[green_on_black]

It seems to me that an estimation of any computable number would scale with at least but not equal to O(n) of its denominator. Otherwise, it would be non-computable (please correct me, this is just my intuition). For 355/113 to be more than 16x closer to pi than 22/7 would be a matter of course. Furthermore, 1/791 = 1/7 * 1/113, and 355 is the floor of 22 * 7/113, as one might except of a slightly closer approximation. Not to undervalue 355/113. But (personally) intuitively, it's not particularly mysterious.

[sgtnoodle]

For irrational numbers, you mean? A rational number is one that can be perfectly represented by a fraction. I'm not sure I follow your use of big-O notation here, but are you saying that the bound of the error of a rational approximation of an irrational number is inversely proportional to the denominator?

[alanbernstein]

How does that explain the much bigger error in the approximation with an incremented denominator, 358/114?

[eru]

You might find a comparison to the continued fraction for the golden ration enlightening.