ApproximatingPi

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 $\pi$ can be approximated as 22/7, but did you know that a better approximation is $\frac{355}{113}$ ? 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?

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?

Approximating Pi

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