We're gonna do randomised Fibonacci sequences. So, we're gonna take a Fibonacci sequence, we're gonna randomise it ....with this coin.
Now, I think we should first do a recap of Fibonacci sequences first, just to see what the original Fibonacci sequence is.
So, your Fibonacci sequence
starts with a 1, and a 1, and then the next term is the sum of the previous two.
We're gonna add those together
So 1 + 1 makes 2, 1 + 2 makes 3,
2 and 3 make 5, 8, 13,
21 - and so on.
So, if you want to write that down as a little formula
You'd write it like this: if you want to find the next Fibonacci number, you just add the previous two
Okay
So, couple of well-known facts about Fibonacci numbers as well;
If you take two consecutive Fibonacci numbers, and we'll divide the big one by the smaller one,
then you'll see, and it's been proven for a long time, that this will tend to a number, which is:
1.618033988...something, something, something
And people will know that this is a special number,
This is called the golden ratio, which is this symbol, the golden ratio. But you can use that to...estimate.
Let's say you've a really big Fibonacci number and
You don't know what it is - you can use the golden ratio.
So the idea is, every time, we're sort of multiplying by the golden ratio to get the next one up
Less true at the beginning, becomes more true the biggest these numbers are
(Brady: so you can, kind of like, reverse engineer your numbers?)
Yeah, yeah, you can work out -
the millionth
Fibonacci number, right
So if you wanted to work out the millionth Fibonacci number, what you're doing is multiplying by the golden ratio - a million times.
So, let's say, approximately
equal to
golden ratio, which is this symbol, to a million. And I can tell you what that is -
I've actually worked this out - 4.4 x 10, and it's going to be a huge number: 208,987
So this a is really big Fibonacci number, but this an estimate, 'cause it is a bit rough
If you wanna see what it actually is, the actual millionth Fibonacci number is...
1.95 x 10 to the power 208,987.
So not quite there, but the size of these numbers is spot on. So that was just a little recap of Fibonacci.
'Cause what I wanna do now is,
randomise this! We all, awh Fibonacci we know everything about it! Oh, boring, so predictable.
So now, let's use my coin to randomise this a little bit. With this idea, I'm gonna create
a random Fibonacci sequence, the next one is going to be the sum, or the difference, of the previous two numbers.
So, we're gonna start in the same way, with a 1 and a 1. Now, what is the next value?
We're gonna use my coin to find out
(Brady: So, normally it would have been 2?)
So this is, this, if it was the ordinary Fibonacci sequence we would add these together,
we'd get a 2. I'm gonna flip my coin. I might take the sum, I might take the difference.
So let's see, what it's going to be.
(Brady: You've got negative on one side...Ahh yeah, okay.)
See I, I prepared. I came prepped!
Ready? All right, so let's do - is it a sum or difference...And I've got difference!
(Brady: Difference. Okay, so that's - naught is the next number?)
So zero is the next number. Let's do the next number. Is it going to be a sum or a difference?
It's gonna be a sum! So I've got 1 now, add to the previous two to give me, 1.
Let's see what I get next, let's try it.
[slaps hand]
Okay, I got a sum so we're got back to...
another 1, there. [slaps hand]
Okay, I got a plus - so that's a 2. Let's keep going, is it a sum or difference?
Okay, we've now got a difference, so we're gonna do 2 - 1, which gives me 1. Let's do a few more.
We've got a plus now so that's going to be, a 3. 2 and a 1.
Oh, and a difference here - so 3 - 1 = 2. So it's unpredictable.
I don't know what's going to happen next. And, if you don't mind I want to really make that point.
If I did it again...I'm going to get a different sequence. But is there anything I can say about them?
So, I don't know if these are going to - you know,
go off to infinity - like the Fibonacci sequence just gets bigger and bigger and bigger, goes off to infinity.
I don't know if it's going to go to - maybe it just goes off to minus infinity, I don't know that
Maybe it, kind of hovers around zero? Maybe the pluses and minuses cancel out.
Or maybe it's unpredictable, maybe it's just chaotic and I don't know what's going to happen next.
So the surprise is - I can make some predictions. I can make some long term predictions,
for example, I can tell you what the millionth random Fibonacci number is going to be
because
like the original Fibonacci sequence, there is a growth ratio.
Just the same way,
I want to show you that, so it's: 1.1319882487943....something, something, something.
So this is our constant and it's going on forever. Which means we can use this
to predict the millionth random number. So, let's do it. Random Fibonacci number one million.
Same kinda thing that I did with the Fibonacci numbers. It's gonna be, approximately,
this constant: oh, I'll just do 1.319...you get the idea. To the power: one million
Which is:
some big number. 8.3 x 10 to the 53,841
I will tell you, this is a statement about the size of the number - not the sign of the number.
So I don't know if it's a plus or a minus number. So it might be a plus or minus.
So this might be going off to positive infinity, and there's no reason why it can't go down to negative infinity;
but they are growing, and the size of these numbers are growing,
and, in the long term, we can predict how much it will grow by. If you wanna put it another way,
you could say it like this:
If you take the nth random Fibonacci number, and make it positive, and we took the nth root,
this is tending towards our magic number here. 1.1319 and stuff, as
n tends to infinity.
Although, well I'll put a little thing here
"Almost surely"
That's a funny expression I've slipped in. This tends to this constant "almost surely"
Because the original Fibonacci sequence
would count as a randomised Fibonacci sequence *if* I was just getting plus
every time I flipped my coin: plus, plus, plus, plus.
So, that counts and that has a growth rate which is the golden ratio - it's not this value.
But the chances of me getting plus, plus, plus; over and over again so that I've got the Fibonacci sequence
becomes increasing small.
Err, so if you look at all the possible random Fibonacci sequences
things like the original Fibonacci sequence
almost never happens. I'm not going to say it's impossible, but it's *almost* never going to happen. If you did this for real,
almost 100% of the time you're getting a
properly random sequence of pluses and minuses which tend towards this growth constant.
Another example of what wouldn't work, is if I had a pattern. If I had something like
plus, minus, minus as a pattern, and it went plus, minus, minus. So if I started with a 1 and a 1
it would be plus, minus, minus as a pattern
which would go plus, would make a 2. Then a minus; would be 2 - 1, so that should be a 1.
Then a minus again 1 - 2, I've got -1
Then we go for a plus which gives me a 0
Then we'll do a minus
which is going to give me a 1 and another minus gives me a 1. And I get back
to the original two numbers and then that pattern will continue, so that's not growing at all.
That's just oscillating up and down, so that doesn't have a growth rate, because it's not growing anywhere.
You won't really see this coming out though, until you're using...
Until you've done this for about...two hundred thousand? times
So you have to go quite far and then you start to see that this starts to tend to these numbers.
(Brady: You don't think someone watching is going to make some computer simulator that will do this two hundred thousand times?)
And that's what they did!
That's exactly what they've done.
So when they were trying to calculate these numbers they simulated it. And they could see that in the long term,
this was the growth rate.
In the 60s they showed that these randomised Fibonacci sequences do grow.
I think that's a surprise in itself because, you know,
they've got pluses and minuses don't really cancel out, turns out they don't, so these sequences are growing.
So they could prove that it was growing, but they didn't know what the growth rate was.
So, that took, awh till
1999, so we're talking 30 years later
and, it was calculated what this growth rate was, and that, was really hard.
Proper advanced mathematics, proper advanced computing; but what they did calculate was
this part of it:
1.13198824
and that's as far as they could go 'cause it's really hard to calculate. And that was 20 years ago.
And in the 20 years since, we've worked out 5 more digits and that's as far as we've got.
(Brady: Audrey. Hang on)
Brady: Audrey! James: Hello! D'you want to hear about applications, of Fibonacci sequences?
Audrey you're not supposed to be in here. Out. Come on, out you go.
Out
Come on
Out you go - She was interested!
Out you go [laughs]