Fibonacci's 'Numbers': The Man Behind The Math

The Latin phrase filius bonacci, in the first line of the Liber Abaci manuscript (above), gave rise to Leonardo da Pisa's modern nickname, Fibonacci. Click Here For A Closer Look National Library of Florence hide caption

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The Man of Numbers

Fibonacci's Arithmetic Revolution

by Keith Devlin

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A page from the Liber Abaci manuscript. Leonardo da Pisa wrote symbolic calculations in the margin to illustrate the methods described in the text. Click Here For A Closer Look i i

A page from the Liber Abaci manuscript. Leonardo da Pisa wrote symbolic calculations in the margin to illustrate the methods described in the text. Click Here For A Closer Look Siena Public Library hide caption

itoggle caption Siena Public Library
A page from the Liber Abaci manuscript. Leonardo da Pisa wrote symbolic calculations in the margin to illustrate the methods described in the text. Click Here For A Closer Look

A page from the Liber Abaci manuscript. Leonardo da Pisa wrote symbolic calculations in the margin to illustrate the methods described in the text. Click Here For A Closer Look

Siena Public Library

Though generations of schoolchildren have cursed arithmetic, the world was a much more inconvenient place without it. Before the advent of modern arithmetic in the 13th century, basic calculations required a physical abacus.

But then came a young Italian mathematician named Leonardo da Pisa — no relation to da Vinci — who, in 1202, published a book titled Liber Abaci. That's Latin for "Book of Calculation."

And though it doesn't necessarily sound like an overnight best-seller, it was a smash hit. Liber Abaci introduced practical uses for the Arabic numerals 0 through 9 to Western Europe. The book revolutionized commerce, banking, science and technology and established the basis of modern arithmetic, algebra and other disciplines.

Weekend Edition "Math Guy" Keith Devlin tells the story of this arithmetic revolution in his new book, The Man of Numbers. Numerals 0 to 9 had been around in Hindu and Arabic cultures for centuries, but the problem was, Europeans didn't really do business with the numbers.

"They recorded everything in good old Roman numerals and if they wanted calculations, they went down the street to someone who was adept at using a physical abacus," Devlin tells NPR's Scott Simon. "It was actually a board with lines on it on which you moved pebbles around; it was a crude and inefficient way of doing business."

The first edition of Liber Abaci was a dense, detailed book that was hard for the average person to grasp. So da Pisa released a simplified version to reach the traders and commercial people of Pisa — and the result spread around the world.

"Within a few decades of Liber Abaci appearing you've got what may have been 1,000 or more different people writing practical arithmetic textbooks," says Devlin. "Ordinary people who wanted to set up a business — and didn't have a lot of money to pay people to do the accounting for them — could do it for themselves."

The basics of accounting, banking, insurance and double entry bookkeeping all came out of 13th century Pisa, Devlin says. And that was thanks to the new ability to do arithmetic efficiently.

Sure, basic arithmetic may seem a simple thing today, but Devlin says its introduction to the world was comparable to the invention of the computer. Tedious and complicated tasks that required a specialist were suddenly faster and easier — and something you could do for yourself. "[Da Pisa] is Steve Jobs, Bill Gates. It's the computer revolution that we lived through in the 1980s, and the parallels are actually uncanny," says Devlin.

Despite his lasting impact on the modern world, da Pisa is not exactly a household name. But you might recognize him by his nickname: Fibonacci. In addition to writing Liber Abaci, da Pisa also introduced the famous Fibonacci sequence to Western Europe. (Remember that one from high school math? It starts with 0 and then 1, and then every subsequent number is the sum of the two numbers that precede it.) The name was given to him by a historian in the 19th century who read the phrase filius Bonacci — "son of Bonacci" — at the beginning of Liber Abaci and gave da Pisa his moniker.

Although Fibonacci can take credit for practical arithmetic in the Western world, Devlin says that even without him, it's unlikely that people would have had to rely on the abacus forever.

"One of the things about almost all of mathematics is that it will eventually surface and get used," Devlin says. "It's a matter of who does it and when."

Excerpt: 'Man Of Numbers'

Your Days Are Numbered

Try to imagine a day without numbers. Never mind a day, try to imagine getting through the first hour without numbers: no alarm clock, no time, no date, no TV or radio, no stock market report or sports results in the newspapers, no bank account to check. It's not clear exactly where you are waking up either, for without numbers modern housing would not exist.

The fact is, our lives are totally dependent on numbers. You may not have "a head for figures," but you certainly have a head full of figures. Most of the things you do each day depend on and are conditioned by numbers. Some of them are obvious, like the ones listed above; others govern our lives behind the scenes. The degree to which our modern society depends on numbers that are hidden from us was made clear by the worldwide financial meltdown in 2008, when over-confident reliance on the advanced mathematics of futures predictions and the credit market led to a total collapse of the global financial system.

How did we — as a species and as a society — become so familiar with and totally reliant on these abstractions our ancestors invented just a few thousand years ago? As a mathematician, this question had puzzled me for many years, but for most of my career as a university professor of mathematics, the pressures of discovering new mathematics and teaching mathematics to new generations of students did not leave me enough time to look for the answer. As I grew older, however, and came to terms with the unavoidable fact that my abilities to do original mathematics were starting to wane a bit — a process that for most mathematicians starts around the age of forty (putting mathematics in the same category as many sporting activities) — I started to spend more time looking into the origins of the subject I have loved with such passion since I made the transition from "It's boring" to "It's unbelievably beautiful" around the age of sixteen.

For the most part, the story of numbers was easy to discover. By the latter part of the first millennium of the Current Era, the system we use today to write numbers and do arithmetic had been worked out — expressing any number using just the ten numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and adding, subtracting, multiplying, and dividing them by the procedures we are all taught in elementary school. (Units column, tens column, hundreds column, carries, etc.) This familiar way to write numbers and do arithmetic is known today as the Hindu-Arabic system, a name that reflects its history.

Prior to the thirteenth century, however, the only Europeans who were aware of the system were, by and large, scholars, who used it solely to do mathematics. Traders recorded their numerical data using Roman numerals, and performed calculations either by a fairly elaborate and widely used fingers procedure or with a mechanical abacus. That state of affairs started to change soon after 1202, the year a young Italian man, Leonardo of Pisa — the man who many centuries later a historian would dub "Fibonacci" — completed the first general purpose arithmetic book in the West, Liber abbaci, that explained the "new" methods in terms that ordinary people could understand — tradesmen and businessmen as well as schoolchildren. While other lineages can be traced, Leonardo's influence, through Liber abbaci, was by far the most significant and shaped the development of modern western Europe.

Leonardo learned about the Hindu-Arabic number system, and other mathematics developed by both Indian and Arabic mathematicians, when his father brought his young son to join him in the North African port of Bugia (now Bejaïa, in Algeria) around 1185, having moved there from Pisa to act as a trade representative and customs official. Years later, Leonardo's book not only provided a bridge that allowed modern arithmetic to cross the Mediterranean, it also bridged the mathematical cultures of the Arabic and European worlds, by showing the west the algebraic way of thinking that forms the basis of modern science and engineering (though not our familiar symbolic notation for algebra, which came much later).

What Leonardo did was every bit as revolutionary as the personal computer pioneers who in the 1980s took computing from a small group of "computer types" and made computers available to, and usable by, anyone. Like them, most of the credit for inventing and developing the methods Leonardo described in Liber abbaci goes to others, in particular Indian and Arabic scholars over many centuries. Leonardo's role was to "package" and "sell" the new methods to the world.

The appearance of Leonardo's book not only prepared the stage for the development of modern (symbolic) algebra, and hence modern mathematics, it also marked the beginning of the modern financial system and the way of doing business that depends on sophisticated banking methods. For instance, Professor William N. Goetzmann of the Yale School of Management, an expert on economics and finance, credits Leonardo as the first to develop an early form of present-value analysis, a method for comparing the relative economic value of differing payment streams, taking into account the time-value of money. Mathematically reducing all cash flow streams to a single point in time allows the investor to decide which is the best, and the modern version of the present-value criterion, developed by the economist Irving Fisher in 1930, is now used by virtually all large companies in the capital budgeting process.

Excerpted from Man of Numbers: Fibonacci's Arithmetic Revolution by Keith Devlin. Copyright 2011 by Keith Devlin. Reprinted with permission of Bloomsbury Publishing.

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