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 \(\sqrt{2} = a/b\) 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 \(\sqrt{2}\) is irrational.

Added on 06 July 2018: Nicolas Miari wrote in a comment on Quora:

First time I’ve seen a “Visual proof by infinite descent”!

In that case, you will have fun in turning this diagram into an infinite descent proof of irrationality of \(\sqrt{2}\):

The quickest proof by infinite descent is perhaps proof of irrationality of the Golden Ratio (I found it in a beautiful little book by Tim Gowers Mathematics: A Very Short Introduction).

By definition, the rectangle \(ABCD\) is a golden rectangle, if after cutting off the square \(B’BCC’\), the remaining rectangle \(ADC’B’\) is similar to the original one, \(ABCD\), see the picture:

The ratio of lengths of sides of a golden rectangle is called the golden ratio.

Theorem. The golden ratio is irrational.

Proof.  If the golden ratio were rational, a “golden rectangle” could be drawn on square grid paper, as on picture above.  After cutting a square from it we get a smaller “golden rectangle” drawn on square grid paper. By principle of infinite descent, this is impossible – hence the golden ratio is irrational.

Hmm, we did not even care about the numeric value of the golden ratio  \(\dots\)