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FIBONACCI NUMBERS

FIBONACCI NUMBERS. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 , 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 570288, 9227465, 14930352, 2415781, 39088169 . {.

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FIBONACCI NUMBERS

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  1. FIBONACCI NUMBERS 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 570288, 9227465, 14930352, 2415781, 39088169 .. {                   0,                       if  n = 0, F(n) =        1,                       if  n = 1,                   F(n-1) + F(n-2),  if  n > 1 Presented By : SoumyaGulati Class IX–A Roll No. 23

  2. About Fibonacci : • Born in Pisa, Italy in 1175 AD • Full name was Leonardo Pisano • Grew up with a North African education under the • Moors • Traveled extensively around the Mediterranean • coast • Met with many merchants and learned their systems • of arithmetic • Realized the advantages of the Hindu-Arabic system Fibonacci’s Mathematical Contribution : Books and Letters • Wrote five mathematical works ::: Four books and One preserved letter • Liber Abbaci (The Book of Calculating) • Practicageometriae (Practical Geometry) • Flos • Liber quadratorum (The Book of Squares) • A letter to Master Theodoruswritten around 1225

  3. FIBONACCI’S MATHEMATICAL CONTRIBUTION • Introduced the Hindu-Arabic number system into Europe based on ten digits and a decimal point • Europe previously used the Roman number system consisted of Roman numerals • Persuaded mathematicians to use the Hindu-Arabic number system I = 1 V = 5 X = 10 L = 50 C = 100 D = 500 M = 1000 1 2 3 4 5 6 7 8 9 0 AND

  4. THE FIBONACCI NUMBERS • Were introduced in “The Book of Calculating” • Series begins with 0 and 1 • Each subsequent number is the sum of the previous two. • So now our sequence becomes 0,1, 1, 2. • The next number will be 3. • Pattern is repeated over and over In mathematical terms, the sequence F nof Fibonacci numbers is defined by the recurrence relation F n = Fn-1 + Fn-2with seed values F0 = 0 and F n = 1 N is a Fibonacci number if and only if 5 N2 + 4 or 5 N2 – 4 is a square number

  5. THE FIBONACCI NUMBERS The first 21 Fibonacci numbers F nfor n = 0, 1, 2, ..., 20 are : The sequence can also be extended to negative index nusing the re-arranged recurrence relation which yields the sequence of "negafibonacci" numbers[ satisfying Thus the complete sequence is

  6. FIBONACCI’S RABBITS Suppose a newly-born pair of rabbits (one male, one female) are put in a field. • Rabbits are able to mate at the age of one month so that at the end of its second month, a female can produce • another pair of rabbits. • Therefore the number of rabbits per month = • End of the first month = 1 pair • End of the second month = 2 pair • End of the third month = 3 pair • End of the fourth month = 5 pair • 5 pairs of rabbits produced in one year. 1, 1, 2, 3, 5, 8, 13, 21, 34, …

  7. FIBONACCI’S HONEY BEES The number of ancestors of honey bees at each generation follows the Fibonacci series!

  8. FIBONACCI’S NUMBERS IN PASCAL’S TRIANGLE • In Pascal’s Triangle, the entry is sum of the two numbers either side of it, but in the row above “shallow” diagonal sums in Pascal’s Triangle are the Fibonacci numbers • Fibonacci numbers can also be found using a formula. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1

  9. FIBONACCI’S NUMBERS AND PYTHAGORUS TRIANGLES • Every successive series of four Fibonacci numbers can be used to generate Pythagorean triangles • Fibonacci numbers 1, 2, 3, 5 produce Pythagorean ∆ with sides 5, 12, 13 • First side (a) of Pythagorean triangle = 12 • Second side (b) of Pythagorean triangle = 5 • Third side (c) of Pythagorean triangle = 13 METHOD : Any four consecutive Fibonacci numbers F n, Fn+1, Fn+2 and Fn+3 can also be used to generate a Pythagorean triple Multiply the two middle or inner numbers (here 2 x 3 = 6); Double the result ( 6 x 2 = 12). [SIDE a] a = 2 Fn+1 x Fn+2 Multiply together the two outer numbers ( 1 x 5 = 5). [SIDE b] b = F n x Fn+3 The hypotenuse is found by adding together the squares of the inner two numbers (here 22=4 and 32=9 and their sum is 4+9=13). [SIDE 3] c = (Fn+1 )² x (Fn+2 )²

  10. FIBONACCI’S NUMBERS IN NATURE • Fibonacci spiral found in both snail and sea shells. A tiling with squares whose sides are successive Fibonacci numbers in length A Fibonacci spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling; this one uses squares of sizes 1, 1, 2, 3, 5& 8.

  11. FIBONACCI’S NUMBERS IN PLANTS • Sneezewort (Achilleaptarmica) shows the Fibonacci numbers This plant in particular shows the Fibonacci numbers in the number of "growing points" that it has. Suppose that when a plant puts out a new shoot, that shoot has to grow two months before it is strong enough to support branching. If it branches every month after that at the growing point, we get the picture shown here.

  12. FIBONACCI’S NUMBERS IN LEAVES OF PLANTS • Plants show the Fibonacci numbers in the arrangements of their leaves • Three clockwise rotations, passing five leaves • Two counter-clockwise rotations

  13. FIBONACCI’S NUMBERS IN PINE CONE PLANT • Pinecones clearly show the Fibonacci spiral

  14. FIBONACCI’S NUMBERS IN TECHNOLOGY • Fibonacci spirals can be made through the use of visual computer programs. • Each sequence of layers is a certain linear combination of previous ones.

  15. FIBONACCI’S NUMBERS IN FLOWERS & SEEDS ON FLOWER HEADS Lilies and Irises= 3 petals Buttercups and Wild Roses = 5 petals Arrangement of seeds on flower heads 55 spirals spiraling outwards & 34 spirals spiraling inwards Corn marigolds 13 petals Black-eyed Susan’s 21 petals

  16. FIBONACCI’S NUMBERS IN FRUITS & VEGETABLES If we cut a fruit or  vegetable we will often find that the number of sections is a Fibonacci number:  Section of Banana 3 Pineapple scales have Fibonacci spirals in sets of 8, 13, 21 Section of Apple 5 Bananas have 3 or 5 flat sides Fibonacci spiral can be found in cauliflower

  17. FIBONACCI’S NUMBERS IN OUR HAND • The Fibonacci numbers can be found in the human hand and fingers • Every human has 2 hands, which contain 5 fingers • Each finger has 3 parts separated by 2 knuckles All of these numbers fit into the sequence. However we need to keep in mind, that this could simply be coincidence

  18. FIBONACCIS’S NUMBERS IN GRAPHS • The Fibonacci numbers arise from the golden section • The graph shows a line whose gradient is Phi • First point close to the line is (0, 1) • Second point close to the line is (1, 2) • Third point close to the line is (2, 3) • Fourth point close to the line is (3, 5) • The coordinates are successive Fibonacci numbers

  19. MAKING MODEL ON FIBONACCI NUMBERS

  20. BIBLIOGRAPHY http://www.mathsisfun.org http://www.wikipedia.com http://www.mcs.surrey.ac.uk http://www.evolutionoftruth.com http://www.pass.maths.org.uk http://www.braungardt.com http://www.sigmaxi.org http://www.violin.odessa.ua

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