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FIBONACCI NUMBERS

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FIBONACCI NUMBERS

Chaitanya khurana

Class 9th

Roll No ---08

Bhaskar House

Step By Step School, Noida

The Fibonacci numbers can be rather simply

defined by the following:

- 1. Start with the numbers 1 and
- 2. Add them together to make the next number.
- 3. Then form the next number in sequence by adding the previous two together.

- More formally: [F1=1; F2=2], [F3=3], [Fn=Fn-1+Fn-2]. The resulting sequence begins like this:
- F1 to F19 : 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765..........
- The (recurrence) formula for these Fibonacci numbers is:
F(0)=0, F(1)=1, F(n)=F(n-1)+F(n-2) for n>1.

Fn Number Fn Number Fn Number Fn Number

F0 0 F13 233 F30 832040 F39 63245986

F1 1 F14 377 F27 196418 F40 102334155

F2 1 F15 610 F28 317811

F3 2 F16 987 F29 514229

F4 3 F17 1597 F30 832040

F5 5 F18 2584 F31 1346269

F6 8 F19 4181 F32 2178309

F7 13 F20 6765 F33 3524578

F8 21 F21 10946 F34 5702887

F9 34 F22 17711 F35 9227465

F10 55 F23 28657 F36 14930352

F11 89 F24 46368 F37 24157817

F12 144 F25 75025 F38 39088169

- Leonardo Pisano Bigolloalso known as Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or, most commonly, simply Fibonacci, was an Italian mathematician, considered by some "the most talented western mathematician of the MiddleAges.
- Fibonacci is best known to the modern world for the spreading of the Hindu–Arabic numeral system in Europe, primarily through the publication in 1202 of his Liber Abaci (Book of Calculation), and for a number sequence named the Fibonacci numbers after him, which he did not discover but used as an example in the Liber Abaci.

- Leonardo Fibonacci was born around 1170 to GuglielmoBonacci, a wealthy Italian merchant. Guglielmo directed a trading post (by some accounts he was the consultant for Pisa) in Bugia, a port east of Algiers in the Almohaddynasty’s sultanate in North Africa(now Bejaia, Algeria). As a young boy, Leonardo traveled with him to help; it was there he learned about the Hindu–Arabic numeral system.
- Recognizing that arithmetic with Hindu–Arabic numerals is simpler and more efficient than with Roman numerals, Fibonacci traveled throughout the Mediterranean world to study under the leading Arab mathematicians of the time. Leonardo returned from his travels around 1200. In 1202, at age 32, he published what he had learned in Liber Abaci( Book of Abacus or Book of Calculation), and thereby popularized Hindu–Arabic numerals in Europe.

- The Fibonacci numbers are Nature's numbering system. They appear everywhere in Nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind.

- Phyllotaxis is the study of the ordered position of leaves on a stem. The leaves on this plant are staggered in a spiral pattern to permit optimum exposure to sunlight. If we apply the Golden Ratio to a circle we can see how it is that this plant exhibits Fibonacci qualities. Click anywhere on the slide to see larger picture

- In the case of tapered pinecones or pineapples, we see a double set of spirals – one going in a clockwise direction and one in the opposite direction. When these spirals are counted, the two sets are found to be adjacent Fibonacci numbers.

- As well, many flowers have a Fibonacci number of petals. Some, like this rose, also have Fibonacci Spiral, petal arrangements.

- Branching plants also exhibit Fibonacci numbers. Again, this design provides the best physical accommodation for the number of branches, while maximizing sun exposure

- we have 8 fingers in total, 5 digits on each hand, 3 bones in each finger, 2 bones in 1 thumb, and 1 thumb on each hand.

2cm

- Even the length of the bones of ones finger is in Fibonacci number system

3cm

5cm

8cm

How is all this possible ???

- http://www.goldenmeangauge.co.uk/golden.htm
- http://ccins.camosun.bc.ca/~jbritton/goldslide/jbgoldslide.htm
- http://library.thinkquest.org/C005449/renaissance.html
- http://goldennumber.net/
- http://www2.spsu.edu/math/tile/aperiodic/penrose/euclid.htm
- http://library.thinkquest.org/27890/mainIndex.htm
- http://educ.queensu.ca/~fmc/october2001/GoldenArt.htm
- http://mathforum.org/dr.math/faq/faq.golden.ratio.html
- http://www.q-net.net.au/lolita/symmetry.htm
- http://ccins.camosun.bc.ca/~jbritton/goldslide/jbgoldslide.htm
- http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html
- http://savvyclasses.wikispaces.com