Preparing to Fibonacci Day: Post Script

We have spent the last couple of weeks looking and some of the many ways the Fibonacci sequence has been applied to real life, to mathematics, to nature (rabbits, bees, flowers, pineapples, pinecones, etc.) to music, to fashion, to furniture, and to art.  All of this started with a simple problem about raising rabbits.

Leonardo of Pisa, or Leonardo Fibonacci, is best known for the Fibonacci sequence and this one problem in his book "Liber Abaci".  And we are still overlooking his greatest achievement.

His book showed us how to do a new form of arithmetic.  First by introducing the use of Hindu-Arabic numerals (which replaced Roman numerals).  Second by showing us new methods of performing the calculations.

If you are not seeing it yet, just think about doing long division with Roman numerals.  The changes that Leonardo of Pisa inspired created a great change in how Europeans did mathematics - including long division.

I encourage teachers who want to show what this man did to give a pop quiz on 11/23/2014.  It should be a quiz on long division, and using Roman numerals.

JUST KIDDING!!!  (maybe)


David

Preparing for Fibonacci Day: What Number Come Before the 1 and the 1?

The Fibonacci sequence “starts” with 1, and 1 (or some say 0, 1, and 1), and each number after that is obtained by adding the prior two numbers.  But is it possible that there are number that can come before the 1, and 1?

___, 1, 1

What number could go in the blank spot so that the first two number add up to the third number?  Something plus one equals one?

OK, I see a hand up in the back – just shout it out.

“Zero!”

Yep, 0 + 1 = 1.  What would be a number that I could add to 0 to get the first 1?

“One!”

1, 0, 1, 1, …

Let’s keep working backwards.  What number could I add to one in order to get 0?

“Negative one!”

That’s right – a negative one.

…, -1, 1, 0, 1, 1, 2, 3, 5, 8, …

So the answer is Yes – there are numbers that can come before the Fibonacci numbers – but what kind of pattern do they follow?

We could keep on going like this, but there is another way of getting the same answers.  Going forward we add two consecutive Fibonacci number to get the next number.  If we want to go backwards we subtract the two numbers to get the previous number.

1 – 1 = 0, 1 – 0 = 1, 0 – 1 = -1, 1 – (-1) = 2, -1 – 2 = -3, 2 – (-3) = 5, -3 – 5 = -8.  Now what do we have?

…, -8, 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, …

Do you see a pattern?

Could you start with -8 and 5 and get the original Fibonacci sequence?  Would it work if you started with 8 and -5?  Are there numbers that come before the -8?

These numbers are known as the “NegaFibonacci” numbers.

David

Solutions to Friday's Brain-O Quiz

Solution to Fridays Brain-O Quiz.

1.  Easy, just push the cork in.

2.  Corn on the cob.

3.  “One Word”

4.  Electric trains don’t make smoke.

5.  (66/6) + 6 – 6 = 11

David

Preparing for Fibonacci Day: Fibonacci and the Golden Ratio

One of the more well know properties of the Fibonacci sequence is its relationship to the Golden Ratio (or Phi):

If you take the ratio of two consecutive Fibonacci number (put the bigger number on top, and the smaller one on the bottom) you will get a fraction that approximates the Golden Ratio.  The larger the Fibonaccci numbers are, the closer your approximation will be.

This is true, but what most people don’t know is that this property is not unique to the Fibonacci sequence.  If you start with numbers other than 1 and 1, you will the sequence will still begin to approximate the value of the golden ration closer and closer as the numbers get bigger.  And you don’t have to pick integers – you can choose fractions, decimals, or even negative numbers.

For example let’s look at the Lucas sequence.  The Lucas sequence begins with 2 and 1 (instead of 1 and 1), but otherwise works just like the Fibonacci sequence.

The 41^st Lucas number divided by the 40^th Lucas number is:

370248451 / 228826127 = 1.61803398874989480550007298773185983259682579865541...

The 41^st Fibonacci number divided by the 40^th Fibonacci number is:

165580141 / 102334155 =
1.61803398874989489090910068099941803398874989489090...

Both of these are the same up to 14 decimal places!  I don’t think I will ever need to be more accurate, but If I do I can do it.

If I use the equation listed above for the definition of the Golden Ration (Phi) is:

1.61803398874989484820458683436563811772030917980576...

So why did I pick the Lucas sequence?  Well, the Lucas sequence hides another secret about Phi that the Fibonacci sequence does not have.  In the Lucas sequence the first two terms are 2 and 1.  We will call them L₀ = 2, and L₁ = 1.  L₂ = 3, which is equal to Phi² rounded to the nearest integer.  L₃ = 4, which is Phi³ rounded to the nearest integer.  In fact, this pattern continues to at least the 53^rd term.  I suspect that it continues further, but my spreadsheet program is only accurate to 12 digits.

REFERENCES:

http://en.wikipedia.org/wiki/Fibonacci_number

http://en.wikipedia.org/wiki/Golden_ratio

http://en.wikipedia.org/wiki/Lucas_number

https://www.youtube.com/watch?v=dTWKKvlZB08&list=UUoxcjq-8xIDTYp3uz647V5A

https://www.youtube.com/watch?v=PeUbRXnbmms&list=UUoxcjq-8xIDTYp3uz647V5A

David

Preparing for Fibonaci Day: Fibonacci Tartan and Bagpipe Music

Field Trip!  Please go watch this Numberphile Video at: http://www.youtube.com/watch?v=e4sF_Z5oJek.

I can’t add anything to this presentation, other than I cannot find a source to order either Fibonacci Tartan fabric or garments made out of Fibonacci Tartan.  Though it would be kewl to have a tartan for mathematicians, especially Italian mathematicians.  (I personally am not Italian, and I’m not aware of any Italian ancestors in my family tree.  But my daughter was born in Italy.)

David

P.S.  The word “Kewl” is pronounced like the words “cool” and “kool”, but has nothing to do with temperature or a child’s soft drink mix.

Preparing for Fibonacci Day - Pisano Periods

Field Trip!  Go watch the Numberphile Video: http://www.numberphile.com/videos/pisano_period.html

The Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …

Many mathematicians have studied this integer sequence for hundreds of years and found several interesting patterns.  One of them is a pattern called Pisano periods.

It says that the m^th Fibonacci number (or F(m)) evenly divides the n^th Fibonacci number (or F(n)) if m evenly divides n.  So the 3^rd Fibonacci number (or F(3) which equals 2) divides every 3^rd Fibonacci number ((F(6), F(9), F(12), etc.)

1 1 2 1 3 2 4 3 5 5 6 8 7 13 8 21 9 34 10 55 11 89 12 144

2 is the 3rd Fibonacci number.  Every 3rd number after 2 is divisible by 2 (it’s and even number). 3 is the 4th Fibonacci number.  Every 4th number after 3 is divisible by 3. 5 is the 5th Fibonacci number.  Every 5th number after 5 is divisible by 5. 8 is the 6th Fibonacci number.  Every 6th number after 8 is divisible by 8. 13 is the 7th Fibonacci number.  Every 7th number after 13 is divisible by 13. 21 is the 8th Fibonacci number.  Every 8th number after 21 is divisible by 21. 34 is the 9th Fibonacci number.  Every 9th number after 34 is divisible by 34. 55 is the 10th Fibonacci number.  Every 10th number after 55 is divisible by 55. 89 is the 11th Fibonacci number.  Every 11th number after 89 is divisible by 89. 144 is the 12th Fibonacci number.  Every 12th number after 144 is divisible by 144. Etc.

David