Download Fibonacci Sequence http://www.youtube.com/watch?v

Fibonacci Sequence
http://www.youtube.com/watch?v=eeAGrFBDehE
http://www.youtube.com/watch?v=dr20wqKYYtQ
The Great Fibonacci Sequence is everywhere, just look and see!!!
The Fibonacci Sequence is the series of numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
The next number is found by adding up the two numbers before it.




The 2 is found by adding the two numbers before it (1+1)
Similarly, the 3 is found by adding the two numbers before it (1+2),
And the 5 is (2+3),
and so on!
Example: the next number in the sequence above would be 21+34 = 55
It is that simple!
Here is a longer list:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584,
4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, ...
Can you figure out the next few numbers?
Makes a Spiral
When you make squares with those widths, you get a nice spiral:
Do you see how the squares fit neatly together?
For example 5 and 8 make 13, 8 and 13 make 21, and so on.
The Rule
The Fibonacci Sequence can be written as a "Rule"
First, the terms are numbered from 0 onwards like this:
n=
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
...
xn =
0
1
1
2
3
5
8
13
21 34 55 89 144 233 377 ...
So term number 6 is called x6 (which equals 8).
Example: the 8th term is
the 7th term plus the 6th term:
x8 = x7 + x6
So we can write the rule: The Rule is xn = xn-1 + xn-2



where:
xn is term number "n"
xn-1 is the previous term (n-1)
xn-2 is the term before that (n-2)
Example: term 9 would be calculated like this:
x9 = x9-1 + x9-2 = x8 + x7 = 21 + 13 = 34
Golden Ratio
And here is a surprise. If you take any two successive (one
after the other)Fibonacci Numbers, their ratio is very close
to the Golden Ratio "φ" which is approximately 1.618034...
In fact, the bigger the pair of Fibonacci Numbers, the closer the
approximation. Let us try a few:
A
B
B/A
2
3
1.5
3
5
1.666666666...
1.6
5
8
8
13
1.625
...
...
...
144
233
1.618055556...
233
377
1.618025751...
Note: this also works if you pick two random whole numbers to begin the sequence, such as 192 and
16 (you would get the sequence 192, 16, 208, 224, 432, 656, 1088, 1744, 2832, 4576, 7408, 11984,
19392, 31376, ...):
A
B
B/A
192
16
16
208
13
208
224
1.07692308...
224
432
1.92857143...
0.08333333...
...
...
7408
11984
...
1.61771058...
11984
19392
1.61815754...
...
...
...
It takes longer to get good values, but it shows you that it is not just the Fibonacci Sequence that can
do this!
Using The Golden Ratio to Calculate Fibonacci Numbers
And even more surprising is that we can calculate any Fibonacci Number using the Golden
Ratio:
The answer always comes out as a whole number, exactly equal to the addition of the
previous two terms.
Example:
When I used a calculator on this (only entering the Golden Ratio to 6 decimal places) I got
the answer 8.00000033. A more accurate calculation would be closer to 8.
A Pattern
Here is the Fibonacci sequence again:
n=
0
1
2
3
4
5
6
7
xn =
0
1
1
2
3
5
8
13 21 34 55 89 144 233 377 610 ...
There is an interesting pattern:
8
9
10 11
12
13
14
15
...



Look at the number x3 = 2. Every 3rd number is a multiple of 2 (2, 8, 34, 144,
610, ...)
Look at the number x4 = 3. Every 4th number is a multiple of 3 (3, 21, 144, ...)
Look at the number x5 = 5. Every 5th number is a multiple of 5 (5, 55, 610, ...)
And so on (every nth number is a multiple of xn)…it is beauty, isn’t it?
Terms Below Zero
The sequence can be extended backwards!
Like this:
n=
...
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
...
xn =
...
-8
5
-3
2
-1
1
0
1
1
2
3
5
8
...
(Prove to yourself that adding the previous two terms together still works!)
In fact the sequence below zero has the same numbers as the sequence above zero,
except they follow a +-+- ... pattern. It can be written like this:
x−n = (−1)n+1 xn
Which says that term "-n" is equal to (−1)n+1 times term "n", and the
value (−1)n+1 neatly makes the correct 1,-1,1,-1,... pattern.
History
Fibonacci was not the first to know about the sequence, it was known in India hundreds of
years before!
About Fibonacci The Man
His real name was Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in Italy.
"Fibonacci" was his nickname, which roughly means "Son of Bonacci".
As well as being famous for the Fibonacci Sequence, he helped spread through Europe the
use of Hindu-Arabic Numerals (like our present number system 0,1,2,3,4,5,6,7,8,9) to
replace Roman Numerals (I, II, III, IV, V, etc). That has saved us all a lot of trouble! Thank
you Leonardo.
Golden Ratio
The golden ratio (symbol is the Greek letter "phi" shown at left) is
a special number approximately equal to 1.618
It appears many times in geometry, art, architecture and
other areas.
The Idea Behind It
If you divide a line into two parts so that:
the longer part divided by the smaller
part
is also equal to
the whole length divided by the longer
part
then you will have the golden ratio.
Beauty
This rectangle has been made using the Golden Ratio, Looks
like a typical frame for a painting, doesn't it?
Some artists and architects believe the Golden Ratio makes
the most pleasing and beautiful shape.
Do you think it is the "most pleasing rectangle"?
Maybe you do or don't, that is up to you!
Many buildings and artworks have the Golden
Ratio in them,
such as the Parthenon in Greece,
but it is not really known if it was designed that
way.
The Actual Value
The Golden Ratio is equal to: 1.61803398874989484820... (etc.)
The digits just keep on going, with no pattern. In fact the Golden Ratio is
known to be an Irrational Number. Calculating It
You can calculate it yourself by starting with any number and following these steps:



A) divide 1 by your number (=1/number)
B) add 1
C) that is your new number, start again at A
With a calculator, just keep pressing "1/x", "+", "1", "=", around and around. I started with
2 and got this:
Number
1/Number
Add 1
2
1/2=0.5
0.5+1=1.5
1.5
1/1.5 = 0.666...
0.666... + 1 = 1.666...
1.666...
1/1.666... = 0.6
0.6 + 1 = 1.6
1.6
1/1.6 = 0.625
0.625 + 1 = 1.625
1.625
1/1.625 = 0.6154...
0.6154... + 1 = 1.6154...
It is getting closer and closer!
But it takes a long time to get even close, however there are better ways and it can be
calculated to thousands of decimal places quite quickly.
Drawing It
Here is one way to draw a rectangle with the Golden Ratio:




Draw a square (of size "1")
Place a dot half way along one side
Draw a line from that point to an opposite corner (it
will be √5/2 in length)
Turn that line so that it runs along the square's side
Then you can extend the square to be a rectangle with the
Golden Ratio.
The Formula
Looking at the rectangle we just drew, you can see that there is a simple formula for it. If
one side is 1, the other side will be:
The square root of 5 is approximately 2.236068, so The Golden Ratio is approximately
(1+2.236068)/2 = 3.236068/2 = 1.618034. This is an easy way to calculate it when you need it.
Fibonacci Sequence
There is a special relationship between the Golden Ratio and the Fibonacci Sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
(The next number is found by adding up the two numbers before it.)
And here is a surprise: if you take any two successive (one after the other) Fibonacci
Numbers, their ratio is very close to the Golden Ratio.
The Most Irrational ...
I believe the Golden Ratio is the most irrational number. Here is why ...
One of the special properties of the Golden Ratio is that it can be
defined in terms of itself, like this:
(In numbers: 1.61803... = 1 + 1/1.61803...)
That can be expanded into this fraction that goes on for ever
(called a"continued fraction"):
So, it neatly slips in between simple fractions.
Whereas many other irrational numbers are reasonably close to rational numbers (for example Pi =
3.141592654... is pretty close to 22/7 = 3.1428571...)
Other Names
The Golden Ratio is also sometimes called the golden section, golden
mean, golden number, divine proportion, divine section and golden
proportion.