4 ms·
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-comput
by green_on_black 4y ago
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 4y agoFor 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?
- eru 4y agoYou might find a comparison to the continued fraction for the golden ration enlightening.
- alanbernstein 4y agoHow does that explain the much bigger error in the approximation with an incremented denominator, 358/114?