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It’s all About Fibonacci Numbers. Learn anything you need and want to know about Fibonacci. What is it? Who is it? Why is it important? By: Amaris Parris

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Who is Fibonacci? Leonardo was born in Pisa, Italy. His father Guglielmo was nicknamed Bonaccio ("good natured" or "simple"). Leonardo's mother, Alessandra, died when he was nine years old. Leonardo was posthumously given the nickname Fibonacci (derived from filius Bonacci, meaning son of Bonaccio). Leonardo became an amicable guest of the Emperor Frederick II, who enjoyed mathematics and science. In 1240 the Republic of Pisa honoured Leonardo, referred to as Leonardo Bigollo, by granting him a salary. In the 19th century, a statue of Fibonacci was constructed and erected in Pisa. Today it is located in the western gallery of the Camposanto, historical cemetery on the Piazza dei Miracoli.

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What is a Fibonacci Number? The Fibonacci number sequence (1,2,3,5,8,13,21,34,55,89,144,...) is constructed by adding the first two numbers to arrive at the third. The ratio of any number to the next number is 61.8 percent, which is a popular Fibonacci retracement number. The inverse of 61.8 percent is 38.2 percent, also used as a Fibonacci retracement number. It is the ratio of the Fibonacci sequence that is important and valuable, not the actual numbers in the sequence. The Fibonacci number sequence (1,2,3,5,8,13,21,34,55,89,144,...) is constructed by adding the first two numbers to arrive at the third. The ratio of any number to the next number is 61.8 percent, which is a popular Fibonacci retracement number. The inverse of 61.8 percent is 38.2 percent, also used as a Fibonacci retracement number. It is the ratio of the Fibonacci sequence that is important and valuable, not the actual numbers in the sequence.

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What is A Fibonacci Sequence? A tiling with squares whose sides are successive Fibonacci numbers in length A tiling with squares whose sides are successive Fibonacci numbers in length In mathematics, the Fibonacci numbers are a sequence of numbers named after Leonardo of Pisa, known as Fibonacci. Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been previously described in Indian mathematics. In mathematics, the Fibonacci numbers are a sequence of numbers named after Leonardo of Pisa, known as Fibonacci. Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been previously described in Indian mathematics. The first number of the sequence is 0, the second number is 1, and each subsequent number is equal to the sum of the previous two numbers of the sequence itself. In mathematical terms, it is defined by the following recurrence relation: The first number of the sequence is 0, the second number is 1, and each subsequent number is equal to the sum of the previous two numbers of the sequence itself. In mathematical terms, it is defined by the following recurrence relation:

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Examples Of Fibonacci Sequence Probably most of us have never taken the time to examine very carefully the number or arrangement of petals on a flower. If we were to do so, we would find that the number of petals on a flower, that still has all of its petals intact and has not lost any, for many flowers is a Fibonacci number. Probably most of us have never taken the time to examine very carefully the number or arrangement of petals on a flower. If we were to do so, we would find that the number of petals on a flower, that still has all of its petals intact and has not lost any, for many flowers is a Fibonacci number. 3 petals: lily, iris 5 petals: buttercup, wild rose, larkspur, columbine (aquilegia) 8 petals: delphiniums 13 petals: ragwort, corn marigold, cineraria, 21 petals: aster, black-eyed susan, chicory 34 petals: plantain, pyrethrum 55, 89 petals: michaelmas daisies, the asteraceae family Some species are very precise about the number of petals they have - e.g. buttercups, but others have petals that are very near those above, with the average being a Fibonacci number.

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Another Example Of A Fibonacci Sequence The original problem that Fibonacci investigated, in the year 1202, was about how fast rabbits could breed in ideal circumstances. "A pair of rabbits, one month old, is too young to reproduce. Suppose that in their second month, and every month thereafter, they produce a new pair. If each new pair of rabbits does the same, and none of the rabbits dies, how many pairs of rabbits will there be at the beginning of each month?" At the end of the first month, they mate, but there is still one only 1 pair. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs. The number of pairs of rabbits in the field at the start of each month is 1, 1, 2, 3, 5, 8, 13, 21, etc.

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Fibonacci Facts The Fibonacci sequence first appeared as the solution to a problem in the Liber Abaci, a book written in 1202 by Leonardo Fibonacci of Pisa to introduce the Hindu- Arabic numerals used today to a Europe still using cumbersome Roman numerals. The original problem in the Liber Abaci asked how many pairs of rabbits can be generated from a single pair, if each month each mature pair brings forth a new pair, which, from the second month, becomes productive. The resulting Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,..., have been the subject of continuing research, especially by the Fibonacci Association, publisher of the Fibonacci Quarterly since 1963. The Fibonacci numbers are found to have many relationships to the Golden Ratio F = (1 + /5)/2, a constant of nature and a value which fascinated the ancient Greeks, appearing throughout Greek art and architecture. One can verify with a hand calculator that the ratio of Fn+1 to Fn is approximated by 1.6180339..., which is the decimal equivalent of the Golden Ratio. Aside from an extraordinary wealth of mathematical relationships, there are numerous applications of the Fibonacci numbers in the most diverse fields. The Fibonacci Quarterly (FQ) has served as a focal point for widespread interest in the Fibonacci and related sequences, especially with respect to new results, challenging problems, and innovative proofs. The following references for Fibonacci number theory and applications in botany, biology, physics, music, and art provide an anthology of attractive mathematical ideas, selected from an abundance of material. Articles from all back issues of FQ are available from Bell & Howell Information and Learning, Article Clearinghouse Division, 300 North Zeeb Road, Ann Arbor MI 48106-1346.

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Interesting Facts It is an infinite sequence. It is an infinite sequence. Neighbouring Fibonacci numbers have no common factors. A proof of this is: “if a and b have a common factor then it must also be a factor of a + b if a and b have a common factor then it is also a factor of b-a if a and b have no common factor, then neither do b and a+b for if b and a+b had a common factor, then their difference would too and the difference is just a. So in any Fibonacci series which starts with a and b that have no common factors, then neither do any of the other number pairs in the series. Since 1 and 1 have no common factor, then no neighbouring pairs in the series will.” This proof is taken from http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html Neighbouring Fibonacci numbers have no common factors. A proof of this is: “if a and b have a common factor then it must also be a factor of a + b if a and b have a common factor then it is also a factor of b-a if a and b have no common factor, then neither do b and a+b for if b and a+b had a common factor, then their difference would too and the difference is just a. So in any Fibonacci series which starts with a and b that have no common factors, then neither do any of the other number pairs in the series. Since 1 and 1 have no common factor, then no neighbouring pairs in the series will.” This proof is taken from http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html Every number is a factor of other Fibonacci numbers, for example third number is a multiple of 2, every 4th is a multiple of 3, every 5th is a multiple of 5,every sixth is a multiple of 8. Every number is a factor of other Fibonacci numbers, for example third number is a multiple of 2, every 4th is a multiple of 3, every 5th is a multiple of 5,every sixth is a multiple of 8. The ratio of Fibonacci numbers converges to phi, which is 1.618 (to 3 decimal places). The ratio of Fibonacci numbers converges to phi, which is 1.618 (to 3 decimal places). Take 3 consecutive numbers in the Fibonacci sequence, square the middles number, multiply the first and third number, the difference between the two answers is always one. Take 3 consecutive numbers in the Fibonacci sequence, square the middles number, multiply the first and third number, the difference between the two answers is always one. Take 4 consecutive numbers, multiply the two outside and two inside numbers. The first product will be either one more or one less than the second. Take 4 consecutive numbers, multiply the two outside and two inside numbers. The first product will be either one more or one less than the second.

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Here Are Some Links To Take A Quiz And See How Much You Know! Have Fun! http://www.funtrivia.com/playquiz/quiz15029911369b0.html http://www.funtrivia.com/playquiz/quiz15029911369b0.html http://www.funtrivia.com/ http://www.funtrivia.com/ http://www.funtrivia.com/quizzes/ http://www.funtrivia.com/quizzes/ http://www.funtrivia.com/trivia-quiz/General/The-Fibonacci- Sequence-150299.html http://www.funtrivia.com/trivia-quiz/General/The-Fibonacci- Sequence-150299.html http://www.funtrivia.com/trivia-quiz/General/The-Fibonacci- Sequence-150299.html

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Hope You Have A Better Understanding On Fibonacci Numbers!!! Written By: Amaris Parris Of Class 803- Technology

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