These are the simple facts that answer the question What are the Fibonacci numbers?
Below are the opening numbers of the Fibonacci sequence. If you want more numbers just keep adding the previous two numbers to get the next in the sequence.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. and so on…….
Formula: Add the last two numbers in the sequence to get the next
0=1=1. 1+1=2. 1+2=3. pick any two consecutive numbers in the sequence, add them together and you will get the next number in the sequence and so on…. 55+89=144.
Is that it?!
If you don’t want to know anymore, don’t need to read any further. But there is quite a bit more to know.
Understanding the Fibonacci sequence is more than just the numbers. Simply put, These numbers create a pattern that once you recognise you will begin to see it everywhere.
And that is the beginning of why the Fibonacci numbers are so special. But let’s finish our examination of the numbers in the sequence first.
(Then I can tell you what really the Fibonacci sequence is. And when I get to that we won’t be talking about digits at all)
What to do with the Fibonacci numbers?
They are just numbers right? Correct. But look closely at the process of arriving at the next number in the sequence. Nothing comes from outside of the list of numbers you already have. No number is new, it is only the sum of two numbers already in the sequence. Only numbers within the sequence are used to generate the next number.
The sequence creates itself.
If there is one thing you remember from this article, remember that. It is a self accumulating sequence of numbers.
What are the Fibonacci numbers and Rabbits all about?
With the discussion on what are the Fibonacci numbers comes a mention of the Rabbit Riddle. The Rabbit Riddle really is a rabbit riddle. I hurt my head trying to follow it. But I offer a couple of tips here:
- Forget all your common sense questions like ‘what if only males are born?’ or ‘surely some would die’. In this thought experiment all Rabbits live and all mature pairs reproduce another pair (one male and one female) within each step of the chart.
- Think of each pair as a single unit (this is the bit that got me confused for ages) we are counting PAIRS of rabbits as a unit, not single rabbits.
Ok so here’s the chart. (I’ve left out 0 just for space reasons) Small rabbits are the newly born ones which take one full step of the chart to mature to an adult pair. The mature pair will always breed another pair at every step.
It is a rabbit family tree. The numbers on the right hand side are the total number of pairs at each step. Starting with 0 or 1 these turn out to be the Fibonacci numbers.
The resulting rabbit population arises from what is already in existence in the original rabbits. No new rabbits arrive from outside this family. It is a self accumulating growth process. It is the passing of time and rate of increase that are the variables. But keep them the same for each step and you arrive at the chart above. This TIME and RATE OF INCREASE are also at play in shell formation.
There are several activities you can do around this/with this chart and the Fibonacci numbers. More on that on the Fibonacci Rabbit Riddle page. And if you want to know where this riddle first appeared you can read about that here.
Now we look again at something interesting we can do with the numbers of the Fibonacci sequence.
But what really are the Fibonacci numbers?
Write out all the numbers in the sequence. Now write out the numbers again underneath the first set and nudge them all along by one position to the right. You now have stacked pairs of consecutive numbers. Next, draw a line in between the upper and lower number to make it look like a fraction.
Fibonacci numbers as fractions and decimal
Now treat each pair like a fraction by dividing the top number by the lower number. Write down each decimal number you get. Eg : divide 13 by 8 to get 1.625. Or 34 divide by 21 to get 1.619. Do it with all the pairs you can. The further along the row of fractions, the closer the result is to 1.618. The first few in the row of fractions are wide of this but come in closer quickly in the sequence. If your calculator screen was wide enough the digits after the decimal would go on forever never reaching an end.
This decimal is ɸ Phi.
ɸ is a relationship.
A relationship expressed as a decimal number. 1.618 using the symbol ɸ
By the way, you can do this starting with ANY number. Because it is a relationship it is not specific to the digits used. I’ll say that again, It is NOT specific to the digits used. It is a RELATIONSHIP. Understanding this relationship is what’s important when looking for Fibonacci sequence illustrated in nature
So here is the fraction exercise we just did above again. Except this time starting with numbers 21 and 22. Start with 21+22=43. And you have the first three numbers in the sequence. Now follow the formula of adding the last two together to get the next in the sequence……22+43=65, 43+65=108 etc.
Write out two lines of the numbers you have in your sequence. Stack them one above the other and move the lower line to the right by one space. Now treat the stacked numbers as a fraction. Divide the top number by the lower and you will arrive at a decimal number. Do this for each stacked pair. As the sequence moves further along it becomes closer to 1.618. Exactly as happened in the numbers from the Fibonacci sequence.
The Fibonacci numbers are in a relationship to each other
If you want to know more about this relationship between the numbers; The Divine Proportion, is one name for it. you can read more on that in the article What is the Fibonacci Sequence?
Ok, that is as far as I go with the actual digits that make up the Fibonacci sequence. I can’t do anymore mathematical expressions of it. There are equations of it but I’m really not the person to explain equations. It can also be geometry. But what I’m good at is pattern. How the Fibonacci numbers become a pattern and that pattern in nature is really what I want to show you.