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Math around Us: Fibonacci Numbers

Math around Us: Fibonacci Numbers. John Hutchinson March 2005. Leonardo Pisano Fibonacci. Born: 1170 in (probably) Pisa (now in Italy) Died: 1250 in (possibly) Pisa (now in Italy). What is a Fibonacci Number?. Fibonacci numbers are the numbers in the Fibonacci sequence

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Math around Us: Fibonacci Numbers

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  1. Math around Us:Fibonacci Numbers John Hutchinson March 2005

  2. Leonardo Pisano Fibonacci Born: 1170 in (probably) Pisa (now in Italy) Died: 1250 in (possibly) Pisa (now in Italy)

  3. What is a Fibonacci Number? Fibonacci numbers are the numbers in the Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, . . . , each of which, after the second, is the sum of the two previous ones.

  4. The Fibonacci numbers can be considered to be a function with domain the positive integers. Note that FN+2 = FN+1+ FN

  5. Note Every 3rd Fibonacci number is divisible by 2. Every 4th Fibonacci number is divisible by 3. Every 5th Fibonacci number is divisible by 5. Every 6th Fibonacci number is divisible by 8. Every 7th Fibonacci number is divisible by 13. Every 8thFibonacci number is divisible by 21. Every 9th Fibonacci number is divisible by 34.

  6. Sums of Fibonacci Numbers

  7. Sums of Fibonacci Numbers

  8. F1+ F2+ F3 + … + FN = FN+2 -1

  9. Sums of Squares

  10. Sums of Squares

  11. The Formula F12 + F22 + F32 + …+ Fn2 = FnX FN+1

  12. Another Formula FN+I = FI-1FN + FIFN+1

  13. Pascal’s Triangle

  14. Sums of Rows The sum of the numbers in any row is equal to 2 to the nth power or 2n, when n is the number of the row. For example:20 = 121 = 1+1 = 222 = 1+2+1 = 423 = 1+3+3+1 = 824 = 1+4+6+4+1 = 16

  15. Add Diagonals

  16. Pascal’s triangle with odd numbers in red.

  17. 1-White Calla Lily

  18. 1-Orchid

  19. 2-Euphorbia

  20. 3-Trillium

  21. 3-Douglas Iris

  22. 3&5 - Bougainvilla

  23. 5-Columbine

  24. 5-St. Anthony’s Turnip (buttercup)

  25. 5-Unknown

  26. 5-Wild Rose

  27. 8-Bloodroot

  28. 13-Black-eyed Susan

  29. 21-Shasta Daisy

  30. 34-Field Daisy

  31. Dogwood = 4?????

  32. Here a sunflower seed illustrates this principal as the number of clockwise spirals is 55 (marked in red, with every tenth one in white) and the number of counterclockwise spirals is 89 (marked in green, with every tenth one in white.)

  33. Sweetwart

  34. Sweetwart

  35. "Start with a pair of rabbits, (one male and one female). 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; and3. no rabbit dies.How many pairs of rabbits will there be after each month?"

  36. Let’s count rabbits

  37. Let’s count tokens A token machine dispenses 25-cent tokens. The machine only accepts quarters and half-dollars. How many ways can a person purchase 1 token, 2 tokens, 3 tokens, …?

  38. 25C Q 1 50C QQ-H 2 75C QQQ-HQ-QH 3 100C QQQQ-QQH-QHQ-HQQ-HH 5 125C QQQQQ-QQQH-QQHQ-QHQQ-HQQQ-HHQ-HQH-QHH 8 Count them

  39. 89 Measures Total Gets loud here 55 Measures 34 Measures Strings remove mutes Replace mutes 34 Measures 21 Measures 13 21 Measures 13 8 21 Theme Texture First Movement, Music for Strings, Percussion, and Celeste Bela Bartok

  40. The Keyboard

  41.                        <>                                                  <>           <>

  42. The hand

  43. Ratios of consecutive

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