Take any long piece of English text — a novel, a newspaper archive, a pile of tweets — and count how many times each word appears. Then sort the words from most common to least common. That sorted list is the key to everything in this book.

The word in first place is almost always the. In second place, usually of or and. Count them up in a large corpus of English and you'll typically find something like: "the" appears about 70,000 times per million words, "of" about 36,000 times, "and" about 28,000 times. Keep going and the numbers keep dropping — but not randomly. They drop in a strikingly predictable way.

To describe this pattern precisely, we need two simple terms. Rank is a word's position in the sorted list — 1st, 2nd, 3rd, and so on, from most frequent to least. Frequency is how many times that word actually occurs. So "the" has rank 1 and some frequency $f_1$; "of" has rank 2 and frequency $f_2$; and so on. A rank-frequency distribution is just the full table pairing each rank with its frequency — the raw material Zipf worked from.

Here's the pattern. In 1949, linguist George Zipf noticed that if you multiply a word's rank by its frequency, you get roughly the same number no matter which word you pick. The most frequent word (rank 1) appears about twice as often as the second-ranked word, about three times as often as the third-ranked word, about ten times as often as the tenth-ranked word, and so on. In symbols:

$$f(r)\approx \frac{C}{r}$$

Here $f(r)$ is the frequency of the word at rank $r$, and $C$ is just a constant — roughly the frequency of the most common word, since at $r=1$ the formula gives $f(1)=C$. This is Zipf's Law: frequency is inversely proportional to rank. Double the rank, and the frequency roughly halves.