"'The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0.... any number may be written, as is demonstrated below.--Leonardo Fibonacci (CA. 1170s-1240s)'
"With the above words, Leonardo of Pisa ...also known as Leonardo Fibonacci, began his first and best-known book, "Liber abaci" (Book of the abacus), published in 1202. At the time the book appeared, only a few privileged European intellectuals who cared to study the translations of the works of al-Khwarizmi and Abu Kamil knew the Hindu-Arabic numerals we use today. Fibonacci, who for a while joined his father, a customs and trading official, in Bugia (in present-day Algeria), and later traveled to other Mediterranean countries (including Greece, Egypt, and Syria), had the opportunity to study and compare different numerical systems and methods for arithmetical operations. Upon concluding that the Hindu-Arabic numerals, which included the place-value principle, were far superior to all other methods, he devoted the first seven chapters of his book to explanations of Hindu-Arabic notation and its use in practical applications.
"In Bugia (now called Bejaia), in Algeria, Fibonacci became acquainted with the art of the nine Indian figures, probably with, in his words, the "excellent instruction" of an Arab teacher....
"In many cases, Fibonacci gave more than one version of the problem, and he demonstrated an astonishing versatility in the choice of several methods of solution. In addition, his algebra was often theoretical, explaining in words the desired solution rather than solving explicit equations, as we do today. Here is a nice example of one of the problems that appear in "Liber abaci"...:
"A man whose end was approaching summoned his sons and said: 'Divide my money as I shall prescribe.' To the eldest son, he said, 'You are to have 1 bezant [a gold coin first struck in Byzantium] and a seventh of what is left.' To his second son he said, 'Take 2 bezants and a seventh of what remains.' To the third son, 'You are to take 3 bezants and a seventh of what is left.' Thus he gave each son 1 bezant more than the previous son and a seventh of what remained, and to the last son all that was left. After following their father's instructions with care, the sons found that they had shared their inheritance equally. How many sons were there and how large was the estate?
"For the interested reader, I present both the algebraic (modern) solution and Fibonacci's rhetorical solution to this problem in Appendix 6."
p.92-95, "Son of Good Nature," THE GOLDEN RATIO, by Mario Livio (Random House, Inc. NY: 2003)
"'The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0.... any number may be written, as is demonstrated below.--Leonardo Fibonacci (CA. 1170s-1240s)'
"With the above words, Leonardo of Pisa ...also known as Leonardo Fibonacci, began his first and best-known book, "Liber abaci" (Book of the abacus), published in 1202. At the time the book appeared, only a few privileged European intellectuals who cared to study the translations of the works of al-Khwarizmi and Abu Kamil knew the Hindu-Arabic numerals we use today. Fibonacci, who for a while joined his father, a customs and trading official, in Bugia (in present-day Algeria), and later traveled to other Mediterranean countries (including Greece, Egypt, and Syria), had the opportunity to study and compare different numerical systems and methods for arithmetical operations. Upon concluding that the Hindu-Arabic numerals, which included the place-value principle, were far superior to all other methods, he devoted the first seven chapters of his book to explanations of Hindu-Arabic notation and its use in practical applications.
"In Bugia (now called Bejaia), in Algeria, Fibonacci became acquainted with the art of the nine Indian figures, probably with, in his words, the "excellent instruction" of an Arab teacher....
"In many cases, Fibonacci gave more than one version of the problem, and he demonstrated an astonishing versatility in the choice of several methods of solution. In addition, his algebra was often theoretical, explaining in words the desired solution rather than solving explicit equations, as we do today. Here is a nice example of one of the problems that appear in "Liber abaci"...:
"A man whose end was approaching summoned his sons and said: 'Divide my money as I shall prescribe.' To the eldest son, he said, 'You are to have 1 bezant [a gold coin first struck in Byzantium] and a seventh of what is left.' To his second son he said, 'Take 2 bezants and a seventh of what remains.' To the third son, 'You are to take 3 bezants and a seventh of what is left.' Thus he gave each son 1 bezant more than the previous son and a seventh of what remained, and to the last son all that was left. After following their father's instructions with care, the sons found that they had shared their inheritance equally. How many sons were there and how large was the estate?
"For the interested reader, I present both the algebraic (modern) solution and Fibonacci's rhetorical solution to this problem in Appendix 6."
p.92-95, "Son of Good Nature," THE GOLDEN RATIO, by Mario Livio (Random House, Inc. NY: 2003)