### From Benford to Erdös

**Jad: **Hello, I'm Jad Abumrad.

**Robert: **I'm Robert Krulwich.

**Jad: **This is Radiolab.

**Robert: **We're still talking about numbers and now we're going to switch [laughs] though, it may fatigue some of us. If you think about them a little differently, if you learn to embrace them and give them a bit of a hug, wonderful things can happen. I'm going to introduce you to a nosy man named the Mark Nigrini.

**Mark: **I'm an associate professor at the School of Business at the College of New Jersey.

**Robert: **Has a really heavy New Jersey accent, but what he really likes to do-

**Jad: **What kind of accent was that?

**Robert: **That was-

**Mark: **I originally grew up in Cape Town, South Africa.

**Robert: **South African. He likes to play detective and the clues he looks for are numbers.

**Mark: **I can't walk past a number without just wondering about it. What went into that number? How did it get there?

[music]

**Mark: **For example, after I finished filling up at a gas station, sometimes I would just walk around and look at the dollar amounts on the pump.

**Robert: **He peeks in at the pump right next door.

**Mark: **It's rather amazing, you can almost tell who's been there before. If you see a number like $1.40, then you know, "Teenager with no money."

**Jad: **Why explain that?

**Robert: **That's all the kid can afford.

**Mark: **Quite right. Sometimes I'll see $10.04 and I'll say, "You meant to do $10, but you're a bit slow today [chuckles]."

**Jad: **You'd go to the gas pumps and they tell you all little short stories?

**Mark: **Yes.

**Robert: **His favorite story that numbers tell actually starts back in 1938. Imagine an office.

**Mark: **In Schenectady, New York. At the GE research laboratories.

**Robert: **In that office is a man and he's sitting at his desk.

**Mark: **Mr. Frank Benford.

**Robert: **Mr. Frank Benford is a physicist. He's doing some difficult calculations. He's hunched over a book, probably actually one of the most boring books you could imagine. This is a book of logarithmic tables.

**Jad: **What are logarithmic tables?

**Mark: **Log tables were a very convenient way of doing multiplication in the early part of the last century.

**Robert: **Remember, this is before there were calculators. If you wanted to multiply something like 145×3,564, you could just go to this book and look it up. It starts with numbers you might want to multiply by 1, 100 on the first pages, then 101 and 102, up to 200 and 300. Then the back of the book is like 900s. The further you go, the higher and higher the numbers you use to multiply.

**Mark: **That's right.

**Robert: **Our Benford fellow is sitting there doing these calculations and he's looking at the numbers, flipping through the book.

**Jad: **He's staring at the pages.

**Robert: **He notices something weird.

**Mark: **He noticed that the first few pages were more worn than the last few pages.

**Robert: **Meaning more smudgy and darker and oily, as if he was using the front of the book-

**Mark: **More than the last few pages.

**Robert: **He wondered, "Why is this happening?"

**Mark: **"Strange."

**Robert: **"I'm not aware of favoring one part of the book over the other. Am I doing something a little odd? Maybe it's something bigger." That's when it hit him. He thought, "Maybe in this world, they are more numbers with low first digits than with high first digits."

**Jad: **What?

**Robert: **More numbers that start with one or two than numbers that start with seven, eight, or nine.

**Jad: **Just because his book is worn?

**Robert: **That's what started him thinking, so here's what he did.

**Steve: **He compiled some tens of thousands of statistics.

**Robert: **That's Steve Strogatz, mathematician at Cornell University.

**Steve: **Just anything he could think of that was numerical. Molecular weights of different chemicals, baseball statistics, census data.

**Mark: **The revenues of all the companies listed on the main stock exchanges in America.

**Robert: **Everywhere he looked in all these different categories it seemed, yes, there were more numbers beginning with one and twos and eights and nines.

**Jad: **Wait, really?

**Robert: **Yes. This has been checked out again and again and again, and it's true. Size of rivers.

**Mark: **Earthquakes, and things like that.

**Steve: **Populations, or number of deaths in a war, areas of counties.

**Mark: **Streamflow data.

**Jad: **What if you were to, say, get all the people in New York together and look at their bank accounts?

**Mark: **Bank account balances follow Benford's Law nearly perfectly.

**Jad: **Meaning that if you just look in at the amount of money that people have, matter of fact, in all the bank accounts, you'll find they begin with one more often than they begin with two?

**Mark: **Perfectly, yes. Actually they begin with one 30.1% of the time. They'll begin with a two 17.6% of the time. They'll begin with a 3, 12.5% of the time.

**Jad: **That's a big difference. Why would three be-- I'm sorry, keep going.

**Mark: **The poor nine would only occur as a first digit 4.6% of the time, which actually would make the one approximately six times as likely as the 9. It's quite amazing.

**Robert: **That is more than quite amazing. That's deeply suspicious.

**Steve: **This is crazy what I'm telling you, and I can't give you good intuition, why it's true.

**Robert: **Steve and Mark and many, many, many mathematicians would tell you, despite what you may think, there is a preference, a deep preference in the world it seems, for number sequences that start with ones and then twos and then threes.

**Jad: **Robert? So what?

[laughter]

**Robert: **This is not just a mathematical curiosity Jad. No, no, no. There is something you can do with this information.

**Jad: **What?

**Robert: **Well, when Mark Nigrini first ran across Benford's Law, he thought, "Maybe I can use this law to bust people."

**Mark: **For payroll fraud, tax return fraud.

**Jad: **You thought, "Hey, we can use this to catch a thief?"

**Mark: **That's right.

**Jad: **How?

**Robert: **Well, Nigrini figured if you look at a bunch of numbers, at bank statements or expense reports and so on, and you see that the numbers in the business do not match the natural pattern of Benford's law, she numbers don't begin with ones more than the twos, twos more than with threes, and so on, then you could say, "Hey, this is not natural. This may not be true. This may be fraud." He started giving lectures on the idea that Benford easy way to catch thieves. The only problem was-

**Mark: **They didn't quite believe Benford's law, which means the rest of my talk. Isn't going to go anywhere.

[music]

**Robert: **It is now my great pleasure to introduce you to one of the most fabulous people I've ever had the name to say, Darrell D. Dorrell.

**Jad: **It's alliterate too.

**Robert: **Alliterative heaven.

**Jad: **It's palindromic.

**Darrell: **It's Darrell Dorrell.

**Jad: **Darrell?

[laughter]

**Robert: **I should say, Darrell is, what does he call it? He calls it-

**Darrell: **I'm a Forensic Accountant.

**Robert: **Forensic accountant. Which means his job is to examine numbers and figures to see if someone is stealing.

**Darrell: **It's an investigative process.

**Robert: **While at first he was unsure about Benford's Law-

**Darrell: **A little bit skeptical.

**Robert: **One day-

**Darrell: **I happened to talk to one of my neighbors who was a retired statistics professor, and he said, "Benford's, I have my students do that proof every year." He actually wrote out the proof for me. It's immutable.

[music]

**Darrell: **It's a mathematical law.

**Robert: **Now it's one of his favorite tools of his trade.

**Darrell: **We have a case right now underway, relatively small company, family shareholders. There are four of them. One of them feels like she has been misrepresented as a shareholder.

**Robert: **Meaning she thinks these other three guys might be stealing?

**Darrell: **Yes.

**Robert: **I know you can't tell us what this business is doing, but is it a-?

**Darrell: **Let's say it's a regulated business.

[laughter]

**Darrell: **It's a business that each of you purchase on a regular basis through your local governmental authority.

**Robert: **Trash collection or sewage?

**Jad: **Recycling?

**Darrell: **Sure.

**Robert: **Anyway, this one woman thought she was being cheated.

**Darrell: **She got an attorney involved. The attorney requested data, so we have seven years' of income tax returns.

**Robert: **That's all he had, just tax returns?

**Darrell: **Yes.

**Robert: **He edited them all into the computer.

**Darrell: **Aggregate them, run Benford's and boom. Clicked on the graph. We instantly saw, bingo. For a couple of the years coincident with when the dispute began, the way they've reported their taxes violates Benford's.

**Robert: **Very suspicious.

**Darrell: **Blew out the Benford's pattern.

**Jad: **You mean there were too many nines on the tax returns?

**Robert: **Meaning if you looked at the tax returns of this company, you will see a pattern that isn't natural, exactly. Not enough ones and too many sevens, eights and nines. Now you have to convince detectives and then lawyers and then judges that this is real evidence of wrongdoing, but they've not heard of this thing. Don't know about it.

**Darrell: **Benford's, as a practical tool, has probably been around maybe 10 years, maybe 15 at the outset.

**Speaker 1: **Please welcome Darrell Dorrell.

**Darrell: **I'm in a conference now with about 700 people.

[applause]

**Darrell: **Nice to see all of you here. I've spoken four times and each time I've asked about Benford's, "Who's heard of it?" "Who's familiar with Benford's law?" Maybe, maybe 5% of the people.

**Speaker 2: **Can you just look at pennies?

**Speaker 3: **A couple observations, Darrell.

**Robert: **They're asking, "Do judges allow Benford's in as evidence to suggest that someone has committed the crime?"

**Speaker 4: **Is there case law out there that actually cite the use of Benford's law?

**Robert: **Daryl tells them, "Yes. You can use this evidence in court."

**Darrell: **Federal, state and local, farm the experiences we've had.

**Robert: **Then he tells them stories. Like the case of the CEOs stealing money to buy-

**Darrell: **Automobiles, firearms, artworks, jewellery. Run Benford's and, boom. Your CEO is in federal prison.

**Robert: **The case of the dentist and his wife.

**Darrell: **She began having an affair with a guy who turned out to be a meth dealer, the dentist suspected her of having dipped into the till. Run Benford's and, boom.

**Jad: **Busted.

**Darrell: **She eventually pled.

**Robert: **The guy with a $40 million Ponzi scheme.

**Darrell: **Run Benford's and boom.

**Room: **Well, almost boom. Benford's was an element in all these cases. It wasn't the clincher, but still.

**Darrell: **It is a very compelling argument. In 10 years from now, it'll be the equivalent of a fingerprint.

[music]

**Jad: **You still haven't addressed the central mystery here. Why in the world would there be more ones than nines? Shouldn't they be equi-?

**Robert: **Equi-coincident?

**Jad: **Yes.

**Robert: **The answer is actually very complicated and deeply mathematical. The simple answer is-

**Jad: **Is there an answer though?

**Robert: **Yes, there is an answer and it has to do-

**Jad: **Do you understand the answer?

**Robert: **No. I understand that it has to do with logarithms and the business of doubling and the culture of numbers but if you were to sit me down and, say, explain it to me carefully and well, no. it's just too numeric for me to explain it to you. I will now take a little sidestep to a group of people who would be able to explain it to us if they were in this room, but we didn't find them in this room. We found them in another room.

**Jad: **We're rarely in the same room that they are in.

**Robert: **Let's go with our reporter, Ben Calhoun and meet a crowd of mathematicians. Ben?

**Ben: **Yes.

**Robert: **You decided to I don't know, it was some kind of a busman's holiday. You wanted to go to a math conference?

**Ben: **I did, badly.

**Robert: **What happened?

**Ben: **I went to CUNY, which is the City University of New York. It was a math conference, it was on combinatorial and additive number theory.

**Robert: **A good time had by all.

**Ben: **It goes by the optimistic acronym CANT. I had heard that if I went to this room, there was going to be a bunch of mathematicians from all over the place and they would be able to tell me where they taught, what their name was, but they would have this other way of identifying themselves. They had this number.

**Speaker 5: **My number is two.

**Speaker 6: **Three.

**Speaker 7: **Two.

**Speaker 8: **Three.

**Speaker 9: **Two.

**Speaker 6: **Mine is three actually.

**Ben: **Nice.

**Speaker 6: **I'm really excited about it.

**Robert: **What does that mean, "I'm a two, I'm a three"?

**Ben: **It's an Erdős number.

**Speaker 2: **What's an Erdős?

**Ben: **Erdős is a guy. Your Erdős number is how many steps away you are from this guy, Paul Erdős.

**Robert: **You're going to tell me his story?

**Ben: **Yes. Are you ready?

**Robert: **Okay.

**Paul: **Let me turn off my cell phone so we don't ruin the best take.

**Ben: **That's Paul Hoffman. He wrote a book about Paul Erdős. We start out in Budapest, Hungary in 1913. It's spring. Two math teachers have a son named Paul.

**Paul: **He had two sisters. They were three and five and they had scarlet fever and they died the day he was born. Imagine that, his mother loses her two daughters and gains a son.

**Robert: **Oh, my God.

**Ben: **She was so terrified after that, that Paul would get a fatal disease and die, that she didn't let him leave the house pretty much for the first 10 years of his life. She didn't let him play with other kids really didn't let him go to school, didn't let him go outside.

**Paul: **Also, when he was one and a half, his dad was captured and put in a Soviet prisoner of war camp for six years of his life. Here's this kid at home without other children around, his mother's out teaching mathematics. All the books in the house were math and he taught himself basically to read by looking at these math books. He also said to me, "That numbers became my best friends." Here's a kid whose whole life is mathematics from the beginning.

**Ben: **Let's fast forward. Paul Erdős gets his PhD in his early 20s. This is in the early 1930s. Paul Erdős is Jewish, which means he knows that he's got to get out of Hungary.

**Paul: **He managed to get to the United States.

**Ben: **But he has to leave his family behind.

**Paul: **When the Nazis moved into Budapest, four of his mother's five siblings were killed. His father died as they were herding Jews and trying to move them into the ghetto and he only had his mother left.

**Ben: **She was in Hungary. In 1941, Paul Erdős was at Princeton University. He was just 27 years old, completely cut off from his family. He was lonely, and he was homesick.

**Paul: **I mean, this guy had no conventional friendships. He had no sexual relationships, his only contact with the world was the people he worked with. What's remarkable to me is other people who had been through this life experience might have ended up in a mental institution or worse but he didn't. He turned this inwardness into making mathematics a joyous and social occasion.

**Ben: **He started connecting with people.

**Paul: **I don't get this. What do you mean?

**Ben: **He started traveling. He would hear about somebody who was working on something interesting and he would find a way to get there, show up at their door, and he had this phrase that he would say, "My brain is open." He was there to work with them on whatever it was that they were working on. He just kept moving.

**Paul: **Made a circuit of 25 different countries.

**Ben: **Eventually, he gave up almost all of his possessions and he became essentially homeless.

**Robert: **He had no home?

**Ben: **He had no home. Everywhere he went, people had to put him up. As a house guest, the man was an acquired taste.

**Paul: **He didn't know how to do basic things. He couldn't cook, he couldn't even boil water for tea, he could barely change his clothes.

**Ben: **Erdős didn't know how to tie his own shoes until he was 11.

**Paul: **He had some kind of skin condition so he only wore silk.

**Ben: **Silk clothes, you had to wash.

**Paul: **He went through life this way.

**Ben: **There was the schedule.

**Paul: **He did mathematics 20 to 22 hours a day.

**Ben: **He'd bang pots and pans around in the kitchen at 4:00 AM because he wanted you to come downstairs and do more math.

**Robert: **Why would anybody want a visit from this guy? This sounds like a walking nightmare. You have to cook for him and stay with him and wash his clothes, and tie his shoes.

**Ben: **Regardless of all of it, you wanted him to come see you.

**Robert: **Why?

**Ben: **Because he was just that good.

**Paul: **This was like God coming to visit you. He knew your strengths. He knew how you thought. It was fascinating to watch him. There were times, like I went with him to a math conference. He was there in his hotel room and at one point, there were 10 or 12 mathematicians in the room. Some were sprawled on his bed, some were sitting on the floor, and he'd be working with one for a few minutes. Then he would turn to another and then he'd go back to a third. He was working simultaneously with all these people on different problems.

**Ben: **Paul Erdős wrote more papers and collaborated with more people than any other mathematician who's ever lived.

**Paul: **He did mathematics with anybody, even if the person was a dim bulb in the world of mathematics.

**Robert: **What's this going on?

**Ben: **This is Paul Erdős, on his 80th birthday.

**Paul Erdős: **The only good wish for an old man you can say is easy cure of the incurable disease of life.

**Ben: **Surrounded by-

**Robert: **A lot of people.

**Ben: **-the mathematicians who loved him and put him up in their house.

**Speaker 7: **We want to express all our deep feelings to you and I want to raise this toast for you.

**Joel: **He was a saint.

**Ben: **A saint?

**Joel: **A saint.

**Ben: **That's Joel Spencer. He's a mathematician. He was also friends with Paul Erdős.

**Joel: **Now that he is gone I think of him sometimes in a religious context, because he gave this faith to those of us that are doing mathematics which, after all, if you look at it from the outside, it's a little bit of a strange activity while you put this enormous effort into finding these statements.

**Ben: **Mathematicians will spend years of their lives trying to prove these things that, from the outside, look totally obscure and pointless.

**Joel: **Yet, it was clear working with him, that what we were doing was we were trying to find Truth with a capital T. A truth that transcends our physical universe. I think that's the reason why we'd like to talk about our connection to Paul, because our feeling of mathematics, the feeling for what we want mathematics to be, Paul Erdős was the embodiment of that feeling.

**Ben: **Somewhere along the way, mathematicians started keeping track of their connection to Paul Erdős. That's what Erdős numbers actually are. If you published a paper with Paul Erdős, your Erdős number is one. If you published a paper with someone else, and they published a paper with Paul Erdos, then your Erdos number is two and so on and so on.

**Robert: **This is all the people that Paul Erdős in some way has touched?

**Ben: **All the people who are connected to him through their ideas.

**Jerry: **There are about 500 people with Erdős number one and about 8,000 people with Erdős number two.

**Ben: **This is Professor Jerry Grossman. He's at the University of Oakland and Michigan. What he did was he took each ring of Erdős numbers and he charted it out.

**Jerry: **These rings get larger and larger for a while. Erdős number three has about 34,000 people in it. At least 84,000 with Erdős number four, then it starts decreasing.

**Robert: **84,000, that's a lot of people. If you go ring upon ring upon ring and you do the whole deal, how many people did this man in the end influence?

**Jerry: **I think it's about 200,000 Mathematicians.

**Robert: **200,000?

**Jad: **200,000. Picture that for a second. It's like a Solar System with more than 200,000 mathematicians all orbiting around Paul Erdős

**Ben: **Your Erdős number is?

**Speaker 7: **One.

**Ben: **Icorn is number two. Your Erdős number is?

**Speaker 8: **Two.

**Speaker 9: **I wrote a paper with my adviser and the other students and she had written a paper with a Mathematician who had written the paper with Erdős.

**Speaker 10: **My Erdős number is three.

**Ben: **Your Erdős number is?

**Speaker 11: **One.

**Speaker 12: **Everybody with Erdos number that they got that.

**Ben: **Everybody in this room knows their number.

**Speaker 13: **I would be very surprised if there are people who don't know.

[music]

**Robert: **Ben Calhum's Erdős number is 00.5778/B-16. Coming up, a story from our friend Steve Strogac, the mathematician from Cornell who tells about a friendship he has. A very precious friendship with his math teacher. So it's all about Mathematician but this is a very unusual friendship.

**Jad: **I'm Jad Abumrad.

**Robert: **Robert Krulwich.

**Jad: **Stick around.

**Darrell: **This is Darrell Dorrell. I'm a forensic accountant in Portland, Oregon. On the web at the AECF org. This is NPR, National Public Radio.

**Speaker 14: **That's right. Good job. Wait. Come back.

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