The square root of 2 can be written as an infinite continued fraction.
√2 = 1 +
1
2 +
1
2 +
1
2 +
1
2 + ...
The infinite continued fraction can be written, √2 = [1;(2)], (2) indicates that 2 repeats ad infinitum. In a similar way, √23 = [4;(1,3,1,8)].
It turns out that the sequence of partial values of continued fractions for square roots provide the best rational approximations. Let us consider the convergents for √2.
1 +
1
= 3/2
2
1 +
1
= 7/5
2 +
1
2
1 +
1
= 17/12
2 +
1
2 +
1
2
1 +
1
= 41/29
2 +
1
2 +
1
2 +
1
2
Hence the sequence of the first ten convergents for √2 are: