√2 is irrational

People long believed every length could be written as a ratio of two integers. But Baudhāyana, writing his Śhulbasūtra (a manual for building fire altars) around 800 BCE, met lengths that would not fit any fraction. The Greeks met the same problem a few centuries later.

A square of side 1 has diagonal d with 1² + 1² = d², so d² = 2 and d = √2.

Can √2 be written as p/q? No. Numbers on the number line that cannot be written as a ratio of integers are irrational numbers.

The first proof is credited to Hippasus, of the Pythagorean school. It is a proof by contradiction: assume the opposite of what you want, and show it leads to something impossible. Like a detective saying, "Suppose he was at home. Then the car could not be outside. But it was. So he was not at home."

The proof, step by step:

• assume √2 = p/q in lowest terms (p and q co-prime);
• square: 2 = p²/q², so 2q² = p²;
• so p² is even, which means p is even: write p = 2k;
• then 2q² = 4k², so q² = 2k², and q is even too.

But now p and q share the factor 2, and we said they had no common factor. Every step was correct, so the starting assumption must be false: √2 is irrational.

The same method proves √3, √5 and √7 irrational (use 3, 5 or 7 in place of 2), and √10 (use 2 or 5). The key fact each time: if a prime divides p², it divides p.