The Fibonacci numbers again start:
1, 1, 2, 3, 5
and the rule is: if each number here
equals the one before it
So, if this was the nth number
and we're gonna call that x n
for the nth number
it equals the one before it
plus the one before that
so that equals
x n minus one
plus x n minus two
So that's our finding thing
and what we're trying to find
ultimately
is what is the ratio between the two of the numbers
and importantly, because
we're looking for the ratio
it is approaching the limit
it doesn't matter which two terms
it should be the same ratio, give or take
and
if you do infinitely many the same ratio for all of them
but, rrrrhhhhh
So. Uh. For any two of the terms
so what we can say now
is the ratio between
x to the n and the one before that
which was x to the n minus one. The one before.
should equal the same as the ratio
between x n minus one to the one before that
which is x n minus two
and we're gonna call it
What letter would you like Brady?
awa aya a
any any Greek letter?
Do you have a favorite?
Brady: uhhh....
delta?
Guy: DELTA!
ok, so lets
do a capital delta
because its quicker to draw
so we're trying [to come up] with this ratio, delta is.
and we know its the ratio between any two
consecutive terms
now we know
according to our relationship which generates this sequence
that x n equals the two before it
added together
so I'm gonna put that in
intstead of
so all I've done is substitute that in
but now I could start simplifying
because I've got the same term
there twice
so, in fact, if I divide through
that's one plus
x to the n minus two
over x to the n minus one
which still equals
x to the n minus one over
x to the n minus two
which still equal to the ratio
delta we're looking for
so now you'll go
hang on a second
that is the same as this but upside down
and in fact that equals that ratio delta
so I can now change this to be
one plus
well that's the inverse of this ratio
so that's one over delta
equals and that there is the delta
and so the relationship
for the ratio of any sequence
where you get the next term
from adding the two previous ones
one plus the inverse of that ratio
equals the ratio
Now what ratio could that possibly be?
well let's work it out!
because we can turn this
into a
quadratic?
If I multiply through by triangle
it equals that plus one
equals triangle squared
and so delta squared minus delta
minus one equals zero
So, all I've done is rearrange that formula
now we can solve this using our friend the quadratic equation
cuz we know
that if you have a
times x squared plus
b x plus c
equals zero
then x equals negative b
plus or minus
the square root of b squared minus four a c
over two a
so we can substitute in to use the quadratic formula
So we now know delta equals
negative b, which is the coefficient there
which will be one
plus or minus the square root of b squared --is going to be one
minus four times one, times negative one, so it's going to be
plus four. Two times one is two. So that's going to equal...
Now I'm only going to use the positive for now.
One plus root 5 divided by two, which is the golden ratio.
If you actually work that out, that's where 1.6180... comes from.
And in fact, that is one of the definitions of the golden ratio
And so that's why, for any starting term, it always approaches the same sequence
If you use the negative, you get the negative inverse of the golden ratio; another wonderful property of it.
It I'd use the negative... If I'd used equals one minus root five over two
and that equals -0.6180...
So they're the two solutions to that [particular] equation
So, that's why it works of any sequence where you add two terms to get the next one
Brady: So the fact... The Fibonacci numbers meant nothing there?
Guy: The Fibonacci numbers are one of [an] infinite family of such numbers
because for any two whole numbers you want to start with, the exact same thing works out
In fact, why use whole numbers? Go bananas! At no point did I specify whole numbers here.
It's just, whatever you start with, add two terms, you get the next one. You always get the golden ratio
That's not some special property Fibonacci numbers have.
All of them have got that property. We want a better link.
What we really want is a sequence of numbers which have a link the the golden ratio
that other sequences don't get. We want something above this.
Brady: The Brady numbers!
Guy: Well, the Brady numbers are wonderful, but I can go slightly better
So, I'm going to show you the Lucas numbers.
I, personally, am a massive golden ratio septic. I think it's hugely overrated...