# What are the fibonacci numbers

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To carry out their calculations, merchants in the early 13th century used an abacus or a system called finger reckoning. Commerce changed when Leonardo of Pisa — known today as Fibonacci — published the first arithmetic textbook. Mathematician Keith Devlin talks about the history of arithmetic and his new book "The Man of Numbers: Fibonacci's Arithmetic Revolution."

IRA FLATOW, host: You're listening to SCIENCE FRIDAY. I'm Ira Flatow. Let me hit the way-back machine and travel back to the 13th century. We see the Mediterranean as a hub of international commerce, merchants trading colorful dyes and spices from the Far East, goats from North Africa, barrels of Tuscan wine. Commerce is everywhere.

But imagine today doing all that trading without modern arithmetic. How are merchants calculating their bills, tracking their inventory, with just simple additions and subtractions, not to mention the exchange rates, the interest rates. Most businessmen recorded figures using an abacas, theirfingers or Roman numerals. As you know, these are all very inefficient systems. Try running your business this way today.

Well, then a young man from Pisa appears on the scene and changes arithmetic forever. His name, Leonardo. He publishes a book that introduces a simple way to do arithmetic using just 10 numbers - hey, zero through nine - and something to write with.

We know him today as Fibonacci, and the story of this mathematical revolution is told in my next guest's new book, "The Man of Numbers: Fibonacci's Arithmetic Revolution." Joining me now is Keith Devlin. He's a mathematician at Stanford University. You also know him as NPR's WEEKEND EDITION's math guy. Welcome back to SCIENCE FRIDAY, Keith.

KEITH DEVLIN: Hi, Ira, nice to be back on the show again.

FLATOW: Nice to have you. This is not a Leonardo to be confused with da Vinci, right?

DEVLIN: No, although their paths at least their heritages do intertwine a couple of hundred years later.

FLATOW: I'll bet. Tell us what it was like. Was it really that crude math back then, or no, there was no math, there was just a little basic arithmetic.

DEVLIN: It really - in fact, you know, you were talking about international trade. It's true it was international, but trade really was two guys with goods on a camel, because it was impossible to run a commercial empire, as the Italians were desperate to do at the beginning of the 13th century, without arithmetic.

And so they did indeed use finger-counting, finger calculation. They used an abacas board - the abacas with beads on a wire, that's a Chinese invention that came into Europe much later. But it was mechanical calculations. You store the results in Roman numerals.

It was doable, but first of all, most people couldn't use the abacas board or the finger-counting very well themselves. They had to pay somebody else to do it for them.

And then the results were recorded in Roman numerals. So if two people disputed the calculation, there was no audit trail like there is when we do arithmetic today. You couldn't check the working. You had to go right back to the beginning and do the whole calculation again.

So it was incredibly inefficient, and it kept commerce from really developing, and it just restricted it to basically two guys with goods on a camel traveling around.

(SOUNDBITE OF LAUGHTER)

DEVLIN: I guess two camels.

FLATOW: I guess so. What did Leonardo write in his book about arithmetic?

DEVLIN: So the - what we now take for granted, and the fact that we take it so much for granted shows how pervasive was the revolution he started in the 13th century - that method of doing arithmetic was developed in India in the first six or seven centuries of the current era.

And then the traders in the Arabic-speaking world that were plying their goods between Europe and North Africa down to the Orient on the Silk Route, they picked up this method from the Indians. They started to use it in their trade and commerce. Some of the Arabic-speaking scholars developed it. Algebra grew out of their interest in this. The Arabic mathematicians developed modern algebra or the beginnings of modern algebra.

So by the end of the 12th century, this new method of doing arithmetic had found its way to the shores of North Africa, and it was heavily used by the traders in the Arabic-speaking world. But then Leonardo, as a teenager, goes over from Pisa, where he's been brought up, lands in Bugia in North Africa to join his father, who is a successful international businessman who has gone out from Pisa to Bugia to represent Italian traders.

Leonardo, who's got a good head for figures, he was sort of mathematically gifted, he would turn out to be extremely gifted, he sees this method being used. Since he's grown up in the hub of commerce, namely Pisa, when he looks at this method, he sees something that nobody else, including his father, had seen: the enormous potential this has to change commerce for everybody, because he says - and this is where he begins to sound like Steve Jobs and Bill Gates - he says: If I package this and present it so that ordinary people, ordinary businesspeople, can understand it, everyone can do their own arithmetic, everyone has the potential to become an international trader. It was a personal computing revolution that was about to happen, and Leonardo was about to launch it.

FLATOW: 1-800-989-8255 is our number, if you want to talk about Leonardo. When did he become known as, you know, Fibonacci?

DEVLIN: Well, that was in the 19th century. A historian just gave him that name. In Leonardo's day, they didn't really have surnames. So I guess that it was - when this historian was writing about him, his name was Guillome Libre(ph), early in the 19th century, he made up a surname.

In fact, when Leonardo writes his book "Liber Abaci," at the beginning of the book, and they say in Latin he calls himself Filius Bonacci, the son of Bonacci, which really he meant the family of Bonacci, and so this guy Libre, this historian in the early 19th century, takes that first Filius Bonacci, shortens it to Fibonacci and generates a nice modern-sounding, Italian-sounding surname for Leonardo.

FLATOW: And that stuck with him forever.

DEVLIN: It stayed with him (unintelligible) that name and the Fibonacci Sequence has been associated with him forever. But Leonardo really had nothing to do with the Fibonacci Sequence.

FLATOW: He didn't? It's named after him, and he had nothing to do with it?

DEVLIN: It had been around for hundreds of years. In his book, "Liber Abaci," because he's trying to teach this method to everybody, he fills it with hundreds and hundreds of examples. Most of them are examples about trade and commerce and exchanging goods and divvying out the profits from a trade, currency conversions, hundreds and hundreds of examples like that, which he must have realized would tend to get a bit tedious after a while.

So every now and then he throws in a little cute problem that you can think about in the bathtub, and one of them was about a fictitious rabbit's population. And he throws this as a throw-away problem. That's the problem that gives you the Fibonacci Sequence.

You know, he didn't invent it. He didn't solve it himself. He just took it - it had been around for hundreds of years. And then he goes along his way. I guarantee if Leonardo came onto the scene today, and you said I really like your Fibonacci Sequence, he would look at you and say: What on Earth are you talking about?

You know, he was trying to change the world. He really wasn't interested in populations of rabbits. But you know, history runs in strange ways, and we now associate that sequence with his name. It is a cool sequence.

FLATOW: Yeah. So how fast, how quickly did his new arithmetic catch on?

DEVLIN: Well, remember we are pre-Twitter. We are pre-Facebook. We are pre-everything, really. Manuscripts were copied by hand, and it could take a year or more to copy a manuscript. And "Liber Abaci" was hundreds and hundreds of pages long.

So we have to measure everything in terms of decades. If you measure it in terms of decades, it was a rapid-fire revolution. Within a century - well, certainly within his own lifetime - he became famous, totally famous. The emperor invited him to his court in an early version of "Jeopardy!," where Leonardo had to answer on the spot mathematical problems thrown at him by some of the best mathematicians in Emperor Frederick's court.

So he was a celebrity in his time. He wrote several other books, deep mathematics books, very famous. Within a few years of him producing "Liber Abaci," there began to appear what eventually were hundreds and maybe even thousands of books written in vernacular Italian explaining commercial arithmetic to everybody.

It was a huge bestseller if you could write a book on arithmetic, and these were written in the hundreds, and they circulated initially throughout Italy and then eventually throughout Western and Northern Europe.

FLATOW: What was it? Did he have did he have, you know, an a-ha moment about.

DEVLIN: His a-ha moment was very much when he saw this method being used, and he realized just how efficient it was. And the reason people don't like it today is because it's dull and mechanical. And we don't need to do that because we have machines that do that dull and mechanical thing.

I mean, everyone - my iPhone does the calculations for me these days. But in those days, this was huge because this turned something that was impossible and required fingers and all sorts of time and Roman numerals, and turned it into something which, yes, was dull and mechanical, but you could do it for yourself.

It was so valuable to the Italians, especially the trades and the businesspeople, that they flocked - they tried to get hold of copies of these

books that had been produced. Schools were being set up all over Italy where you could go. There was a charge(ph) - always an adults - and learn the arithmetic that you would need to set up in business for yourself and to become a trader. It was huge.

And within 200 years it was spreading throughout all of Europe, and life changed forever. I mean, modern - you look back to the origins of accounting, banking, international trade and commerce, insurance, it all came out of Italy in the 13th century, and one of the key ingredients for that to happen was that the ordinary person could do arithmetic quickly and efficiently and accurately and in the process leave an audit trail.

That was the key to Italy becoming the dominant world trade group, the world trade country in the 13th century.

FLATOW: And as a mathematician, what caught your interest in this story?

DEVLIN: Oh, I saw a kindred - I'm a - I spent my life doing mathematics. But I've also devoted a lot of time trying to explain mathematics to the general public and trying to find ways to make it accessible, to package it, if you like, and market it in a way that different kinds of ordinary people could understand. Leonardo did the same. He was a great mathematician, much better mathematician than me if we factor in 800 years of history, great mathematician. But he spends a lot of effort, puts in a lot of effort to making that difficult mathematics understandable to the general population.

So I saw in him a kindred spirit 800 years earlier, and I decided - this book, by the way, it took me the longest of any book to write. It took me over 10 years. Over 10 years ago I started - well, it was hard to find the information about the guy. I had to keep going to Italy. That's my story. I also like Tuscan food. But I kept going to Italy. There are worst places to go to do your research.

And so I was going and doing this. It turned out that just when I was starting the research in about 2000, 2001, a historian of mathematics, Raffaella Franci at the University of Siena, made the most amazing discovery in the archives in Florence. She found a copy of a long-lost book of Leonardo that was the key to understanding how Libra Abachi(ph) she came to start this revolution, and that had puzzled historians for 200 years. And as I'm starting my project, this discovery has made, I'd already met this person, Raffaella Franci. She gives me the results of her research. And suddenly I have the basis for writing "The Man of Numbers." It took me a bit longer to put it together, but I was lucky in that I was at the right place at the right time to write this story.

FLATOW: And he was also, because his father was involved in trade himself, right? Sort of an impetus for him to get involved and thinking about this.

DEVLIN: Oh, absolutely. Yeah. It was very much a matter of growing up in the right environment. The curious thing to me, as I was doing this research over 10 years, coming from Stanford in the middle of Silicon Valley, everything I discovered said, boy, this is exactly what happened in the 1980s, with the personal computer revolution. Everything, right down to the data. In fact, I was so struck by the parallels that I went to the publisher with "The Man of Numbers" and said, look, there's a great little subsidiary story comparing Leonardo with Steve Jobs.

So with the publisher's agreement, we actually brought out a companion e-book called "Leonardo and Steve," which tells the parallels, the amazing parallels, between what Leonardo of Pisa did in the 13th century and what Steve Jobs and later Bill Gates did in the 1980s in California.

The parallels are absolutely uncanny. And we decided it would be kind of cute to bring that out as an e-book, which we just did.

FLATOW: 1-800-989-8255. Interesting. But, you know, Steve Jobs may not like to be forgotten for centuries like Fibonacci was, right?

DEVLIN: That's right. The trouble with Fibonacci was the printing press came along 200 years later. Among the first books that were printed were some of these arithmetic - books for - I would almost call them arithmetic for dummies, but they were actually arithmetic for someone who wanted to make a lot of money as a business person. These books were coming out, so when the printing press comes along, printers immediately rushed to start bringing out copies of these books. But they looked for the ones that had been more recently written. They're not going to go back to something 200 years old that was big and written in Latin.

So the moment these come out, everybody forgets the old handwritten manuscripts, and Leonardo's manuscript just starts gathering dust in the monasteries, in the archives, and nobody looks at it again until, in fact, the late 18th century, when a historian comes across a reference in a paper - actually, a book by Luca Pacioli, who was a contemporary of Leonardo da Vinci. He finds this reference to Leonardo of Pisa, starts to dig out the old manuscripts, and discovers the role Leonardo had played. So for many hundreds of years, Leonardo's world had been forgotten. Then, beginning in the late 18th century and onwards, historians began to undercover the story. And as I'd mentioned a moment ago, it wasn't until 2003 when Raffaella Franci put in the last piece of the story that showed us what great things happened as result of Leonardo's work.

FLATOW: Talking with Keith Devlin, the author of "The Man of Numbers: Fibonacci's Arithmetic Revolution" on SCIENCE FRIDAY from NPR. Let me see if I can get a quick call in here before we go. Let's go to Jay in Sacramento. Hi, Jay.

JAY: Well, Hi.

FLATOW: Hi there.

JAY: First of all, thanks very much for making me aware of a book I'm going to have to pick up. But I'm curious about something. There's must be something I'm misunderstanding about Fibonnaci's advancement. The Egyptians, much earlier, were doing amazing architectural and engineering work, building all that stuff, and it must have required advanced ciphering abilities that I would think predate what we're talking about here. What am I not understanding there about Fibonacci's contribution?

DEVLIN: Not just the Egyptians, the Babylonians, the Chinese - there were lots of societies who did amazing work using arithmetic and numbers. And the ancient Greeks before had done great things. What was significant - but in those times, you had to be an expert, you had to devote your life to being a mathematician in order to do that. You had to get to that person. What Leonardo did - and this is why it's like the personal computing revolution, he produced a method whereby everybody could teach themselves or help to learn how to do it themselves.

He took something that had been developed by others and was useful but required experts and made it accessible so anybody could do it for themselves. He basically gave it to the world. That was a huge difference, and that's why it made possible international trade and commerce, the growth of international trading empires and large conglomerates, because they depend upon the fact that every employee does arithmetic on the fly. Remember, there were no computers back then, so it was making it accessible that was the key thing (unintelligible).

FLATOW: Could he have patented it? And you know, today - I'm thinking like a Silicon Valley entrepreneur.

DEVLIN: Believe me, the lawyers would have been swarming all around him if he was doing this today.

FLATOW: Yeah. I mean, so he's sort of, like, left like open source? For anybody to use it?

DEVLIN: It was open source, except you had to get hold of a hand-written copy and it would take a long, you know, a year or more to make - the copies are beautiful. In my book I have some images of some of the 13th century manuscript copies of Leonardo's book, not the original. We don't have one of the - in his hand. They are truly beautiful. I've looked at some of the originals in Siano in Florence, and they're absolutely staggeringly beautiful manuscripts, lovingly written by hand in multicolored inks, delight to look at. But.

FLATOW: Yeah. Did you come across Galileo's finger while you were there?

DEVLIN: No, but I.

FLATOW: I think I saw it in one of the museums.

DEVLIN: Yeah, I was more obsessed with Leonardo.

(SOUNDBITE OF LAUGHTER)

DEVLIN: You know, I mean, I kind of like the guy. He's been - you know, you compare him with Galileo and Copernicus.

FLATOW: Right.

DEVLIN: He's one of those guys.

FLATOW: Right.

DEVLIN: And one of the main reasons I wrote this book was I wanted to put Leonardo on the pedestal he deserves to be as one of the greatest people in the history of the modern world, along with Copernicus and Galileo, neither of whom could have done what Leo did without simple elementary arithmetic at their disposal.

FLATOW: It's an excellent book. It's called "The Man of Numbers: Fibonacci's Arithmetic Revolution." It's WEEKEND EDITION's math guy, Keith Devlin. Good luck with the book. It's a great read.

DEVLIN: Thank you very much, Ira. (Unintelligible).

FLATOW: You've done a lot for advancing his reputation. We're going to take a break. We'll be right back. Stay with us. We'll talk about creating sounds on the computer. I'm Ira Flatow. This is SCIENCE FRIDAY from NPR.

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