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.

Posted by JayDeeKay on 2015-02-17 18:32:48