Consider the numbers 1/3, 7, and negative 15.
To which sets of numbers, listed below,
does each of these numbers belong?
And they give us a bunch of different sets of numbers.
And we could just review these.
The natural numbers, and you'll see
this used in different contexts in different places,
but for the sake of this video, we'll
assume this is counting numbers, and it does not equal 0.
So the natural numbers are 1, 2, 3, 4, and so on and so forth.
We'll assume that the whole numbers are essentially
the natural numbers plus 0.
So the whole numbers are 0, 1, 2, 3, 4, so on and so forth.
The integers we'll assume it's the whole numbers
plus the opposite of the natural numbers,
or the negative version of the natural numbers.
So we would go from very negative values
to maybe negative 4, negative 3, negative 2,
negative 1, 0, 1, 2, 3, 4, so on and so forth.
Rational numbers.
These are numbers that can be expressed
as the ratio of two integers.
So a rational number can be expressed as m
over n, where both m and n are integers.
And of course n cannot be 0, because anything divided by 0
is not defined.
M could be 0, and in fact, 0 is a rational number,
because 0 can be represented as 0 over 1.
Or really 0 over 2, or 0 over 3.
I think you get the idea.
Or 0 over negative 537.
These are all representations of 0,
and you have an integer in the numerator, integer
in the denominator.
So 0 is definitely a rational number.
You just can't have n, you can't have the denominator, be 0.
Irrational numbers are numbers that cannot be expressed
in this way.
And so common versions of irrational numbers
are numbers that you will never see digits that repeat on, go
on and on forever, or have series of sequences of numbers
that keep repeating on and on and on.
And examples of irrational numbers are square root of 2,
or pi.
And the ratio of the circumference, the diameter
of a circle, or you might learn about the irrational number
e, which is 2.71.
And it keeps going, and it doesn't repeat.
And then real numbers are really all of these combined.
If you combine the irrational numbers
and the rational numbers, then you
are talking about the real numbers.
Every real number is either rational or irrational.
So let's classify what we have up here.
1/3 over here-- let me do that in a different color-- so 1/3,
is it a natural number?
Well no, it's not.
It's not just a natural counting number.
Is it a whole number?
No, it's not a whole number, it's in fact a fraction.
Is it an integer?
No once again, it is not an integer.
Is it a rational number?
Well sure it is.
It can be represented as 1 over 3.
Both 1 and 3 are integers.
So 1/3 is rational.
And if something is rational, it cannot be irrational.
There's nothing that's both rational and irrational.
This literally means not rational.
And it is also a real number.
Pretty much all of these are going to be real numbers.
Now, let's try 7 out.
So the number 7, it is a natural number.
It is just a regular counting number.
It is a whole number.
Any natural number is also going to be a whole number.
And anything that is a natural number that's
also going to be a whole number is also going to be an integer.
So this is also an integer.
And anything that is a natural number, a whole number,
and integer, is also going to be a rational number.
And to see that, you can write 7 as 7 over 1.
In fact, you can write any integer
as that number over 1, which hopefully makes it obvious
that that is also a rational number.
And if it's rational, you can't be irrational.
That literally means not rational.
And then of course, you're going to be real.
It's going to be a real number.
And you're going to appreciate later
on in your mathematical careers why we even
go through the trouble of saying it's a real number.
Because as far as it seems right now,
this kind of includes all the numbers.
What you're going to see in the future is,
we're going to start defining numbers that
don't fall into this category.
And they're a little bit philosophically interesting
as well.
But anyway, back to the problem.
We have a negative 15.
Negative 15 is not a natural number, it's a negative number.
So it's not a natural number, natural number
is only the positive integers, we could say.
Whole number, it's not a whole number,
because that's just the natural numbers and 0.
It is an integer.
It is the opposite of positive 15.
It is an integer.
It is rational, it can be represented as 15 over 1.
So it is a rational.
And because it is rational, it cannot be rational.
And of course, it is also real.
And we are done.