Fibonacci Numbers

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

2. Goals • Answer: 12 • Answer: 11.25 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, , 16.875

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. 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 What are Fibonacci Numbers?

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

7. "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?" The Famous Fibonacci Question

8. 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 10 number sequence Answer At the end of the year, 144 rabbits will be produced.

9. 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? At the end of 12 years, 233 calves will be produced. The Cow Question

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. 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. 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. Fibonacci Numbers in Mother Nature

13. TheGoldenRatio is: 1 + 5 1.6180339887… 2 • If you take the ratio of two successive Fibonacci numbers, the quotient gradually approach this GoldenRatio 1/1 = 1,   2/1 = 2,   3/2 = 1·5,   5/3 = 1·666...,  8/5= 1·6,   13/8 = 1·625,   21/13 = 1·61538... The Golden Ratio