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.
It's certainly true for pi (355/113), e (878/323), 2^(1/2) (816/577),
3^(1/2) (989/571), etc. All of these except for pi have multiple
approximations (with denominators less than 1000) that are better than
the five-digit decimal counterparts.
Consider also that some five-digit decimal approximations are actually
quite _bad_ because of rounding (take, say, 3^(1/2)).
--
Erik Max Francis && max@[EMAIL PROTECTED]
&& http://www.alcyone.com/max/
San Jose, CA, USA && 37 18 N 121 57 W && AIM, Y!M erikmaxfrancis
The surface of the Earth is the shore of the cosmic ocean.
-- Carl Sagan
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