Introduction to Fibonacci number In mathematics, the Fibonacci numbers are the numbers in the

1. Introduction to Fibonacci number In mathematics, the Fibonacci numbers are the numbers in the following integer sequence: By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two. In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation: with seed values, The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci. Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian Mathematics. (By modern convention, the sequence begins with F0 = 0. The Liber Abaci began the sequence with F1 = 1, omitting the initial 0, and the sequence is still written this way by some.) • 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, 13, 21, • and 34. See golden spiral.

2. Occurrences in mathematics • The Fibonacci numbers are the sums of the "shallow" diagonals (shown in red) of Pascal’s Triangle • The Fibonacci numbers can be found in different ways in the sequence of binary strings. • The number of binary strings of length n without consecutive 1s is the Fibonacci number Fn+2. For example, out of the 16 binary strings of length 4, there are F6 = 8 without consecutive 1s – they are 0000, 0100, 0010, 0001, 0101, 1000, 1010 and 1001. By symmetry, the number of strings of length n without consecutive 0s is also Fn+2. • The number of binary strings of length n without an odd number of consecutive 1s is the Fibonacci number Fn+1. For example, out of the 16 binary strings of length 4, there are F5 = 5 without an odd number of consecutive 1s – they are 0000, 0011, 0110, 1100, 1111. • The number of binary strings of length n without an even number of consecutive 0s or 1s is 2Fn. For example, out of the 16 binary strings of length 4, there are 2F4 = 6 without an even number of consecutive 0s or 1s – they are 0001, 1000, 1110, 0111, 0101, 1010.

3. Relation of Fibonacci series with flowers (nature) 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, several things would become apparent. First, we would find that the number of petals on a flower is often one of the Fibonacci numbers. One-petalled ...  • white calla and two-petalled flowers are not common.  euphorb Three petals are more common.  Trillium

4. There are hundreds of species, both wild and cultivated, with five petals.  columbine Eight-petalled flowers are not so common as five-petalled, but there are quite a number of well-known species with eight.  bloodroot Thirteen, ...  black-eyed susan

5. Twenty-one and thirty-four petals are also quite common. The outer ring of ray florets in the daisy family illustrate the Fibonacci sequence extremely well. Daisies with 13, 21, 34, 55 or 89 petals are quite common.  shasta daisy with 21 petals Ordinary field daisies have 34 petals ... a fact to be taken in consideration when playing "she loves me, she loves me not". In saying that daisies have 34 petals, one is generalizing about the species - but any individual member of the species may deviate from this general pattern. There is more likelihood of a possible under development than over-development, so that 33 is more common than 35. The association of Fibonacci numbers and plants is not restricted to numbers of petals. Here we have a schematic diagram of a simple plant, the sneezewort. New shoots commonly grow out at an axil, a point where a leaf springs from the main stem of a plant.

6. If we draw horizontal lines through the axils, we can detect obvious stages of development in the plant. The main stem produces branch shoots at the beginning of each stage. Branch shoots rest during their first two stages, then produce new branch shoots at the beginning of each subsequent stage. The same law applies to all branches. Since this pattern of development mirrors the growth of the rabbits in Fibonacci's classic problem, it is not surprising then that the number of branches at any stage of development is a Fibonacci number. Furthermore, the number of leaves in any stage will also be a Fibonacci number.

7. Relation of fibonacci series with humans Our hand shows the Fibonacci Series Each section of your index finger, from the tip to the base of the wrist, is larger than the preceding one by about the Fibonacci ratio of 1.618, also fitting the Fibonacci numbers 2, 3, 5 and 8. By this scale, your fingernail is 1 unit in length. Curiously enough, you also have 2 hands, each with 5 digits, and your 8 fingers are each comprised of 3 sections.  All Fibonacci numbers! The ratio of the forearm to hand is Phi Your hand creates a golden section in relation to your arm, as the ratio of your forearm to your hand is also 1.618, the Divine Proportion.

8. Even your feet show phi The foot has several proportions based on phi lines, including : 1)  The middle of the arch of the foot 2)  The widest part of the foot 3)  The base of the toe line and big toe 4)  The top of the toe line and base of the "index" toe

9. Prepared by: Shubham & Ankit Class :- +1 Drvd.a.v. Centenary public school ,, phillaur (india).