# Tag Archives: continued fractions

## Percentages for sceptics: part III

I wanted to do some self-criticism of my previous two posts in this series:

1. You can calculate the minimum of responses from a single percentage by hand (no need for computer programmes or look-up tables).
2. I’ve made a very rough model to estimate how many people the program typically returns when fed six percentages (as I did several times here).
In between, I’ve collected some links to demonstrate how great continued fractions can be.

Handy calculations

There are many ways of writing real numbers (fractions and irrationals) apart from in decimal notation. You can represent them in binary, for instance $\pi = 11.001001000011\ldots$, or in other bases. These have their uses: there is a formula to calculate the $n$th binary digit of $\pi$ without calculating all the preceding digits.

For our purposes we will use continued fractions. People write $\pi$ as $[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, ...]$: this notation means that 3 is a good first approximation to $\pi$, the well-known $3+\frac{1}{7}=\frac{22}{7}$ is the closest you can be with any fraction $\frac{p}{q}$ with $q \leq 7$. Then $3+\frac{1}{ 7 + \frac{1}{15}}=\frac{333}{106}$ is the best with $q\leq 106$, and the fourth term

$3+\frac{1}{7+\frac{1}{15+\frac{1}{1}}}=\frac{335}{113}$

is a very good approximation, as the next number in the square brackets, 292, is very large (I’ll motivate this observation at the end of the section). The golden ratio is sometimes called the ‘most irrational’ number because it has a continued fraction expansion with all ones, so the sequence converges slowly.