Erik Max Francis wrote:
> Jack Tingle wrote:
>
>> Yesterday being pi-day (3/14 at approximately 1:59:27), I tried a
>> little exercise. The most commonly used approximations for this nasty
>> little number are 3.1416, 22/7, and 355/113. Since pi is 3.14159...
>> 3.1416 is a pretty good approximation, and only requires you to
>> remember five digits. 22/7 only needs 3 digits, while 355/113 needs a
>> prodigious act of memory on SIX whole digits [shocked muttering from
>> audience].
>>
>> Their relative merits, taking 3.14l6 as the baseline, has 22/7 with
>> 172x the error of our baseline. Amazingly, hexadigital 355/113 has
>> only 3.6% of the error of the best 5 digit champion, 3.1416!
>
> Actually, it shouldn't be all that surprising. Consider that 3.1416 is
> just another way of writing 31416/10000. In a sense, that is an
> arbitrary denominator, one chosen just because of our predilection for
> base 10 numbers. When you instead choose a denominator designed to give
> you a good approximation -- even if you use fewer digits -- then chances
> are good you can find a better approximation than that arbitrary one.
All well, good, and interesting, but for practical use, 355/113 seems
the clear winner in utility and parsimony. With 3.1416 as a distant
runner-up. Math in alternate bases is time consuming in the Real World
(TM).
Regards,
Jack Tingle
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