1167 words - 5 pages

Leonardo Pisano(1170-1250) was an Italian number theorist, who was con-sidered to be one of the most talented mathematicians in the Middle Ages.However, He was better known by his nickname Fibonacci, as many famoustheorems were named after it. In addition to that, Fibonacci himself some-times used the name Bigollo, which means good-for-nothing or a traveller. Thisis probably because his father held a diplomatic post, and Fibonacci travelledwidely with him. Although he was born in Italy, he was educated in NorthAfrica and he was taught mathematics in Bugia. While being a 'bigollo', hediscovered the enormous advantages of the mathematical systems used in thecountries he visited.Fibonacci's contributions to mathematics are remarkable. Even in the worldtoday, we still make daily use of his discovery. His most outstanding contributionwould be the replacement of decimal number system. Yet, few people realizedit. Fibonacci had actually replaced the old Roman numeral system with theHindu-Arabic numbering system, which consists of Hindu-Arabic(0-9) symbols.There were some disadvantages with the Roman numeral system: Firstly, it didnot have 0's and lacked place value; Secondly, an abacus was usually requiredwhen using the system. However, Fibonacci saw the superiority of using Hindu-Arabic system and that is the reason why we have our numbering system today.1He had included the explanation of our current numbering system in his book\Liber Abaci". The book was published in 1202 after his return to Italy. It wasbased on the arithmetic and algebra that Fibonacci had accumulated during histravels.In the third section of his book \Liber Abaci", there is a math questionthat triggers another great invention of mankind. The problem goes like this:A certain man put a pair of rabbits in a place surrounded on all sides by awall. How many pairs of rabbits can be produced from that pair in a year if it issupposed that every month each pair begets a new pair, which from the secondmonth on becomes productive? This was the problem that led Fibonacci to theintroduction of the Fibonacci Numbers and the Fibonacci Sequence. What isso special about the sequence? Let's take a look at it. The sequence is listed asSn=f1, 1, 2, 3, 5, 8, 13, 21, 34, 55, g(1)Starting from 1, each number is the sum of the two preceding numbers. Writingmathematically, the sequence looks likeSn=f8 i > 2; i 2 Z; ai = ai2 + ai1 where a1 = a2 = 1g(2)The most important and inuential property of the sequence is that the higherup in the sequence, the closer two consecutive Fibonacci numbers divided byeach other will approach the golden ratio1, ' = 1+p52 1:61803399. The proveis easy. By de nition, we have' = a+ba = ab(3)From '=ab , we can obtain a = b'. Then, by plugging into Equation 3, we willget b'+bb' = b'b . Simplify, we can get a quadratic equation '2 ' 1 = 0.Solving it, ' = 1+p52 1:61803399. The golden ratio was widely used in theRenaissance2 in painting....

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