[MUSIC] Let's discuss real numbers. Informally, a real number is a number with
a decimal representation. And we represent a set of real numbers by a capital R. Now,
there are different subsets of real numbers. The first subset to consider are
what we call the natural numbers. And these are just the counting numbers that
we are used to. one, two, three, four, and so on. And we represent the set of natural
numbers by a capital N. That is N = one, two, three, four and so on. The second
subset to considers is what we call the integers. The integers are all these
natural numbers, together with their negatives and zero. And we represent the
set of integers by a capital Z. So, Z is all the negative natural numbers together
with zero and then, all these natural numbers. The third subset to consider will
be called the rational numbers. And we represent the set of rational numbers by
Q. And these are fractions or ratios of integers. So, a over b, where b is not
zero and a and b are integers. Now, the decimal representation of a rational
number either terminates or repeats. For example, one / two.
This is a rational number and its decimal representation is 0.5 and it stops or
terminates. Whereas, one-third which is also a rational number, has its decimal
representation repeating. It doesn't stop, it repeats. Now, if a real number is not
rational then it's what we call an irrational number. And the set of
irrational numbers we represent by a capital I. So, these are real numbers that
are not rational. So, a real number is either rational or irrational. So, let's
write that up here. Capital R, so, rational numbers together with the
irrational numbers and we have that the natural numbers are contained in the
integers, contained in the rationals, are contained in the reals. Let's look at an
example. Let S be the set and we want to list the subsets of S consisting of the
natural numbers, the integers, the rationals, and the irrationals. Remember,
that the natural numbers are the counting number, one, two, three, four and so on.
So, looking at S, we see we have three here as well as ten, so these will be our
natural numbers. Now, the integers are these natural numbers, together with the
zero, and we do have the zero in our list so let's add zero here as well as any
negative natural number. Looking at S, we see we have this -one here in the front,
but also, we have this minus square root four.
Because of the square root, we might have missed this. But isn't the square root of
four just two? So, this is really just -two, which is an
integer. So, we'll add these to our list as well, -one and negative square root of
four. Now, what about the rationals? All of
these integers are rational numbers. Because think of three, for example, we
can write three as three / one, so we can think of it as a fraction. So, we'll have
all of these integers, three, ten, zero, -one, negative square root four, but what
else? Looking at S, we have this three-fourths here, that will be a
rational number, but also look at this last number here. Again, because of this
square root, it might be misleading. But really, this is the same as square root 81
divided by the square root of four, which is equal to nine / two.
So, it is a rational number. Som we'll add those two values to our list here.
three-fourths and square root 81 / four. Alright. Finally, the irrational numbers
are any number in S that are not in this set here. Namely, negative square root
five pi and square root seven. So, our irrationals are negative square
root five pi and square root seven. And this is how we classify real numbers
into these subsets. Thank you and we'll see you next time.