π (pi) ≈ 3.14159265…, e (Euler's number) ≈ 2.71828182…, φ (the golden ratio) ≈ 1.61803398… All three are irrational numbers, with decimal expansions that go on forever without ever repeating. But split them into finer categories, and π and e belong to a special class called transcendental numbers, while φ doesn't. Why are these irrational numbers treated so differently?
First, let's look at where these numbers come from. π is a circle's circumference divided by its diameter. That ratio stays exactly the same no matter how big or small the circle is: measure the diameter around the circumference, and it fits about 3 times plus a bit more (0.14159… of the way around). e comes from "a limit that keeps growing endlessly." At a bank paying 100% interest per year, getting the interest just once doubles your principal, but the more finely you split up the interest — paying it more often — the larger the final amount grows. Yet no matter how finely you split it, the amount never crosses a certain value — that limiting value is e. As a formula, it's what (1+1/n)ⁿ approaches as n grows infinitely large. φ is the value that the ratio between two neighboring numbers in the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13…) approaches. Divide two large Fibonacci numbers, and the result always gets closer to 1.618…
Now let's see why these three fall into different classes. In mathematics, every number is first split into rational numbers (numbers that can be written exactly as a fraction) and irrational numbers (numbers that can't). Irrational numbers then split into two further branches: an algebraic irrational can be a solution to some 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 the equation x² − 2 = 0, so it's an algebraic number. φ works the same way — writing out its definition as the limit of the Fibonacci ratio turns into the equation x² − x − 1 = 0, so it's irrational, but not transcendental.
π and e, on the other hand, have been proven to never be a solution to any equation with integer coefficients, no matter how you build one. That π is transcendental was proven in 1882 by the German mathematician Lindemann, and that e is transcendental was proven even earlier, in 1873, by the French mathematician Hermite. One reason this proof became so famous is that it settled, in one stroke, an ancient Greek problem over 2,000 years old — "squaring the circle," the question of whether you can construct a square with the same area as a circle using only a straightedge and compass. If π is transcendental, that construction is proven to be impossible in principle.
To sum up, not all irrational numbers are alike. There are "orderly" irrational numbers like √2 or φ that are solutions to some polynomial equation, and there are "completely untethered" irrational numbers like π and e that no polynomial equation can ever pin down. Interestingly, it's also been proven that among all real numbers, transcendental numbers vastly outnumber algebraic ones — though numbers actually proven to be transcendental, like π and e, remain rare enough to count on your fingers.
On our activity page, you can change the radius to see how π emerges as the ratio of a circle's circumference to its diameter, increase n to watch (1+1/n)ⁿ approach e, change the Fibonacci index to see the ratio converge on φ, and press buttons on various numbers to try sorting them into rational numbers, algebraic irrationals, and transcendental numbers yourself.