers in the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13...), and it gets closer and closer to φ ≈ 1.61803398... String together squares sized to Fibonacci numbers, and you naturally get a rectangle close to the golden ratio.
As shown in the diagram below, real numbers split into rational and irrational numbers, and irrational numbers split further into algebraic irrationals and transcendental numbers. Press a button to see which category each number belongs to.