FAQ

FAQ
 * 1.What is Fibonacci numbers?**

http://www.nist.gov/dads/HTML/fibonacciNumber.html
 * Definition:** A member of the sequence of numbers such that each number is the sum of the preceding two. The first seven numbers are 1, 1, 2, 3, 5, 8, and 13. F(n) ≈ round(Φn/√ 5), where Φ=(1+√ 5)/2.
 * Formal Definition:** The nth Fibonacci number is
 * F(n) = F(n-1) + F(n-2), where F(1)=1 and F(2)=1, or
 * F(n) = (Φn - φn)/√ 5, where Φ=(1+√ 5)/2 and φ=(1-√ 5)/2.

2.Who was the Fibonacci?

Fibonacci was a great mathematician. His full name was Leonardo of Pisa. He was born in Pisa in Italia in 1175AD. He was the son of a Persian merchant who also served as a customs officer in North Africa. He travelled widely in Barbary (Algeria) and was later sent on business trips to Egypt, Syria, Greece, Sicily and Provence. Fibonacci was probably the greatest genius of number theory during the 2000 years between Diaphanous and Fermat.

[|http://www.mcs.surrey.ac.uk/Personal/R.Knott/ (Ron Knott's web pages on Mathematics) 
 * 3.What are “Fibonacci rabbits?”**

Now let's consider hypothetical rabbits. Let's say we start with two rabbits. After a time, they produce two new rabbits. Then after a time, these four rabbits produce four more rabbits, etc. The number of rabbits at any given time is the sequence 2, 4, 8, 16, 32, ... (or 21, 22, 23, 24, 25, ...). This is called exponential growth. Now let's say that after our rabbits are born, they are too young to produce more rabbits right away. They have to skip a time period before they start producing pairs of rabbits. Here is a diagram. As you can see, the number of pairs, as time goes on, is the Fibonacci sequence. These rabbits are the famous Fibonacci Rabbits. You can see that this sequence is related to exponential growth. This sequence grows somewhat slower than the doubling shown above. In fact, instead of 21, 22, 23, 24, 25, ..., we can approximate our Fibonacci rabbit population with this series (as we saw above): ø0, ø1, ø2, ø3, ø4, ø5, ... And that is exponential growth. It may be that theorizing about these rabbits is how Fibonacci came up with the concept of Fibonacci numbers. http://www.jimloy.com/algebra/fibo.htm (Jim Loy's Mathematics Page)   **4. Where does Fibonacci number appear**?

The Fibonacci Series is found in Pascal's Triangle
Pascal's Triangle, developed by the French Mathematician Blaise Pascal, is formed by starting with an apex of 1. Every number below in the triangle is the sum of the two numbers diagonally above it to the left and the right, with positions outside the triangle counting as zero. The numbers on diagonals of the triangle add to the Fibonacci series, as shown below. [|http://goldennumber.net/pascal.htm 

**5. What is the easy way to multiply using Fibonacci number?** Let's take the same example: 19x65. This time we take just one number - say 65 - as the head of the right hand column, the left column starting with 1. The second row has 2 on the left and we double 65 to get 130 on the right. Now each successive row is the sum of the previous TWO entries above it, taking each column separately. So since we started with 1 and 2 on the left we will get 3,5,8,... that is, the Fibonacci numbers on the left hand side. Stop when we can find a Fibonacci number which is bigger than the other number in the product - here 19: 1 65+ 2 130 3 195 5 325+ 8 520+ 13 845+ 21 ...... We mark the rows this time by finding those entries in the left column that add up to 19. There many be several ways to do this selection but any will do. Here we have chosen 13+5+1. If we add up the right hand entries on these rows we have: 65+325+845=1235 which is again 19x65.

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibrep.html#fibmult