Tom Apostol in 2000 gave a geometric proof of the irrationality of 2, which for a bright student can even be considered as a “proof without words”:
(The figure has been taken from wikipedia)
The explanation is as follows: Suppose is rational and equals where with . Consider an isoceles right with legs of length and hypotenuse . Draw the blue arcs with center . Observe that is congruent to by SAS, and furthermore note the various lengths given in the figure follow from this fact. Hence we have an even smaller right isosceles triangle , with hypotenuse length and legs . The fact these values are integers even smaller than and and in the same ratio is clear from the figure and thus we contradict the hypothesis that .
The above proof may be aptly summarized in one line: If is in lowest terms then is in lower terms. In this case however the definition of the square root is implicitly used to show that and .