32 THE SQUARE ROOT OF TWO The square root of 2, which was the first irrational to be discovered, was known to the early Pythagoreans, and ingenious methods of approximating to its value was discovered. The best was as follows: Form two columns of numbers, which we will call the a ’s and the b ’s; each starts with 1. The next a , at each stage, is formed by adding the last a and b already obtained; the next b is formed by adding twice the previous a to the previous b . The first 6 pairs so obtained are (1,1), (2,3), (5,7), (12,17), (29,41), (70,99). In each pair, 2a2 b 2 is 1 or 1. Thus b/a is nearly the square root of two, and at each fresh step it gets nearer. For instance, the reader may satisfy himself that the square of 99/70 is very nearly equal to 2. [from Bertrand Russell, History of Western Philosophy ] (a, b), (a’, b’), … a’ = a + b b’ = 2a +b = b/a