Instructor: Hi class.
Today we're going to talk about
the different types
of number sets.
And it's nice to know
what kinds of numbers exist,
and which ones are appropriate
for different types of answers.
For instance, you might get
an answer to a problem
that doesn't make sense,
and you have to change it
into a different kind of number
for it to make sense.
For instance,
there's a statistic
that there are approximately
2.6 children per family,
but it doesn't make sense
to have a .6 child.
So it would make
more sense
to say that there are
two or three children
for most average
American families instead.
So we need to know
what kind of numbers we have.
We're going to take a look
at the most basic type of number,
which is the natural number.
And the natural numbers are numbers
that you would normally use
to count different items.
So these are often
called "counting numbers."
And that's because if you were
to pick up a bag of oranges,
or something,
and count how many are in it,
you would start naturally
with the number 1, then count 2,
and 3, and so on, until you get
how many items are in the bag.
And so natural numbers
are the numbers that you would
naturally count with.
Now when you build
upon that number set,
you get to the number set
that also includes the idea
that if you have a bag,
it could have items in it
that you could count,
or it could be empty.
And if it's empty, you add on
the idea of having the number 0.
And that gives us
these "whole numbers" here.
So those are 0,
plus the counting numbers.
So those would be numbers
like 0, 1, 2, 3, 4, and so on.
Now once we get past the whole numbers
we add on the idea that
not only could you have a bag
with items in it,
or a bag that's empty, you could also
owe somebody something.
And if you, you could either have oranges,
have no oranges,
or owe somebody oranges
if you were counting oranges,
for instance.
And when we get to that,
you get into the idea of the integers.
The integers are the numbers
that most people label
on a number line.
If you make a number line,
you normally draw something like this.
And most people will put 0
somewhere right here.
And going the direction
you read,
you would get larger, and you'd put
the positive counting numbers.
And you would continue on
as far as you needed to.
And then going the other direction
on the number line
are all the negative numbers.
And so normally the numbers
we'd label on those
would be the negative
counting numbers.
And those will start
with -1, -2, -3, -4 and so on.
And so we get
into the idea of
over here that you could
owe somebody items.
Here you could have
no items.
And over here
you could have items.
And these numbers
that I labeled on the number line
are what we call
the integers.
There are numbers
in between all these numbers
that we put
on the number line here.
And those numbers
are fractions of whole amounts,
like a half [1/2],
or one and a half [1 1/2],
or 2.3 and so on.
When we get into the idea
of having numbers in between all these,
we get into the idea
of a rational number.
A rational number is anything
that can be turned into a fraction.
So that would be numbers
like the number 2.
You could put that
as 2 over 1   [2/1].
So that is a fraction.
Something like two fifths
[2/5]   is a fraction.
Something like square root of 9
is actually a fraction
because square root of 9
is 3,
and 3 can be written
as 3 over 1   [3/1].
Something like .25,
that's a decimal that ends.
Any decimal that ends
can be transferred into a fraction.
For instance .25 is the same thing
as one fourth  [.25=1/4].
Also, any decimal that repeats,
like .33333... going on forever,
that's actually one third.
[1/3]
Another way
to write .3 repeated is,
instead of to have a few of them there
with dots on the end,
we could put a bar
over the top
and that means that the 3
repeats forever.
Any decimal that ends
or any decimal that repeats,
so for instance
something like .19 with a bar
would mean that that decimal
is .191919 forever.
So any decimal that ends,
any decimal that repeats,
anything that obviously
is written like a fraction,
or anything that you could put over 1
is also a rational number.
And you can have
positive and negative numbers
for rational numbers.
Now that brings us
to irrational numbers.
Irrational numbers are things
that are not fractions.
So, for instance,
the most popular one
that people are aware of
is the number pi.
We approximate pi
as 3.14,
but it's actually a decimal
that goes on forever
and it never repeats.
And because of that,
it's actually not a fraction.
Another number would be
like square root of 2.
Square root of 3.
Square root of 5.
Notice that I skipped
square root of 4
because the square root of 4
is 2,
and 2 is up here
as one of our fractions.
So basically the square root
of any number
that's not a perfect square
because you can't turn it
into a whole number.
Also there's a number called "e"
which is approximately 2.718.
But just like pi,
it's a number that goes on forever
and never repeats.
It's one that you use
quite a bit
when you're talking about logarithms,
which is in a later class.
So these are the different number sets
that we have here.
And all of these numbers
that you see here
are  what we call
"real numbers."
So everything you see here
is a real number.
Now you may have problems
that ask you what number sets
does this particular number
belong to?
For instance,
if it gives you the number 2,
the number 2
is a natural number.
It's also a whole number.
It's also an integer.
It's also a rational number,
and it is a real number.
So you would list all of those
different types of numbers
for what the number 2 is.
If you were given the number
two thirds  [2/3],
2/3 is a rational number
and it's also a real number.
But it is not
any of the other number sets,
so you would
only give those two values.
If you were given the number -5,
-5 is an integer.
It's also a rational number.
And it's a real number.
So you would list all of those.
And those are the different types
of number sets.
Those will help you
in deciding
if the type of answer
you're getting
makes sense
for your situation.
