the meaning of rﾕ.......ï¾

[Steven D'Aprano]

On Wed, 25 Jul 2012 13:07:03 +1000, Ben Finney wrote:

> Maarten <maarten.sneep at knmi.nl> writes:
> 
>> You just missed it:
>> 22/7
> 
> Which is appropriate, since 22/7 misses π by a wide margin. (355/113 is
> my favourite approximation to π, and is far more accurate.)

Approximation? Pffft. I use the exact value:

π = 16*arctan(1/5) - 4*arctan(1/239)


For those lacking ambition, another approximation is:

π ≈ 9801 ÷ (2206*√2)

Other interesting approximations are:

π ≈ 9/5 + √(9/5)

π ≈ √√(2143/22)

See more here:

http://mathworld.wolfram.com/PiFormulas.html
http://mathworld.wolfram.com/PiApproximations.html

 
> −1 on associating 22/7 as anything to do with π.

I believe that 22/7 is the closest approximation to π with a denominator 
under 100. It is the second convergent of the continued fraction:

π = [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ...]

which means that, like it or not, 22/7 *is* associated with π.



-- 
Steven