>> All right.
So the real number system.
Real number system starts
with natural numbers,
and these are things
that appear in nature.
If you walk outside,
you'll see one, maybe two,
three animals walking around.
You don't see half an
animal walking around.
That would not be natural.
So that's why we call
them "natural numbers";
they appear in nature.
Then, after that, you can
kind of think of it as, oh,
now they discovered
"nothing," they discovered zero.
So now we have to call these
numbers something else,
and they're called
now "whole numbers."
So if you take all the natural
numbers, you now put a zero
in front, they're now
called whole numbers.
Well, then they discovered
the negatives.
This is not the true
progression,
I'm just giving you,
like, a little pattern
with our number system.
So now you take the negatives
and you put those in front,
and now they're called
"integers."
So all the natural
numbers are whole numbers,
and all the whole
numbers are integers.
And then all of a sudden,
here come partial numbers,
like fractions and decimals
that repeat and end.
Those are called
"rational numbers."
So we take all the integers, you
put some fractions and rationals
in there, a lot of them,
we now classify those
as rational numbers.
So that's the progression
so far.
You had the naturals,
you throw on zero,
they're now called whole;
you throw on the integers,
they're now called -- sorry
-- throw on the negatives,
they're now called integers;
throw on the fractions
and the decimals, the
repeating and ending decimals,
they're now called rational.
Well, if something is not
rational, we call it --
you guessed it -- "irrational,"
and that's the numbers
that do not repeat,
they do not end,
that we don't really exactly
know where to put these, right.
You cannot exactly place them
somewhere on a number line
because they do not
repeat, they do not end.
So they're irrational.
We call them -- like our
radicals, if you had square root
of 3 or square root of 2,
these are irrational numbers.
Or pi, there's no
pattern to them.
Okay. So now if you
take all the rational
and all the irrational -- the
rational and irrational --
put them together, they're
now called real numbers
and that's our number system.
So you take the naturals --
that's our counting numbers, 1,
2, 3, 4; throw a 0 on it, it's
now called a whole number;
throw some negatives,
negative whole numbers,
they're are called integers; put
in some fractions and repeating
or ending decimals, those
are called rational;
take the rational and
all the non-repeating
or ending decimals, the
irrationals, put them together
and you get the real
numbers, so that's everything.
All right.
Another way to look
at, as a picture,
if you took the number 2,
that's a counting number,
number 2 is a counting number.
You can also call
it a whole number
because the whole
numbers include all the
counting numbers.
You could also call
it an integer
because the integers include
all the wholes and the counting.
You could also call
it a rational number,
and you can also call
it a real number.
So the number 2 is a counting
number, a whole number,
an integer, a rational
number, and a real number.
The one thing it is
not is irrational.
Number 2 is a nice
rational number.
Okay. So let's look
at these again.
I have a set of numbers up here,
and we're going to place them
where they're supposed to be.
So of these numbers,
which ones are wholes?
Well, 2 is solid, it's a nice
whole; 3 is a whole number.
We also need the number 0.
Remember, the wholes start
with 0; so 0, 1, 2, 3, and 4.
All right.
Integers. Well, we
already know that all
of our whole numbers
are integers.
Well, we can also now include
the negative whole numbers --
so that one right there,
the negative one --
and that is all the integers.
All right.
Irrational.
Irrational are our
radicals and there are pi,
the numbers that do
not repeat or end.
So if you just go through them,
that one ends, it ends, it ends,
it ends, repeats,
ends, does not repeat
or end, that's the radical.
And then pi as well,
that's irrational.
And then we know for rational,
rational's all the numbers
that are not irrational;
so 2, 3, -1, 0.5,
234.12 with A repeat
repetition, and 0, those are all
of our rational numbers.
And then, real, well,
that's all of them.
And that's it.
That's our real number system.
