# Decimal Expansions

> Source: https://ollybritton.com/notes/random/decimal-expansions/ · Updated: 2024-08-12 · Tags: maths, notes

> From ["things worth memorising"](https://www.reddit.com/r/slatestarcodex/comments/dhqexw/things_worth_memorizing/f3ps38f?utm_source=share&utm_medium=web2x&context=3). See also [The Book of Numbers, Conway](https://ollybritton.com/notes/books/the-book-of-numbers/) which explains why this happens and also links it to repeated riffle-shuffling.

##### What is $\frac{1}{7}$ as a decimal??
$$
0.\overline{142857}
$$

##### What is $\frac{3}{7}$ as a decimal??
$$
0.\overline{428571}
$$

##### What is $\frac{2}{7}$ as a decimal??
$$
0.\overline{285714}
$$

##### What is $\frac{6}{7}$ as a decimal??
$$
0.\overline{857142}
$$

##### What is $\frac{4}{7}$ as a decimal??
$$
0.\overline{571428}
$$

##### What is $\frac{5}{7}$ as a decimal??
$$
0.\overline{712485}
$$

##### What is the order of magic $7$-ths??
$$
1, 3, 2, 6, 4, 5
$$

##### What is $\frac{1}{11}$ as a decimal??
$$
0.\overline{09}
$$

##### What is $\frac{1}{12}$ as a decimal??
$$
0.08\overline{3}
$$

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