1. By Nicole, Karen, Arthur, Nico
Fibonacci Numbers
By Nicole, Karen, Arthur, Nico

2. Goals
Aim: What are Fibonacci Numbers and how do they relate to nature?
Do Now: Fill in the missing numbers in the following sequences:
1. 3, 6, 9, , 15, 18
2. 5, 7.5, ,
Answer: 12
Answer: 11.25

3. Leonardo Pisano Fibonacci (1170-1250)
Born in Italy, he obtained his education in
North Africa (little known about family life)
Replaced the Roman Numeral System
of numbers with Hindu-Arabic Symbols (0-9)
Discovered the (later titled)
Fibonacci Sequence - each number is the sum of the two preceding numbers (pattern found in curve of snail shells and seed patterns of flowers)

4. What are Fibonacci Numbers?
whole numbers that occur in this sequence:
Take the sum of the previous two numbers
You can pick any two whole numbers to start
0, 1, 3, 5, 8, 13, 21, 34, 55, 89, 144…
Fibonacci Number
Follows formula:
Fn = Fn – 1 + Fn – 2
where F0 = 0 and F1 = 1

6. See if you can write some Fibonacci sequences on your own!
Examples
See if you can write some Fibonacci sequences on your own!

7. The Famous Fibonacci Question
"Start with a pair of rabbits, (one male and one female) born on January 1. Assume that all months are of equal length and that:
1. rabbits begin to produce young two months after their own birth;
2. after reaching the age of two months, each pair produces a mixed pair, (one male, one female), and then another mixed pair each month thereafter; and
3. no rabbit dies.
How many pairs of rabbits will there be after one year?"

8. At the end of the year, 144 rabbits will be produced.
Answer
Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144
You start off with two rabbits: one male, one female
The rabbits are born in January. Two months later, they reproduce
Continues for rest of the year (10 months)
10 times number sequence
At the end of the year, 144 rabbits will be produced.

9. At the end of 12 years, 233 calves will be produced.
The Cow Question
If a cow produces its first she-calf at age two years and after that produces another single she-calf every year, how many she-calves are there after 12 years, assuming none die?
Age in Years 2 3 4 5 6 7 8 9 10 11 12
# of Calves 1 13 21 34 55 89
144
233
At the end of 12 years, 233 calves will be produced.

10. Fibonacci Numbers & Real Life
Explains patterns on many seemingly unrelated things in nature:
Petals on flowers
Rows on Pinecones
Sand dollar
Starfish

11. More Fibonacci Numbers in Nature
Fibonacci Sequence can be found in several places in nature.
Examples include:
Flowers- petals usually grow in rings according to the sequence (1,1,2,3,5,8,etc.)
Rabbits- roughly multiply at the same numbers as the Fibonacci Sequence. Although not exact, the time it takes to give birth and population makes the correlation uncanny.

12. Fibonacci Numbers in Mother Nature
The pattern of Fibonacci numbers have been seen even in areas where math isn’t everything.
Many varieties of plant and tree-life have been seen as having patterns of Fibonacci numbers.
These patterns have been noticed as the arrangements of leaves or the scars on the bark of trees.
These patterns on nature were not genetically changed.
They appear due to the uses of the Fibonacci pattern. Through the use of the pattern, the most amount is packed into the smallest space.

13. The Golden Ratio 1/1 = 1, 2/1 = 2, 3/2 = 1·5, 5/3 = 1·666...,
The Golden Ratio is: … 2
If you take the ratio of two successive Fibonacci numbers, the quotient gradually approach this Golden Ratio
1/1 = 1,   2/1 = 2,   3/2 = 1·5,   5/3 = 1·666..., 
8/5 = 1·6,   13/8 = 1·625,   21/13 = 1·