Video narration
Can the diagonal of a square be written exactly as a fraction? Take a square with side length one. By the Pythagorean theorem, its diagonal squared is 12 + 12 = 2, so the diagonal is √2. Now assume √2 = p/q, with p as numerator and q as denominator, and the fraction in lowest terms. Square both sides: p2 = 2q2. So p2 is even, which means p is even: an odd number squared is odd. Set p = 2k. Then 4k2 = 2q2, so q2 = 2k2, and q is even too. Now the setup collapses: both p and q are even. That contradicts lowest terms, because both share a factor of two. No such fraction exists. √2 is irrational: it is not equal to any fraction, not merely a decimal that continues. So, can the square's diagonal be exactly written as a fraction? No. That diagonal, and therefore √2, cannot be written as any fraction of integers.