A familiar yet fascinating number: √2. We all know it. But did you know that, once upon a time, mathematicians lost sleep trying to prove why the square root of 2 irrational?
A new proof recently surfaced. Well, “new” might be a stretch—but its approach is definitely different. The classic proof says: suppose √2 = p/q in lowest terms. Turns out, both p and q must be even. That’s a contradiction. Elegant and direct.
This new proof? It starts the same way: assume √2 is rational, so we can write it as p/q. Then we define a set of positive integers that, when multiplied by √2, give natural numbers. Let’s call the smallest such number k.
Now here’s where it gets fun. Take the expression k*(√2 – 1). Crunch the numbers, do the math, and you end up with another natural number. But this new number is smaller than k. Wait… wasn’t k supposed to be the smallest? Boom. Contradiction. That means our assumption—that √2 is rational—goes out the window.
What makes this proof charming? It leans on real numbers, sets, and concepts like infimum. It doesn’t play the “even or odd” game. Instead, it calls the idea of “the smallest element” to the stage and lets the contradiction unravel from there.
Now let’s be honest: this isn’t a proof you show high school students on a sleepy Tuesday morning. You have to accept the existence of real numbers. You have to understand sets. It’s a bit more mature, mathematically speaking.
Still, it’s beautiful. Why? Because reaching the same result through a different path is always worth celebrating in math. Seeing multiple routes stretches how we think. And if you can formalize that route in a system like Isabelle/HOL, then math becomes part art, part engineering.
If you’re into clever and creative proofs like this, you might enjoy browsing this list: 30+ Best Math Proof Books to Learn Mathematical Thinking
Bottom line: √2 is irrational. This time we proved it a different way. Same destination, new perspective. And sometimes, that’s all it takes to keep things exciting.
