Had there been two naturals a and d, such that d/a=SQRT(2), then there would be a square with the edge a and the diagonal d.
Let’s say, that this was the smallest such square of them all. The diagonal d must be an even number since d^2 is even, for it is the sum of two equal natural numbers: a^2 + a^2.
As we know, the diagonals of squares cut each other in half. Therefore the half of the d (which is even) is a natural number, too! The square with the edge d/2 apparently has the diagonal a!
A square with half the area of the original square that has an integer edge and an integer diagonal! This contradicts the above premise.
That is the shortest royal road to understanding why the SQRT(2) isn’t a rational number. At least as far as I am aware of. It’s just a small modification of the Tom Apostol’s proof from the year 2000. In a nutshell, this picture tells most of the story.
Any integer-edge-diagonal-square would imply a half smaller integer-edge-diagonal-square. The smallest one of them would imply the same. Consequently no such square exists. Therefore the square root of 2 can’t possibly be a rational number!
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