Number Sets 1

U12_L1_T1_we1 Number Sets 1

We are asked, what number set
does the number 8 belong to?

So this is actually
a good review

of the different sets of numbers
that we often talk about.

So the first set
under consideration

is the natural numbers.

And these are essentially
the counting numbers,

and you're not counting 0.

So just if you were
actually to count objects,

and you have at
least one of them,

we're talking about
the natural numbers.

So that would be 1, 2,
3, so on and so forth.

So clearly, 8 is
a natural number.

You can count up to 8 here.

You could count 8 objects.

So 8 is a member of
the natural numbers.

The next one we
should consider, let's

consider the whole
numbers right over here.

And I should say
natural numbers.

So let's consider
the whole numbers.

The whole numbers
are essentially

the same thing as
the natural numbers,

but we're now
going to include 0.

So this is 0, 1, 2,
3, so on and so forth.

So clearly, 8 is one
of these as well.

You could eventually
increment your way to 8,

like you're just counting
all of the whole numbers.

Another way to view this is
the non-negative numbers.

So 8 clearly belongs
to this as well.

So let's expand our
set a little bit.

Let's think about integers.

Now these are all the
numbers starting with,

well you could keep counting
down, all the way up

to negative 3, negative
2, negative 1, 0, 1, 2, 3,

and you could just
keep going there.

Now clearly, 8 is
one of these as well.

You can just keep counting to 8.

In fact, let me just
put our check box there.

In general, you
have your integers

that contain both the positive
and the negative numbers and 0,

depending on
whether you consider

that positive or
negative, or neither.

So that's the integers,
right over here.

And then the whole numbers
is a subset of the integers.

So I'll draw it like this.

The whole numbers
are right over here,

that is the whole numbers.

We've now excluded all
of the negative numbers.

So these are all the
non-negative numbers.

All the non-negative
integers, I should say.

So these are the whole numbers.

And then the natural numbers
are a subset of that.

It's essentially everything.

So the only thing that's
in whole numbers that's

not in the natural numbers
is just the number 0.

So this whole area
right here just

corresponds to the number 0.

So it really should
be a bit of a point.

So let me make it clear.

This circle is
the whole numbers,

and then I have the
natural numbers,

which is a subset of that.

Obviously this isn't
drawn to scale.

The natural numbers
is a subset of that.

8 is a member of all of them.

8 is sitting right over here.

So it's a member of the
natural numbers, whole numbers,

and the integers.

Now let's keep expanding things.

Let's talk about
rational numbers.

Now these are numbers that
can be expressed in the form p

over q, where both p
and q are integers.

So can 8 be expressed this way?

Well, you can express
8 as 8 over 1.

Or actually 16 over 2.

Or you could just
keep going, 32 over 4,

you can express it as a
bunch of p's over q's, where

both the p and q are integers.

So it's definitely
a rational number.

And in fact, all of these things
over here are rational numbers.

So let me draw.

So this is all a subset
of rational numbers.

So 8 is definitely a
member of that as well.

Rational numbers, so let me
put the check box over here.

Now what about
irrational numbers?

Irrational numbers.

Well, by definition, these are
numbers that are not rational.

These are numbers that cannot
be expressed in this form,

where p and q are integers.

So if something is rational,
it just cannot be irrational.

So 8 is not a member of
the irrational numbers.

The irrational numbers are
just a completely separate set

over here.

So I would draw it like this.

This area right over
here, this would

be the irrational numbers.

Irrational.

Rational is not a subset of
irrational, they are exclusive.

You can't be in both sets.

So that's irrational
right over there.

And then finally let's ask, is
8 a member of the real numbers?

Now the real numbers are
essentially all of these.

It's combining both the
rational and the irrational.

So the real numbers is all
of this right over here.

And so 8 is clearly
a member of the real.

It's a member of the
real, and within the real,

you either can be rational or
irrational, 8 it is rational.

It's an integer.

It's a whole number.

And it is a natural number.

So it's definitely a
member of the reals.

And just to give
you might be saying,

hey well, what is an
irrational number then?

Can't almost every number
be represented like this?

Or every number you can think
of be represented like this?

And an example of maybe
the most famous example

of an irrational number is pi.

Pi is equal to 3.14159, and
people devote their lives

to memorizing the digits of pi.

But what makes this irrational
is you can't represent it

as a ratio, or as a
rational expression,

of integers, the way you
can for rational numbers.

And this right here
is non-repeating.

And if it was
repeating, you actually

could express it as
a ratio of integers,

and we do that in other videos.

It is non-repeating
and non terminating,

so you never run out digits to
the right of the decimal point.

So this would be an example
of an irrational number.

So pi would sit here
in the irrationals.

Anyway, hopefully you
found that helpful.