JAD ABUMRAD: Hello, I'm Jad Abumrad.

ROBERT KRULWICH: And I'm Robert Krulwich.

JAD: This is Radiolab.

ROBERT: We're still talking about numbers, and now we're gonna switch—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 gonna introduce you to a—well, a nosy man named Mark Nigrini.

MARK NIGRINI: 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 kind of a ...

MARK NIGRINI: I originally grew up in Cape Town, South Africa.

ROBERT: ... South African, yes.

JAD: Oh!

ROBERT: He likes to play detective, and the clues he looks for are numbers.

MARK NIGRINI: I can't walk past a number without just wondering about it. What went into that number? How did it get there? For example, after I finish filling up at a gas station, sometimes I will just walk around and look at the dollar amounts on the pumps.

ROBERT: So he peeks in at the pump right next door.

MARK NIGRINI: And it's rather amazing. You can almost tell who's been there before. If you see a number like $1.40, then you know, "Oh, teenager with no money."

JAD: Why? Explain that.

ROBERT: Because that's all the kid can afford.

MARK NIGRINI: Quite right. Sometimes I'll see $10.04 and I'll say, "Ah, you meant to do $10, but you were a bit slow today." [laughs]

ROBERT: So you'd go to the gas pumps and they tell you a little short story?

MARK NIGRINI: Yes.

ROBERT: And his favorite story that numbers tell actually starts back in 1938. So imagine an office.

MARK NIGRINI: In Schenectady, New York. At the GE Research Laboratories.

ROBERT: And in that office is a man and he's sitting at his desk.

MARK NIGRINI: Mr. Frank Benford.

ROBERT: And Mr. Frank Benford is a physicist, so he's doing some difficult calculations and he's hunched over a book—probably actually the—one of the most boring books you could imagine. This is a book of logarithmic tables.

JAD: What are logarithmic tables?

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

ROBERT: So remember, this is before there were calculators, so if you wanted to multiply something like 145 times 3,564, you could just go to this book and look it up. So it starts with numbers you might want to multiply by: 1 to 100 on the first pages, then 101, 102, up to 200 and 300. And the back of the book is like 900s. The further you go, the higher and higher the numbers you use to multiply.

MARK NIGRINI: That's right.

ROBERT: So our Benford fellow, he's sitting there doing these calculations, and he's looking up the numbers, flipping through the book.

MARK NIGRINI: He's staring at the pages. And ...

ROBERT: He notices something kind of weird.

MARK NIGRINI: ... 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 NIGRINI: More than the last few pages.

ROBERT: And he wondered, "Why is this happening?"

MARK NIGRINI: "Strange."

ROBERT: "I'm not aware of favoring one part of the book over the other. Am I doing something a little odd, or maybe it's something bigger?" And that's when it hit him.

MARK NIGRINI: He thought maybe in this world, there 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 STROGATZ: He compiled some tens of thousands of statistics.

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

STEVE STROGATZ: Just anything he could think of that was numerical: molecular weights of different chemicals, baseball statistics, census data.

MARK NIGRINI: The revenues of all the companies listed on the main stock exchanges in America.

ROBERT: And everywhere he looked in all these different categories, it seemed yes, there were more numbers beginning with one and twos than eights and nines.

JAD: Wait, really?

ROBERT: Oh, yeah. This has been checked out again and again and again, and it's true.

STEVE STROGATZ: Size of rivers.

MARK NIGRINI: Earthquakes, and things like that.

STEVE STROGATZ: Populations, or number of deaths in a war. Areas of counties.

MARK NIGRINI: Stream flow data.

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

MARK NIGRINI: Ah, bank account balances follow Benford's Law nearly perfectly.

ROBERT: 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 NIGRINI: Perfectly, yes. So actually, they begin with one 30.1 percent of the time. They'll begin with a two 17.6 percent of the time. They'll begin with a three 12.5 percent of the time.

JAD: That's a—that's a big difference. Why would three be—I'm sorry, keep going.

MARK NIGRINI: The poor nine would only occur as a first digit 4.6 percent of the time, which actually would make the one approximately six times as likely as the nine. And it is quite amazing.

ROBERT: That is more than quite amazing, that's deeply suspicious!

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

ROBERT: But Steve and Mark and many, many, many mathematicians will 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: Um, Robert?

ROBERT: Mm-hmm?

JAD: So what?

ROBERT: [laughs] Well, 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 NIGRINI: For payroll fraud, tax return fraud.

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

MARK NIGRINI: That's right.

JAD: Huh? 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, so the numbers don't begin with ones more than with twos, and with 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." So he started giving lectures on the idea that Benford is a way to catch thieves. The only problem was ...

MARK NIGRINI: They didn't quite believe Benford's law, which means the rest of my talk isn't going to go anywhere.

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: Darrell D. Dorrell.

ROBERT: Darrell D. Dorrell. Darrell D. Dorrell. Darrell D. Dorrell. Darrell D. Dorrell.

JAD: It's alliterate too.

ROBERT: Alliterative heaven.

JAD: It's like palindromic.

DARRELL D. DORRELL: It's Darrell Dorrell.

JAD: Dorrell!

ROBERT: But I should say, Darrell is—what does he call it? He calls it ...

DARRELL D. DORRELL: 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 D. DORRELL: It's an investigating process.

ROBERT: And while at first he was unsure about Benford's Law ...

DARRELL D. DORRELL: A little bit skeptical.

ROBERT: ... one day ...

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

ROBERT: Huh.

DARRELL D. DORRELL: And it's just—it's immutable. It's a mathematical law.

ROBERT: And now it's one of his favorite tools of his trade.

DARRELL D. DORRELL: 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 D. DORRELL: Mm-hmm. Yes.

ROBERT: I see. I know you can't tell us what this business is doing, but is it a ...

DARRELL D. DORRELL: Let's say it's a regulated business.

ROBERT: [laughs]

DARRELL D. DORRELL: 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 D. DORRELL: Sure.

ROBERT: Anyway, this one woman thought that she was being cheated.

DARRELL D. DORRELL: So she got an attorney involved. The attorney requested data, so we have seven years' of income tax returns.

ROBERT: And that's all he had, just tax returns.

DARRELL D. DORRELL: Mm-hmm. Yes.

ROBERT: So he entered them all into the computer.

DARRELL D. DORRELL: 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: Hmm, very suspicious.

DARRELL D. DORRELL: Yes, blew out the Benford's pattern.

JAD: You mean like 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.

JAD: Ah!

ROBERT: But 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.

DARRELL D. DORRELL: Benford's ...

ROBERT: They don't know about it.

DARRELL D. DORRELL: ... as a practical tool has probably been around maybe 10 years, maybe 15 at the outset.

[ARCHIVE CLIP, announcer: Please welcome Darrell Dorrell.]

DARRELL D. DORRELL: I'm at a conference now with about 700 people.

[applause]

[ARCHIVE CLIP, Darrell D. Dorrell: Nice to see all of you here.]

DARRELL D. DORRELL: I've spoken four times, and each time I've asked about Benford's, who's heard of it ...

[ARCHIVE CLIP, Darrell D. Dorrell: Who's familiar with Benford's law?]

DARRELL D. DORRELL: Maybe, maybe five percent of the people.

[ARCHIVE CLIP, Darrell D. Dorrell: Can you just look at pennies?]

[ARCHIVE CLIP, man: Just a couple observations, Darrell.]

[ARCHIVE CLIP, man: To me, it doesn't make sense to exclude ...]

[ARCHIVE CLIP, woman: Can you use Benford's ...]

ROBERT: And they're asking, "Do judges allow Benford's in as evidence to suggest that someone's committed a crime?"

[ARCHIVE CLIP, woman: Is there case law out there that actually cites the use of Benford's Law?]

ROBERT: And Darrell tells them, "Oh, yeah."

[ARCHIVE CLIP, Darrell D. Dorrell: Yes.]

ROBERT: "You can use this evidence in court."

[ARCHIVE CLIP, Darrell D. Dorrell: Yes, federal, state and local, from the experiences we've had.]

ROBERT: And then he tells them stories. Like the case of the CEOs stealing money to buy ...

DARRELL D. DORRELL: Automobiles, firearms, artworks, jewelry. Run Benford's and boom! Your CEO is still in federal prison.

ROBERT: Or the case of the dentist and his wife.

DARRELL D. DORRELL: 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: Oh, busted!

DARRELL D. DORRELL: She eventually pled.

ROBERT: Or the guy with a $40-million Ponzi scheme.

DARRELL D. DORRELL: Run Benford's and boom!

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

DARRELL D. DORRELL: It is a very compelling argument. And then 10 years from now, it'll be the equivalent of a fingerprint.

JAD: But 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: Well, 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.

JAD: [laughs]

ROBERT: I mean, 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?" Well, I mean, no, it's just too numeric for me to explain it to you. [laughs]

JAD: Okay. All right.

ROBERT: But 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: Yeah. So let's go with our reporter Ben Calhoun and meet a crowd of mathematicians. Ben?

BEN CALHOUN: Yep.

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: [laughs] And what happened?

BEN: Well, I went to CUNY, which is the City University of New York. And it was a math conference called—it was on combinatorial and additive number theory.

ROBERT: Oh! A good time had by all!

BEN: Yeah, it goes by the optimistic acronym CANT.

ROBERT: [laughs]

BEN: So I had heard that if I went to this room, there was gonna 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.

ROBERT: Like?

BEN: They had this number.

[ARCHIVE CLIP, mathematician: My number is two.]

[ARCHIVE CLIP, mathematician: Three.]

[ARCHIVE CLIP, mathematician: Two.]

[ARCHIVE CLIP, mathematician: Yeah.]

[ARCHIVE CLIP, mathematician: Three.]

[ARCHIVE CLIP, mathematician: Three.]

[ARCHIVE CLIP, mathematician: Two.]

[ARCHIVE CLIP, mathematician: Mine is three, actually.]

[ARCHIVE CLIP, Ben Calhoun: Oh, nice!]

[ARCHIVE CLIP, mathematician: Yeah, I'm really excited about it.]

ROBERT: What—what do they mean, "I'm a two, I'm a three?"

BEN: Well, It's an Erdős number.

ROBERT: What's an Erdős?

BEN: Erdős is a guy.

ROBERT: Oh!

BEN: So your Erdős number is how many steps away you are from this guy, Paul Erdős.

ROBERT: So you're gonna tell me his story?

BEN: Yep. Are you ready?

ROBERT: Okay.

PAUL HOFFMAN: 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.

PAUL HOFFMAN: Mm-hmm.

BEN: So we start out in Budapest, Hungary, in 1913. It's spring. Two math teachers have a son named Paul.

PAUL HOFFMAN: And 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: Yeah, 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 HOFFMAN: 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. So 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. And he also said to me that, "Numbers became my best friends." So I mean, here's a kid whose whole life is mathematics from the beginning.

BEN: But 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 gotta get out of Hungary.

PAUL HOFFMAN: And he managed to get to the United States.

BEN: But he has to leave his family behind.

PAUL HOFFMAN: 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: But she was in Hungary. And 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 HOFFMAN: 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. And what's remarkable to me is other people who had been through this kind of life experience might have ended up in a mental institution or worse, but he didn't. He turned this sort of inwardness into making mathematics a joyous and social occasion.

BEN: He started connecting with people.

ROBERT: I don't get this. Like, what do you mean?

BEN: Well, 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."

ROBERT: [laughs]

BEN: And he was there to work with them on whatever it was that they were working on. And he just kept moving.

PAUL HOFFMAN: He 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. So everywhere he went, people had to put him up. And as a houseguest, the man was an acquired taste.

PAUL HOFFMAN: 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 HOFFMAN: He had some kind of skin condition, so he only wore silk.

BEN: Silk clothes you had to wash.

PAUL HOFFMAN: I mean, he went through life this way.

BEN: And there was the schedule.

PAUL HOFFMAN: 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: So 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 HOFFMAN: This was like God coming to visit you. He knew your strengths. He knew how you thought. And it was fascinating to watch him. I mean, there were times, like, I went with him to a math conference. And 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, and 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 HOFFMAN: He did mathematics with anybody, even if the person was a dim bulb in the world of mathematics.

[ARCHIVE CLIP, Paul Erdős: Now let me just thank ...]

ROBERT: What's this we're hearing?

BEN: This is Paul Erdős on his 80th birthday.

[ARCHIVE CLIP, Paul Erdős: The only good wish for an old man you can say is an easy cure of the incurable disease of life.]

[laughter]

BEN: Surrounded by ...

ROBERT: A lot of people. [laughs]

BEN: ... the mathematicians who loved him and put him up in their house.

[ARCHIVE CLIP, mathematician: We want to express all our deep, warm feelings to you, and I want to raise this toast for you.]

[applause]

JOEL SPENCER: He was a saint.

BEN: A saint?

JOEL SPENCER: A saint.

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

JOEL SPENCER: 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 is, if you look at it from the outside, it's a little bit of a strange activity why you put this enormous effort into finding these statements.

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

JOEL SPENCER: And 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 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, and 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 Erdős, then your Erdős number is two and so on and so on.

ROBERT: Oh, so 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 GROSSMAN: 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 in Michigan. And what he did was he took each ring of Erdős numbers and he charted it out.

JERRY GROSSMAN: Erdős number three has about 34,000 people in it. About 84,000 with Erdős number four. Then they start decreasing.

ROBERT: 84,000, that's a lot of people! So if you go ring upon ring upon ring and you do the whole deal, like, how many people did this man, in the end, influence?

JERRY GROSSMAN: I think it's about 200,000 mathematicians.

ROBERT: 200,000?

BEN: 200,000. Picture that for a second. It's like a solar system with more than 200,000 mathematicians all orbiting around Paul Erdős.

[ARCHIVE CLIP, Ben: And your Erdős number is?]

[ARCHIVE CLIP, mathematician: One.]

[ARCHIVE CLIP, mathematician: Infinity. [laughs]]

[ARCHIVE CLIP, Ben: Your Erdős number is?]

[ARCHIVE CLIP, mathematician: Two.]

[ARCHIVE CLIP, mathematician: I wrote a paper with my adviser and the other students, and she had written a paper with a mathematician who had written a paper with Erdős.]

[ARCHIVE CLIP, mathematician: My Erdős number is three.]

[ARCHIVE CLIP, Ben: Your Erdős number is?]

[ARCHIVE CLIP, mathematician: One.]

[ARCHIVE CLIP, mathematician: Everybody with Erdős number one knows that they've got that.]

[ARCHIVE CLIP, Ben: Everybody in this room knows their number.]

[ARCHIVE CLIP, mathematician: I would be very surprised if there are people who don't know. [laughs]]

ROBERT: Ben Calhoun's Erdős number is 00.5778-B-1⁶. Coming up: a story from our friend Steve Strogatz, the mathematician from Cornell who tells about a friendship he has, a very precious friendship with his math teacher. So it's all about mathematicians, but this is a very unusual friendship.

JAD: I'm Jad Abumrad.

ROBERT: I'm Robert Krulwich.

JAD: Stick around.

 

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