Phi
What makes a number? Why does a number excite? What makes a number divine?
The Fibonacci Sequence
Leonardo of Pisa ( c1175- 1220) or Fibonacci first conjectured this series of numbers:
1 ,1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987...
So what I hear you say, all you are doing is adding the previous two numbers to create the nect number in the series! I've seen that in Dan Brown's The Da Vinci Code
That is the last time you will see the name Dan Brown on this web site, this sequence has hidden depths and beauty which was never really explored in the story. Jezza Bear has gone some way to look at this sequence and discover that beatiful number known as Phi
Phi
Let's look at the Fibonacci Sequence again and this time play with the numbers. We are going to divide the first number by the second, the second by the third , the third by the fourth ad infinitum:
- 1/1 = 1.000000
- 2/1 = 2.000000
- 3/2 = 1.500000
- 5/3 = 1.666666
- 8/5 = 1.600000
- 13/8 =1.625000
- 21/13=1.615385
- 34/21=1.619048
- 55/34=1.617647
- 89/55=1.618182
- 144/89=1.617978
- 233/144=1.618056
- 377/233=1.618026
- 610/377=1.618037
- 987/610=1.618033
Note the ratio now levels off to approximately 1.618 this is Phi.
Let's have some fun with Phi
OK, get yourself a calculator, go on, quickly and come straight back:
Try these sums out:
- 1.618 x 1.618 = ?
- (1.618 - 1) x 1.618 = ?
Impress your friends with lightning addition
Some annoying people, just like Jezza Bear, have the ability to add numbers very quickly in their head. This is normally down to knowing a few clever techniques, or more irritatingly that they are very clever. The Fibonacci sequence allows you to add quickly without much effort. The challenge is to calculate the sum of all the Fibonacci numbers from the first to the nth number. Let's see if you can do it?
- Calculate the first 10 Fibonacci numbers.
Answer
1 + 1 +2+3+ 5+8+13+ 21+ 34+ 55=143
The answer is simply (n +2)th - 1. Therefore the tenth Fibonacci number is 144 subtract 1 and you get 143. This is true for what ever value you want in the sequence.
Urrgh, Eleven is the Sin
Let's look at the sum above us again:
1 + 1 +2+3+ 5+8+13+ 21+ 34+ 55=143.
143 is divisible by 11 (143/11= 13) This is true for any consecutive 10 Fibonacci Numbers e.g
55 + 89 + 144 +233 + 377 + 610 +987 + 1597 +2584 +4181 = 10587
10587 is divisible by 11 (10587/11=987)
"HOLD UP," I hear you shout "something strange is going on here Jezza." You would be right, if you examine these two examples you will see something else. The sum of any 10 consecutive numbers is always 11 times the seventh number in that series. Again you can use this to amaze your friends when doing speed maths.
