Editing ApproximatingPi

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[[[>50 In 1706 mathematician William Jones, born
in the small village of Llanfihangel Tre'r Beirdd
on Anglesey, became the first person to use the
16th letter of the Greek alphabet to represent
the ratio of the circumference of a circle to
its diameter. Previously the ratio was known as
the Ludolphian number, after Ludolph van Ceulen,
a German mathematician. ]]]
We all know that EQN:\pi can be approximated as 22/7, but
did you know that a better approximation is EQN:\frac{355}{113} ?
That's better than one part in 10 million, which seems
unreasonably good.

Questions:
* Where does that come from?
** Continued fractions
* Can every number be approximated?
** Yes
* Why are some numbers better approximated than others?
** Because some are "closer" to rationals.
* Are there any with especially bad approximations?
** Yes, the Golden Ratio is especially bad.
** To see this, write down the continued fraction with /*no*/ large numbers:
*** [1;1,1,1,1,...]
** What does that evaluate to?

See also Proof By Contradiction.
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CategoryMaths