FIBONACCI NUMBERS

Kevin George Michael SvoyskyJason Lombino Catherine Lee

Fibonacci numbers

• Named after Leonardo Pisano who was considered to be the greatest mathematician of the Middle Ages

• Sequence named by Eduardo Lucas• Leonardo Pisano stated as studying the nine

Indian figures and then “translated” to Italy

The Rabbit Problem How Many Pairs of Rabbits Are Created by One Pair in One YearA certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also. Because the above written pair in the first month bore, you will double it; there will be two pairs in one month. One of these, namely the first, bears in the second montth, and thus there are in the second month 3 pairs;of these in one month two are pregnant and in the third month 2 pairs of rabbits are born, and thus there are 5 pairs in the month;...there will be 144 pairs in this [the tenth] month;to these are added again the 89 pairs that are born in the eleventh month; there will be 233 pairs in this month.To these are still added the 144 pairs that are born in the last month; there will be 377 pairs, and this many pairs are produced from the abovewritten pair in the mentioned place at the end of the one year. You can indeed see in the margin how we operated, namely that we added the first number to the second, namely the 1 to the 2, and the second to the third, and the third to the fourth and the fourth to the fifth, and thus one after another until we added the tenth to the eleventh, namely the 144 to the 233, and we had the abovewritten sum of rabbits, namely 377, and thus you can in order find it for an unending number of months.

The method Leonardo used to teach the sequence in his Liber Abaci book.http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibBio.html

The Sequence0 01 12 13 24 35 56 87 138 219 3410 5511 8912 14413 23314 37715 61016 98717 159718 258419 418120 6765

IN LIFE

The Golden Ratio

• Can be found by dividing each Fibonacci number by the one before it

• This eventually settles down to about the golden number• It has a value of about 1.618034 and is often shown as the Greek

letter Phi (Φ)

• The total length is to the length of the longer part as the length of the longer part is to the length of the shorter part (a+b/a = a/b) only with a ratio of 1.618034/1

• The Parthenon in Greece has proportions that show the golden ratio

• The golden spiral is very similar to the Fibonacci spiral

Properties