What is Benford's Law?

0:47 Welcome to Matt's Is Not Boring by Kidopoly.com. I'm Mira. And I'm Finn. Before we start, a very happy birthday to Madeline and JP, who are listening together in New Orleans.

1:00 Happy birthday, Madeline and JP. I hope you have the best day ever.

1:05 And if you haven't already, tap follow wherever you're listening so you never miss an episode. Madeline and JP, since it is your birthday, I want you to imagine we are going on a giant scavenger hunt across the whole world, gathering up all the numbers we can find.

1:22 Wait, what do you mean by gathering up numbers?

1:26 I mean real numbers that measure real things. The lengths of every river on Earth, the populations of every town, the distance to every star, the street addresses of every house in your city.

1:40 Whoa, that is a massive pile of numbers! I am picturing a mountain of little pieces of paper. Millions and millions of them.

1:48 Exactly. Now, I want you to look only at the very first digit of each of those numbers. Just the first number on the left. It could be a one, a two, a three, all the way up to a nine.

2:02 Okay, so if a river is 3,000 miles long, I only care about the three.

2:08 You got it! Now, if you tossed all those first digits into a hat and pulled one out, which digit do you think would be the most common?

2:17 Well, they should all be the same, right? Like rolling a huge nine-sided dice. There should be just as many nines as there are ones or fives.

2:26 That is exactly what everyone expects, but it is entirely wrong. In the real world, the numbers are playing a completely different game. To understand this strange game, we have to travel back in time to the United States in the year 1881.

2:43 1881? That is a long time ago. What happened?

2:48 There was a brilliant astronomer named Simon Newcomb. He spent his days looking at the stars, but to do his calculations, he had to use a heavy, leather-bound book from the library.

3:00 Because they did not have calculators back then, right? They had to look up the maths in big books. Exactly.

3:08 This book was entirely filled with tables of numbers. Page after page of nothing but numbers. But Simon Newcomb noticed something really weird about the book itself. He noticed how worn out it was. That is true. But it wasn't the whole book that was falling apart. The first few pages, where all the numbers started with a one, were incredibly grubby.

3:44 The edges were worn, dark, and practically tearing from being flipped so much.

3:49 Wait, really? What about the other pages?

3:52 As he flipped further into the book, to the numbers starting with two, then three, the pages got cleaner. And the very last pages, for the numbers starting with an eight or a nine?

4:05 Don't tell me. Were they perfectly clean?

4:08 They were almost pristine. It looks like nobody ever needed to look up numbers that started with a nine.

4:15 But that makes no sense. Why would people in 1881 only want to look at numbers that start with a one? Were they just bored of the bigger numbers?

4:24 No, they weren't choosing the numbers. The numbers they needed to look up, the real-world measurements they were doing, just happened to start with a one way more often. Simon Newcomb wrote about his grubby pages in a mathematics journal. But then, for over 50 years, pretty much nobody paid attention.

4:44 Aw, that's a shame. It's such a cool mystery.

4:47 It didn't stay lost forever. Fast forward to 1938, a physicist named Frank Benford noticed the exact same thing. Did he find another grubby library book? He sure did. And Frank Benford decided he wasn't just going to write a short note about it. He was going to test it to see if it was really true across the whole world.

5:10 He collected over 20,000 different numbers. He looked at the surface areas of rivers, the addresses on streets, the populations of cities. He just gathered piles and piles of real data.

5:24 20,000 numbers without a computer. He must have been writing for days.

5:30 He was incredibly stubborn about proving it. And what he found was so amazing that today this idea is named after him. It is called Benford's Law.

5:41 Benford's Law. I like how that sounds. But what exactly does the law say? How many numbers actually start with a one?

5:49 If numbers were fair, every digit from one to nine would turn up about 11% of the time. But under Benford's law, about 30.1% of all numbers start with a 1.

6:03 Wait, 30%? That is like 3 out of every 10 numbers in the world starting with a 1. Exactly. And it gets crazier.

6:13 The number 2 shows up as the first digit less often, about 17% of the time. And it keeps dropping as you go up.

6:21 So what about the number 9? The one on the totally clean pages of the book?

6:25 Numbers starting with a 9 show up only about 4.6% of the time. That is fewer than 1 in 20.

6:34 Hold on, that can't be right. Why would the universe just decide to favour the number one? That feels impossible.

6:41 It does feel impossible, but there is a really beautiful reason for it. It has to do with how things grow in the real world. For Benford's Law to work, the numbers have to spread out over several different sizes. Things that start small and can grow really, really big. Like what? Give me a picture. Picture a tiny town with exactly 1,000 people living in it.

7:07 The population starts with a one. For that town's population to start with a two, the town has to completely double in size. It has to grow all the way to 2,000 people. That could take decades.

7:22 Oh I see. It takes a long time to get another thousand people. So the number is stuck starting with a one for years and years.

7:30 Exactly. But now imagine that same town has grown much bigger to 8,000 people. To change its first digit to a 9, it only needs to add 1,000 more people.

7:43 Wait, when it already has 8,000 people, adding 1,000 is super fast. It's just a tiny fraction of the town.

7:51 Yes, it rockets through the 8's and 9s. But once it hits 10,000 people, boom, the first digit is a 1 again, and it has to double all over again to hit 20,000.

8:04unknown Ha!

8:05 That is brilliant! The numbers get stuck in the ones and then zoom through the bigger digits.

8:10 You've got it, but remember, this only works for things that spread across multiple sizes. It wouldn't work on the ages of kids in your class or telephone numbers.

8:22 Because telephone numbers don't start small and grow into massive telephone numbers. They just stay the same size.

8:28 Now that you know about Benford's law, you actually have a kind of superpower. People use this pattern to spot made-up numbers in the real world.

8:37 Made up numbers? Why would anyone make up numbers?

8:41 Sometimes people try to invent data to trick people. But human brains are really bad at making things look truly natural. If someone asks you to make up a list of hundreds of random numbers, you will probably try to mix them up evenly. You'll sprinkle in some sevens, a few fours, a couple of nines, so it looks fair.

9:03 Yeah, if I made them all start with a one, someone would think I was just being lazy.

9:08 Exactly. But real life isn't evenly mixed. Real life follows Benford's law. So if researchers look at a giant list of numbers and they see equal amounts of ones and nines, they know somebody made it all up.

9:22 That is so cool. It's like a secret lie detector built into the universe.

9:27 It really is. And the most beautiful part is how deeply this law is woven into mathematics itself. It doesn't just work on things people build or measure. If you look at the famous Fibonacci sequence, that magical list of numbers that shape spirals in pine cones and sunflowers, those numbers follow Benford's law perfectly.

9:48 So it's in the rivers, it's in the towns, and it's even hiding inside sunflowers. I am never going to look at a list of numbers the same way again.

9:58 And whenever you see a heavy book at the library with grubby pages at the front, you'll know exactly what's going on. Alright, Finn, time for the quiz. Are you ready?

10:07 Oh, I am completely ready. Let's do this.

10:11 Okay, first question. Who was the astronomer who first noticed the grubby library pages in 1881? Next one, what is this mathematical idea called? Question three. Roughly what percentage of these numbers start with a one? Alright, here's number four. Does this rule work on telephone numbers? Last question.

10:50 How many real world numbers did Frank Benford test? Great job! Let's see how you did. The answer to question one is Simon Newcomb. For question two, it is called Benford's Law. Number three, the answer is about 30.1%. Question four. No, it does not work on telephone numbers because they don't spread out over several sizes.

11:22 And the final answer is he tested about 20,000 numbers.

11:27 It blows my mind that something as simple as the very first digit of a number has this huge, invisible pattern wrapping around the whole world.

11:37 That is the beauty of mathematics. It takes something you see every single day and suddenly shows you it is full of secrets. If you loved learning about Benford's Law today, please leave us a five-star review. Just scroll down on your app and tap the stars.

11:54 Yeah, and hit follow so new episodes turn up automatically. Oh, and we have sister shows too. History, science and philosophy's not boring. Go give those a listen and a follow as well.

12:06 And if you want a shout out on the show, or just want to tell us what you're noticing in the world, email us at hello at kidopoly.com. See you next time on Maths Is Not Boring. Bye everyone.