Two Proofs of the Irrationality of the Square Root of 2

The classical algebraic proof and a geometric proof

A consequence of the Pythagorean theorem

The Pythagorean theorem states that the square constructed on the hypotenuse of a right triangle (side c of the triangle in the following image) equals the sum of the squares constructed on the legs (sides a and b in the image). From this identity, it follows that c is equal to the square root of a² + b².

Given any unit of measurement, if the two legs of a right triangle are both 1 unit long, according to the Pythagorean theorem, the square constructed on the hypotenuse c must be equal to 1² + 1², that is, 2. It follows that c is equal in this case to the square root of 2.

Since sides a and b are equal, we can build a square with side 1, whose diagonal has a length equal to the square root of 2.