after a little fooling around, here's a something that i found somewhat interesting:
let S be the set of all "sqare" numbers {s such that s = n^2, where n in N}
let T be the set of all "triangular" numbers {t such that t = (m^2 + m)/2, where m in N}
let W be the intersection of S and T, whose elements w satisfy both w = n^2 and w = (m^2 + m)/2 where both n and m are in N.
now take the ith element of the set W, wi which satisfies wi = ni^2 and w = (mi^2 + mi)/2
it can be shown that:
[math]\lim_{i\to\infty} \frac {m_i}{n_i} = \sqrt{2}[/math]
since ni and mi are integers, it is almost imaginable that sqrt(2) can be expressed as a ratio of integers, as long as the integers are infinte. at least now we know the ratio of these infinite integers.