# Numbers begin with 1 more often than other numerals

A mathematical theory called "Benford's Law" predicts that in a set of numbers, numbers whose first digit is "1" will turn up more frequently than numbers that start with other digits. Benford, a GE physicist in 1938, formulated his law after discovering that GE's book of logarithm tables was substantially more worn on pages of logarithms corresponding to numbers starting with 1.

Because numbers beginning with 1 turn up so often, it's possible to catch cheaters (tax cheats, homework cheats, etc) by checking to see if the numbers they make up skew to ones that begin with the numeral 1 more frequently than other numbers.

It's not perfect, of course (the 1998 NYT article notes that people on \$25 dinner allowances often submit receipts for \$24.90), but it is fascinating.

"If we think of the Dow Jones stock average as 1,000, our first digit would be 1.

"To get to a Dow Jones average with a first digit of 2, the average must increase to 2,000, and getting from 1,000 to 2,000 is a 100 percent increase.

"Let's say that the Dow goes up at a rate of about 20 percent a year. That means that it would take five years to get from 1 to 2 as a first digit.

"But suppose we start with a first digit 5. It only requires a 20 percent increase to get from 5,000 to 6,000, and that is achieved in one year.

"When the Dow reaches 9,000, it takes only an 11 percent increase and just seven months to reach the 10,000 mark, which starts with the number 1. At that point you start over with the first digit a 1, once again. Once again, you must double the number — 10,000 — to 20,000 before reaching 2 as the first digit.

"As you can see, the number 1 predominates at every step of the progression, as it does in logarithmic sequences."

(via Digg)

Update: Christian sez,

The Benford's Law story has some bad math in it. I'm not arguing with the validity of Benford's Law. I just hate to see bad math put up as truth.

Quote from article:
"Let's say that the Dow goes up at a rate of about 20 percent a year. That means that it would take five years to get from 1 to 2 as a first digit."

Actually, it only takes four years. At a rate of 20 percent a year, by year four the Dow would be 2073.6

Quote from article:
"When the Dow reaches 9,000, it takes only an 11 percent increase and just seven months to reach the 10,000 mark, which starts with the number 1."

An 11 percent increase doesn't get the Dow to 10,000. An 11 percent increase of 9000 is 9990, not 10000 or more as the quote states.

Update 2: Ben sez,

The math in the article isn't as bad as one reader suggests.

True, it would take 4 as opposed to 5 years for the Dow Jones to reach 2000 at 20 percent. I assume that the author was saying that the first number starting with 2 would be the fith in the sequence

1000, 1200, 1440, 1728, 2074…

And it is the sequence we are interested in.

The second perceived fault is due to routine rounding off. 10000 is an 11.111111…% increase over 9000, which rounds down to 11% very nicely. At that rate it would take 6.934595174188633 (let's call it 7) months to reach 10000.

Update 3: Alvy points us to an excellent piece on Benford's Law at Steven Wolfram's Mathworld.