I just want to point out that when we've been working with these sequences of
numbers, we've actually been doing mathematics. So, for instance, when we talked
about the sequence of odd numbers and we said what's the next number if the first
three numbers are 1
3 and 5,
and then we fill in the next number and say it's 7,
and then the next after that is 9,
so on and so forth, we're actually doing mathematics and that kind of reasoning we use
is called inductive reasoning.
Now, we go down to the sequence of squares--same thing:
1, 4, 9--you ask yourself what's the next number in that sequence,
so you look at this pattern right here, and then that method of reasoning that you use
is called inductive reasoning, and you say, well this looks like 1², 2²,
3²,
this must be 4², so it's going to be 16.
The one after that then is going to be 5², or 25.
Okay, so that's called inductive reasoning, when we reason like that and
it's actually doing mathematics, and what you want to notice when you do that is
that it's kind of a different state of mind.
When you're in the state of mind, you're looking at these numbers, trying to fill in the
next numbers here,
you don't notice yourself as being separate from your surroundings, you
don't notice time going by,
it is actually a pleasant experience. It is only when you come out of it and maybe
ask yourself, did you do as well as somebody else, or you didn't get this,
or you got it kind of quickly, that you put
some kind of value judgment on it, but the actual process of doing mathematics
is actually a
very enjoyable process. Let's look at the next sequence.
Fibonacci sequence,
named for the mathematician Fibonacci, it's his sequence, let's write it down.
It also starts with a 1,
the next number in sequence is also a 1,
the number after that is 2,
and the number after that is 3.
So, here you're using your inductive reasoning to try to think of what comes next
in the sequence. You're doing mathematics, it's pleasant experience.
You might come out of it and tell yourself, oh you're not getting this, and you put something kinda negative
on it, but that's not the mathematics, that's what you're
bringing to it.
So, we look at this pattern right here, this isn't quite as easy to get as the other
ones, I'll give you the next number, 5.
Still thinking about what comes next? Next number is 8,
and the next number after that: 13.
So how do you get the consecutive numbers of the Fibonacci sequence?
It always starts with two 1s, and then you add the previous two to get the next one.
So, 1 + 1 = 2,
1 + 2 = 3,
2 + 3 = 5,
3 + 5 = 8, so on and so forth.
So, that's the Fibonacci sequence, and the method of reasoning we used to
get those numbers in the sequence is called inductive reasoning.
Now, it turns out that the Fibonacci sequence has all kinds of applications
around the world
that we live in we. Things just sort of align themselves according to Fibonacci sequence.
You wouldn't think so when you first see it, but it's just
all over the place.
I want to show you my favorite application of the Fibonacci sequence,
the family tree of a male honeybee.
Now, a male honey bee has one parent: it's mother. A female honeybee has two parents,
a mother and father.
The female honeybee comes from a fertilized egg, a male honeybee comes from
an unfertilized egg, so let's ask the question: how many bees are in each generation
of the family tree of a male honeybee?
So here's my male honeybee right here,
that male has only one parent,
its mother,
but that mother has two parents, a mother and a father.
So, a female bee
and a male bee.
That female has two parents,
a female
and a male,
but that male has only one parent,
a female.
Right? That females, two parents,
that male, one parent.
That female, two parents,
and so, how many bees in the family tree of a male honeybee? In the first generation, 1.
Next one, 1.
Next one, 2.
The one after that, 3 bees.
The one after that,
5 bees.
Using inductive reasoning we don't have to actually fill out the rest of this.
we've seen that pattern and we know it follows the Fibonacci sequence, so
how many bees in the next generation of the family tree of this male honey bee?
8, and we could work it out by actually putting them down there to prove
that to ourselves.
So, Fibonacci sequence: all kinds of applications all around the world, we're
going to see a few more as we go through this course,
but the thing that I mainly wanted to point out here is that even though
these number sequences seem easy, you're actually doing mathematics
and I also want to point out that that's a pleasant experience.