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