[Math] Prove the ratio of the length and width of the rectangle is rational.

Question

Assume there is a rectangle be combined by finite squares, and the small squares are not of equal size. Also, the lengths of the squares may be irrational.

The question is "Can we know the ratio of the length and width of this rectangle is rational ?"

I guess the answer is "yes!"(by considering many cases). However, I have no idea to prove it.

Answer 1

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.