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, ...
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Can you figure out the next few numbers?
Makes A Spiral
When we make squares with those widths, we get a nice spiral:
Do you see how the squares fit neatly together?For example 5 and 8 make 13, 8 và 13 make 21, và so on.
The Rule
The Fibonacci Sequence can be written as a "Rule" (see Sequences & Series).
First, the terms are numbered from 0 onwards lượt thích 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 isthe 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 is calculated lượt thích this:
x9= x9−1 + x9−2
= x8 + x7
= 21 + 13
= 34
Golden Ratio
And here is a surprise. When we 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... | |
5 8 | 1.6 | |
8 13 | 1.625 | |
... ... | ... | |
144 233 | 1.618055556... | |
233 377 | 1.618025751... | |
... ... | ... |
We don"t have lớn start with 2 & 3, here I randomly chose 192 và 16 (and got 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 | ||
208 224 224 432 | 1.92857143... | |
... ... | ... | |
7408 11984 | 1.61771058... | |
11984 19392 | 1.61815754... | |
... ... | ... |
It takes longer to get good values, but it shows that not just the Fibonacci Sequence can bởi this!
Using The Golden Ratio to lớn Calculate Fibonacci Numbers
And even more surprising is that we can calculate any Fibonacci Number using the Golden Ratio:
xn = φn − (1−φ)n√5
The answer comes out as a whole number, exactly equal khổng lồ the addition of the previous two terms.
Example: x6
x6 = (1.618034...)6 − (1−1.618034...)6√5
When I used a calculator on this (only entering the Golden Ratio to lớn 6 decimal places) I got the answer 8.00000033 , a more accurate calculation would be closer to 8.
Try n=12 & see what you get.
You can also calculate a Fibonacci Number by multiplying the previous Fibonacci Number by the Golden Ratio và then rounding (works for numbers above 1):
Example: 8 × φ = 8 × 1.618034... = 12.94427... = 13 (rounded)
Some Interesting Things
Here is the Fibonacci sequence again:
n = | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | ... |
xn = | 0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 | 89 | 144 | 233 | 377 | 610 | ... |
There is an interesting pattern:
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).
Notice the first few digits (0,1,1,2,3,5) are the Fibonacci sequence?
In a way they all are, except multiple digit numbers (13, 21, etc) overlap, like this:
Terms Below Zero
The sequence works below zero also, lượt thích 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 lớn yourself that each number is found by adding up the two numbers before it!)
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", & the value (−1)n+1 neatly makes the correct +1, −1, +1, −1, ... Pattern.
History
Fibonacci was not the first lớn 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 & 1250 in Italy.
"Fibonacci" was his nickname, which roughly means "Son of Bonacci".
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As well as being famous for the Fibonacci Sequence, he helped spread Hindu-Arabic Numerals (like our present numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) through Europe in place of Roman Numerals (I, II, III, IV, V, etc). That has saved us all a lot of trouble! Thank you Leonardo.
Fibonacci Day
Fibonacci Day is November 23rd, as it has the digits "1, 1, 2, 3" which is part of the sequence. So next Nov 23 let everyone know!