Prove the ratio of the length and width of the rectangle is rational.

The problem is equivlaent to the following statement:

A rectangle with sides 1 and $x$, where $x$ is irrational, cannot be "tiled" by finitely many squares.

It turns out this is a well known problem and the the proof is copied below from the following source:

http://circuit.ucsd.edu/~yhk/ece269-win18/pdfs/matousek.pdf

However I could not find the name of the author.