Numerical+Numbers


 * Numerical Numbers **

How did we become so familiar with, and so reliant on, these abstractions that our ancestors invented just a few thousand years ago? By the latter part of the first millennium AD, the system we use today to write numbers and do arithmetic had been worked out – expressing any number using just the 10 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 primary school. This familiar way to write numbers and do arithmetic is known as the Hindu-Arabic system, a name that reflects its history. Before the 13th century, however, the only Europeans aware of this system were, by and large, scholars, who used it solely to do mathematics. Traders recorded their data using Roman numerals, and performed calculations either by using their fingers or with a mechanical abacus. That state of affairs started to change soon after 1202, the year a young Italian man, Leonardo of Pisa, whom a historian centuries later would dub "Fibonacci", completed the first general-purpose book of arithmetic in the West. //Liber abbaci// explained the "new" methods in terms understandable to ordinary people – and its influence did as much as any other book to shape the development of modern Western Europe.

Leonardo had learnt about the Hindu-Arabic number system when his father took him to the north African port of Bugia (now Bejaïa, in Algeria) in around 1185. Years later, his book would provide not only a bridge that allowed arithmetic to cross the Mediterranean, but also one between the mathematical cultures of the Arabic and European worlds. It was an act every bit as revolutionary as the one carried out by personal computer pioneers in the Eighties who took computing from a small group of "computer types" and made it available to, and usable by, anyone. Not only did the appearance of //Liber abbaci// prepare the stage for the development of modern algebra and hence modern mathematics, but it also marked the beginning of the modern financial system and the way of doing business that depends on sophisticated banking methods. Until recently, history had relegated Leonardo to a footnote. Indeed, his name is known today primarily in connection with the Fibonacci numbers, a sequence that arises from the solution to the "**Rabbit Problem",** one of many whimsical challenges he put in //Liber abbaci// to break the tedium of the hundreds of practical problems that dominate the book. Nestled between puzzles involving the division of food and money, the rabbit problem involves an attempt to count a growing population. Leonardo did not invent it: it dates back at least to the Indian mathematicians who developed the number system that //Liber abbaci// described. But he realized, as they did, that it was an excellent way to practice how to use the new number system. In what was to become his most famous passage, Leonardo wrote his way into 20th-century popular culture with these words:- "A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also.'' Leonardo wanted the reader to assume that once two rabbits become fertile, they produce off-spring every month. As usual, he explained the solution in full detail, but the modern reader can rapidly discern the solution by glancing at the table Leonardo also presented, giving the rabbit population each month:- One animal at the beginning, then two, then three, then five, then eight, then 13, then 21, then 34, then 55, then 89, then 144. The general rule is that each successive number is the result of adding together the previous two: 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, etc. The numbers generated by this process are known today as the Fibonacci numbers, and were given their name by the French mathematician Edouard Lucas in the 1870s, after his compatriot, the historian Guillaume Libri, gave Leonardo the nickname Fibonacci in 1838.

One main reason why these numbers retain their fascination today is due to the surprising frequency with which they arise in nature. For example, the number of petals on flowers is a Fibonacci number more often than would be expected from pure chance: - An iris has three petals; primroses, buttercups, wild roses, larkspur, and columbine have five; delphiniums have eight; ragwort, corn marigold, and cineria 13; asters, black-eyed Susan, and chicory 21; daisies 13, 21, or 34; and Michaelmas daisies 55 or 89. Sunflower heads, and the bases of pine cones exhibit spirals going in opposite directions: the sunflower has 21, 34, 55, 89, or 144 clockwise, paired respectively with 34, 55, 89, 144, or 233 counterclockwise; a pine cone has eight clockwise spirals and 13 counterclockwise. We live with Leonardo's legacy every day – every time we do something that depends upon the modern arithmetic he brought to the West.
 * All Fibonacci numbers.**