Posted by: Alexandre Borovik | July 5, 2018

What is your favorite visual mathematical proof? Perhaps it is Tennenbaum’s proof of the irrationality of the square root of 2

My answer to a question in Quora, slightly edited:

What is your favorite visual mathematical proof?

Perhaps it is Tennenbaum’s proof of the irrationality of the square root of 2. What follows is reposted from a blogpost by David Richeson at his wonderful blog Division by Zero.

Suppose \(\sqrt2 = a/b\) for for some positive integers a and b. Then \(a^2=2b^2=b^2+b^2\). Geometrically this means that there is an integer-by-integer square (the pink  \(a\times a\) square below) whose area is twice the area of another integer-by-integer square (the blue \(b \times b\) squares).

Assume that our \(a\times\) square is the smallest such integer-by-integer square. Now put the two blue squares inside the pink square as shown below. They overlap in a dark blue square.

By assumption, the sum of the areas of the two blue squares is the area of the large pink square. That means that in the picture above, the dark blue square in the center must have the same area as the two uncovered pink squares. But the dark blue square and the small pink squares have integer sides. This contradicts our assumption that our original pink square was the smallest such square. It must be the case that \(\sqrt2\) is irrational.

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