# A geometric proof for the irrationality of √2

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 $\sqrt{2}$ is rational and equals $m/n$ where $m,n\in\mathbb{N}$ with $(m,n)=1$. Consider an isoceles right $\vartriangle ABC$ with legs of length $n$ and hypotenuse $m$. Draw the blue arcs with center $A$. Observe that $\vartriangle ABC$ is congruent to $\vartriangle ADE$ 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 $\vartriangle CDF$, with hypotenuse length $2n-m$ and legs $m-n$. The fact these values are integers even smaller than $m$ and $n$ and in the same ratio is clear from the figure and thus we contradict the hypothesis that $(m,n)=1$.

The above proof may be aptly summarized in one line:  If $\sqrt{2}=m/n$ is in lowest terms then $\sqrt{2}=(2n-m)/(m-n)$ is in lower terms. In this case however the definition of the square root is implicitly used to show that $2n-m and $m-n.