>I've nothing to add. We seem to use different definitions of
>what a ratio or a rational number is

For the benefit hopefully, perhaps, to some other visitor to this
forum:

1.5 = 1 1/2 = 3/2 = 3 : 2 ratio

1,414285714... = 1 29/70 = 99/70 = 99 : 70 ratio.

1,414201183... = 1 70/169 = 239/169 = 239 : 169 ratio.

1,414213562... = (?2) = e.g. 1 13860/33461 = 47321/33461 =
47321 : 33461 ratio.

1,414225941... = 1 99/239 = 338/239 = 338 : 239 ratio.

1,414141414... = 1 41/99 = 140/99 = 140 : 99 ratio.

1.4 1 2/5 = 7/5 = 7 : 5 ratio.

1.333333333... = 1 1/3 = 4/3 = 4 : 3 ratio.

99/70 x 140/99 = 2. 239/169 x 338/239 = 2.
47321/33461 x 47321/33461 = 2. (= 1 x 1 = 1 : 1 ratio.

Why should the square root be treated different to the others?
And why make life so difficult?

Perhaps I should check out the sci.math after all, what do you think?
Kind regards
Klaus Kastberg