What are the Fibonacci numbers and why are they so important? What do they explain?

Can you please explain in ';layman's terms';? I looked it up on a couple of different sites, but I'm still at a loss. My area of study isn't math, nor is math my strong suit.





Thanks in advance for your help.





PS. This isn't homework or anything like that. I am just interested.What are the Fibonacci numbers and why are they so important? What do they explain?
Fibonacci's sequence is a set of numbers starting with 0 and 1 in which each subsequent number is the sum of the two before it.





0 + 1 = 1


1 + 1 = 2


1 + 2 = 3 and so on...





0,1,1,2,3,5,8,13,21,34...





This sequence of numbers was created by Fibonacci as a representation of the fastest possible growth of a rabbit population. It's importance in math is that it has many interesting and mysterious relationships with nature. For instance, the number of petals on most types of flowers is usually a number that can be found in the Fibonacci sequence.





Also, as you go farther through the sequence, the quotient of one number divided by the number that precedes it grows closer and closer to Phi, an irrational number also known as the ';Golden Ratio';. The value of Phi is approximately 1.618.... and it is a ratio that occurs in many natural things, such as certain proportions of the human body.What are the Fibonacci numbers and why are they so important? What do they explain?
011235813 last two numbers add to make the next.
It was that the Fibanacci sequence occurred in certain math problems, as well as nature. It is important in networking, and here is why...





If you look at a leaf, and the veins in it, you see that those veins distribute in this sequence. In other words, the ';network'; of distribution is according to the Fibinacci sequence.





But the main use is in things related to the ';Golden'; number, which keeps coming up over, and over in math. Here's some links...





http://en.wikipedia.org/wiki/Fibonacci_n鈥?/a>





http://www.mcs.surrey.ac.uk/Personal/R.K鈥?/a>





It is just a simple geometric series that has many applications.





Ron.
Fibonacci numbers are a sequence in which each number is the sum of the two previous numbers.





They're useful because they model certain populations. Let's say you're raising pet things, a creature that multiplies without needing a sexual partner. Things, it turns out, have a baby when they reach maturity, have another baby as soon as the first baby reaches maturity, and have a heart attack as soon as the second baby reaches maturity.





Fibonacci numbers do a good job of showing how your ';things'; population grows. If you think about it, the population growth of ';things'; is not a whole lot different than the human population. It turns out that Fibonacci numbers is useful for a lot of things. The shell of the chambered nautilus, for instance, shows a Fibonaccian growth pattern.





Studying Fibonacci numbers is like popcorn - the more you eat, the more you want.
The Fibonacci numbers or Fibonacci sequence were named after Fibonacci, although he didn't invent them. The seqence starts with 0 and 1 where the previous number in the sequence is added to the current number in the sequence:





0, 1, 1, 2, 3, 5, 8, 13,...





numbers in sequence are:


0 + 1 = 1


1+1=2


1+2=3


3+2=5


etc.





Here is a simple movie that talks about it (it is for kids, but is informative):





http://www.brainpop.com/math/numbersando鈥?/a>
These sequence of numbers are found in nature. Such as how many pinecones there are in each generation, or something like that.
Okay, you need to start with two initial values of 0 and 1. Once you have these, then you add these two numbers together to get the next number (1). So the series so far is:


0, 1, 1





To get the next number in the series, add the last two numbers (1 and 1) to get 2


0, 1, 1, 2





You continue this so that the 5th, 6th, 7th and 8th terms are 3, 5, 8 and 13, so the series thus far is:


0, 1, 1, 2, 3, 5, 8, 13





The next number in the series would be 21 (8 + 13).


They can go on forever.





The reason this is so important, is that this series has a tendency of occuring in nature a lot. If you've done a lot of searching on the Internet, one of the most 'famous' occurrences of this is the nautilus sea shell.





Also, the ratio between the last two terms will eventually become 1.618, which is known as the golden ratio. This number also happens very often in nature. For example, this is the ratio between the distance from the head to the fingertips and the distance from the head to the belly button. And this is just the beginning.





I hope this helps!
Fibonacci numbers


0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584.......





F(0) = 0


F(1) = 1


F(n) = F(n-1) + F(n-2)





Check the link for details.
The others have explained the series quite well. One of the places where Fibonacci numbers show very clearly is if you look at the seed pattern in a sunflower. The number of clockwise spirals and the number are counter-clockwise spirals are sucessive Fibonacci numbers.