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 : Soumya Gulati Class IX–A Roll No. 23

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) Practica geometriae (Practical Geometry) Flos Liber quadratorum (The Book of Squares) A letter to Master Theodorus written around 1225

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

F n = Fn-1 + Fn-2 with seed values F0 = 0 and F n = 1 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 n of Fibonacci numbers is defined by the recurrence relation F n = Fn-1 + Fn-2 with 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

THE FIBONACCI NUMBERS The first 21 Fibonacci numbers F n for n = 0, 1, 2, ..., 20 are : F₀ F₁ F₂ F₃ F₄ F₅ F₆ F₇ F₈ F₉ F₁₀ F₁₁ F₁₂ F₁₃ F₁₄ F₁₅ F₁₆ F₁₇ F₁₈ F₁₉ F₂₀ 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 The sequence can also be extended to negative index n using the re-arranged recurrence relation which yields the sequence of "negafibonacci" numbers[ satisfying Thus the complete sequence is F−8 F−7 F−6 F−5 F−4 F−3 F−2 F−1 F0 F1 F2 F3 F4 F5 F6 F7 F8 −21 13 −8 5 −3 2 −1 1 3 8 21

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, …

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

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

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 𝑎 2 + 𝑏 2 = 𝑐 2 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

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.

FIBONACCI’S NUMBERS IN PLANTS Sneezewort (Achillea ptarmica) 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.

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

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

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.

Arrangement of seeds on flower heads 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

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 Apple 5 Section of Banana 3 Pineapple scales have Fibonacci spirals in sets of 8, 13, 21 Bananas have 3 or 5 flat sides Fibonacci spiral can be found in cauliflower

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

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

MAKING MODEL ON FIBONACCI NUMBERS

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