Transcript Fib_Revised_1_

Fibonacci
Numbers
By Nicole, Karen,
Arthur, Nico
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
 Answer: 12
 2. 5, 7.5,
, 16.875
 Answer: 11.25
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)
» whole numbers that occur in this sequence:
0, 1, 3, 5, 8, 13, 21, 34, 55, 89, 144…
Fibonacci
Number
» Take the sum of the previous two numbers
» You can pick any two whole numbers to start
» Follows formula:
Fn = Fn – 1 + Fn – 2
where F0 = 0 and F1 = 1
Examples
See if you can write some
Fibonacci sequences on your own!
"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?"
» 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
At the end of the year, 144
rabbits will be produced.
» 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
# of
Calves
1
2
5
8
13 21 34 55 89 144 233
At the end of
12 years, 233
calves will be
produced.
7
8
9
10 11
12
Fibonacci Numbers
& Real Life
Explains patterns on
many seemingly unrelated
things in nature:
• Petals on flowers
• Rows on Pinecones
• Sand dollar
• Starfish
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.
» 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.
» The Golden Ratio is:
1+
5
1.6180339887…
2
» If you take the ratio of two successive Fibonacci
numbers, the quotient gradually approach this
Golden Ratio
1/
1 = 1,
2/
8/
5 = 1·6,
1 = 2,
13/
3/
2 = 1·5,
8 = 1·625,
21/
5/
13
3
= 1·666...,
= 1·61538...