You can approximate irrational square roots with rational numbers using linear interpolation between the integers as follows:

√1 = 1

√2 ≈ 4/3

√3 ≈ 5/3

√4 = 2

√5 ≈ 11/5

√6 ≈ 12/5

√7 ≈ 13/5

√8 ≈ 14/5

√9 = 3

etc...

How good an approximation is this?

For large numbers, might the previous or next fraction be a better approximation?

Answer to the question: in general, the rational approximations achieved by linear interpolation can always be improved on with continued fractions (other 4, 9, 16, etc., of course) - precisely as Charlie has observed.