There is the binomial approximation (1 - x)[suP]a[/suP] ≈ 1 - ax
In the case of sqrt(0.111111) we have:
sqrt(0.111111) = (1/9 - 1/9000000)[suP]1/2[/suP]
=1/3(1 - 1/1000000))[suP]1/2[/suP]
≈ 1/3(1 - 1/20000000) = 0.3333333333… - 0.0000001666666666…
= 0.3333331666666…
So that explains the “1666” bit.
The “24999” bit will be explained by the next term in the binomial expansion, and so on.