The numbers that you're producing can be better produced by the following Python code: from fractions import Fraction as frac def continued_fraction(x,

The numbers that you're producing can be better produced by the following Python code: from fractions import Fraction as frac def continued_fraction(x, n=10): """Represent x as a continued fraction out to n places.""" last = int(x) out = [] for i in range(n): x = 1/(x - last) last = int(x) out.append(last) return out def bras(x, n=10): """Compute the best rational approximations to x.""" base = int(x) nums = continued_fraction(x, n) S = lambda depth, i = 0: \ frac(1, nums[i]) if depth == 0 else 1 / (nums[i] + S(depth - 1, i + 1)) return ', '.join(str(x) for x in (base,) + tuple(base + S(k) for k in range(n))) from decimal import Decimal print(bras(Decimal('3.1415926535897932384626433832795028841971693993'))) # 3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, 208341/66317, # 312689/99532, 833719/265381, 1146408/364913, 4272943/1360120 These are the best rational approximations to pi, which are determined by the continued fraction of pi: pi ~= 3 + 1/(7 + 1/(15 + ...)) pi = 3 + [7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ...] Large numbers are opportunities to "grab a lot of extra digits" and thus the large qualities come when we truncate after 7, 15, 292, and 14. You may wonder what the "most irrational" number is, in the sense that all of its best-rational-approximations are of low quality. That distinction belongs to: phi = (1 + Decimal(5).sqrt()) / 2 # golden ratio continued_fraction(phi) # [1, 1, 1, 1, 1, 1, 1, 1, 1, 1] bras(phi) # 1, 2, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, 89/55, 144/89 Those last numbers you may recognize from your intro to programming as the Fibonaccis.