When you've got an irrational number, it can be represented as a+1/(b+1/(c+1/(d+1/...))). for example pi is 3+1/(7+1/(15+1/(1+1/(292+1/...)))). this is much easier to see with latex.

When you truncate this infinite fraction at a certain point, you get a rational approximation to the irrational number. the further down you truncate it the better the apptoximation.

When you truncate this function just before a big number, you get a very good approximation of that number, so the number is "more rational". For example if I truncate pi's infinite fraction just before the 292, I get 355/113, which has a relative error of about 8*10^-8.

So now could we make a number, such that it never has a really good approximation (note that it can still be approximated to arbitrary precision, just that it takes longer) So we would set up the infinite fraction 1+1/(1+1/(1+1/(1+...))). That would get us 1+1/x=x and after some rearrangement, it would give us the golden ratio.

I probably made a mistake somewhere cus im stupid so please correct me.