cover

The Sneaky One

Why do some numbers show up more in real life?
Open a book of facts about rivers and mountains and countries, and ~~something weird happens~~. The first digit of each

Open a book of facts about rivers and mountains and countries, and something weird happens. The first digit of each number โ€” the leftmost one โ€” isn't random at all. About 30% of the time, it's a 1. Only 5% of the time, it's a 9. If numbers were fair, each digit would show up about 11% of the time. But real-world numbers play favorites.

This pattern is called ++Benford's Law++, after a physicist who noticed his logarithm tables were **more worn out at the

This pattern is called Benford's Law, after a physicist who noticed his logarithm tables were more worn out at the beginning than the end. People looked up small leading digits way more often. It turns out this isn't a coincidence โ€” it's how numbers that grow and shrink in the real world actually behave.

~~Here's why.~~ Imagine a town that starts with 100 people. To get a leading digit of 2, it has to **double** โ€” reach 20

Here's why. Imagine a town that starts with 100 people. To get a leading digit of 2, it has to double โ€” reach 200. That takes time. But once it's at 200, it only needs to grow 50% more to hit 300. Then just 33% more to reach 400. The jumps get easier. So numbers spend MORE time in the 100s than the 900s.

**This same pattern** shows up everywhere things grow or shrink naturally: *populations of cities, lengths of rivers, pr

This same pattern shows up everywhere things grow or shrink naturally: populations of cities, lengths of rivers, prices of stocks, sizes of files on your computer, even the numbers in your bank account over time. Anything that multiplies or divides by percentages follows the rule.

Think of it like this. You're climbing a staircase where **each step is 10% higher** than the last. You take *seven step

Think of it like this. You're climbing a staircase where each step is 10% higher than the last. You take seven steps to go from 100 to 200, but only four steps to go from 800 to 900. You're literally spending more staircase-time in the low numbers.

++Benford's Law++ is so reliable that accountants use it to catch fraud. If someone invents fake expense reports, they t

Benford's Law is so reliable that accountants use it to catch fraud. If someone invents fake expense reports, they tend to spread the leading digits evenly โ€” lots of 7s and 8s and 9s. Real expenses? Loaded with 1s and 2s. The numbers snitch on the cheater.

It doesn't work for everything. Your height won't follow ++Benford's Law++ โ€” humans don't range from **1 foot to 900 fee

It doesn't work for everything. Your height won't follow Benford's Law โ€” humans don't range from 1 foot to 900 feet. The rule needs data that spans multiple orders of magnitude, where things can be ten times bigger or smaller than each other, growing or shrinking naturally over time.

~~So the next time~~ you see a list of populations or distances or prices, *check the first digits*. You'll find **1 is

So the next time you see a list of populations or distances or prices, check the first digits. You'll find 1 is the secret favorite, quietly winning a game you didn't know the numbers were playing. The universe just counts that way.

How was this book?

A Wonderleaf Book

The Sneaky One

โ€” Why do some numbers show up more in real life? โ€”

Wonderleaf Editions
โ€” ex libris โ€”
A Wonderleaf Book

The Sneaky One

Why do some numbers show up more in real life?

Wonderleaf Editions ยท MMXXVI
Scene 1
Open a book of facts about rivers and mountains and countries, and ~~something weird happens~~. The first digit of each
The Sneaky One2
Scene 1

Open a book of facts about rivers and mountains and countries, and something weird happens. The first digit of each number โ€” the leftmost one โ€” isn't random at all. About 30% of the time, it's a 1. Only 5% of the time, it's a 9. If numbers were fair, each digit would show up about 11% of the time. But real-world numbers play favorites.

3The Sneaky One
Scene 2
This pattern is called ++Benford's Law++, after a physicist who noticed his logarithm tables were **more worn out at the
The Sneaky One4
Scene 2

This pattern is called Benford's Law, after a physicist who noticed his logarithm tables were more worn out at the beginning than the end. People looked up small leading digits way more often. It turns out this isn't a coincidence โ€” it's how numbers that grow and shrink in the real world actually behave.

5The Sneaky One
Scene 3
~~Here's why.~~ Imagine a town that starts with 100 people. To get a leading digit of 2, it has to **double** โ€” reach 20
The Sneaky One6
Scene 3

Here's why. Imagine a town that starts with 100 people. To get a leading digit of 2, it has to double โ€” reach 200. That takes time. But once it's at 200, it only needs to grow 50% more to hit 300. Then just 33% more to reach 400. The jumps get easier. So numbers spend MORE time in the 100s than the 900s.

7The Sneaky One
Scene 4
**This same pattern** shows up everywhere things grow or shrink naturally: *populations of cities, lengths of rivers, pr
The Sneaky One8
Scene 4

This same pattern shows up everywhere things grow or shrink naturally: populations of cities, lengths of rivers, prices of stocks, sizes of files on your computer, even the numbers in your bank account over time. Anything that multiplies or divides by percentages follows the rule.

9The Sneaky One
Scene 5
Think of it like this. You're climbing a staircase where **each step is 10% higher** than the last. You take *seven step
The Sneaky One10
Scene 5

Think of it like this. You're climbing a staircase where each step is 10% higher than the last. You take seven steps to go from 100 to 200, but only four steps to go from 800 to 900. You're literally spending more staircase-time in the low numbers.

11The Sneaky One
Scene 6
++Benford's Law++ is so reliable that accountants use it to catch fraud. If someone invents fake expense reports, they t
The Sneaky One12
Scene 6

Benford's Law is so reliable that accountants use it to catch fraud. If someone invents fake expense reports, they tend to spread the leading digits evenly โ€” lots of 7s and 8s and 9s. Real expenses? Loaded with 1s and 2s. The numbers snitch on the cheater.

13The Sneaky One
Scene 7
It doesn't work for everything. Your height won't follow ++Benford's Law++ โ€” humans don't range from **1 foot to 900 fee
The Sneaky One14
Scene 7

It doesn't work for everything. Your height won't follow Benford's Law โ€” humans don't range from 1 foot to 900 feet. The rule needs data that spans multiple orders of magnitude, where things can be ten times bigger or smaller than each other, growing or shrinking naturally over time.

15The Sneaky One
Scene 8
~~So the next time~~ you see a list of populations or distances or prices, *check the first digits*. You'll find **1 is
The Sneaky One16
Scene 8

So the next time you see a list of populations or distances or prices, check the first digits. You'll find 1 is the secret favorite, quietly winning a game you didn't know the numbers were playing. The universe just counts that way.

17The Sneaky One

~ finis ~

Tiny picture books for big little questions.

โ€” a small constellation of questions โ€”
โœฆWonderleaf
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