### Video Transcript

In this video, we will learn how to
compare and order real numbers.

An ordered set is one in which we
can compare any two elements of the set π and π with one of three possible
outcomes. Either π and π are equal, the
order of π is greater than that of π, or the order of π is greater than that of
π.

Letβs begin by recalling how we can
compare two real numbers on a number line. If π and π are real numbers
represented by the points capital π΄ and capital π΅ on the number line as shown,
then since π΅ lies to the right of π΄ and since this is the positive direction of
the number line, we can say that π is greater than π. Note that this is the same as
saying that π is less than π. The two points could be other way
round as shown on the second number line. In this case, π is greater than π
or π is less than π. Our third possibility is that π΄
and π΅ lie at the same point on the number line. In this case, we have π is equal
to π.

We will now consider a couple of
examples where we need to compare two real numbers given in different forms.

Fill in the blank using less than,
equal to, or greater than. Seven thirtieths what 4.9.

In this question, we need to decide
whether the fraction seven thirtieths is less than, equal to, or greater than
4.9. One way of doing this is by
considering the positions of both real numbers on a number line. Letβs consider the number line
between zero and five as shown. Since 4.9 is already written in
decimal form, we can place this on the number line between the integers four and
five. In the fraction seven thirtieths,
the denominator, 30, is greater than the numerator, seven. This means that seven thirtieths is
less than one.

We can therefore add seven
thirtieths to our number line as shown. It lies between zero and one. Since seven thirtieths lies to the
left of 4.9, we can say it is less than 4.9. This is written with the less than
symbol between seven thirtieths and 4.9. We are able to use this method to
compare any two real numbers using a number line.

Letβs now consider a second example
of this type.

Fill in the blank using less than,
equal to, or greater than. 7.2 what the absolute value of
negative 47 over 38.

We begin by recalling that we can
order numbers based on their position on a number line. One way to do this is to find the
decimal expansions of each number. 7.2 is already written as a
decimal, and we know that this lies between the integers seven and eight as
shown. Next, we recall that taking the
absolute value removes the sign of a number. This means that the absolute value
of negative 47 over 38 is equal to 47 over 38. We could find the value of this
fraction by using a calculator. However, this is not necessary. Noting that 38 goes into 47 once
with a remainder of nine, 47 over 38 can be rewritten as one plus nine over 38. Since nine over 38 lies between
zero and one, then 47 over 38 must lie between the integers one and two. This must therefore also be true of
the absolute value of negative 47 over 38.

We can therefore conclude that
since 7.2 lies to the right of the absolute value of negative 47 over 38, it must be
greater than it. The correct answer is βgreater
than.β 7.2 is greater than the absolute
value of negative 47 over 38.

Before we move on to our next
example, we can use this definition of the comparison of real numbers to define some
useful subsets of the real numbers. Firstly, the set of all positive
real numbers is the set of all real numbers greater than zero. This can be written more formally
as shown. The set of all negative real
numbers is the set of all real numbers less than zero, and this can also be written
in a similar format.

It is worth noting that zero is not
in either of these sets, since we consider zero to not be positive or negative. We can include zero by considering
the nonnegative and nonpositive numbers as follows. The set of nonnegative real numbers
is the set of all real numbers that are not negative. This is given by the union of the
set of all positive real numbers and the set containing zero. In the same way, the set of
nonpositive real numbers is the set of all real numbers that are not positive. This is given by the union of the
set of all negative real numbers and the set containing zero.

We can therefore conclude that all
real numbers are either positive, negative, or equal to zero. On a number line, the negative
numbers lie to the left of zero and the positive numbers lie to the right.

We will now consider an example
where we need to determine whether a given real number is positive or negative.

For a real number π₯, determine
whether π₯ is positive or negative in each of the following cases. Firstly, π₯ is equal to negative
seven; secondly, π₯ is greater than two; and thirdly, negative three is greater than
π₯.

We begin by recalling that positive
numbers lie to the right of zero on a number line, whilst negative numbers lie to
the left of zero. We can therefore determine the
signs of π₯ in each case by considering the possible positions of π₯ on a number
line.

In the first part of the question,
we are told that π₯ is equal to negative seven. We know that negative seven will
lie to the left of zero as shown. And we can therefore conclude that
when π₯ is equal to negative seven, π₯ is negative. In the second part of the question,
we are told that π₯ is greater than two. And this means that π₯ lies to the
right of two on a number line. Since π₯ lies to the right of two
and two lies to the right of zero, we can conclude that π₯ lies to the right of zero
and is therefore positive.

In the final part of the question,
we have negative three is greater than π₯, which can also be read as π₯ is less than
negative three. Marking negative three on our
number line, we know that π₯ lies to the left of this. And since all values to the left of
negative three are negative, we can conclude that if negative three is greater than
π₯, π₯ is negative.

Now that we can compare any two
real numbers, we can use this to order any list of any real numbers. This can be done in one of two
ways: either from least to greatest, which is called ascending order, or from
greatest to least, which is called descending order. A list of real numbers π sub one,
π sub two, and so on, up to π sub π is said to be in ascending order if π sub
one is less than π sub two, and so on, which is less than π sub π. In other words, the numbers are
getting larger. In the same way, a list of real
numbers π sub one, π sub two, and so on, up to π sub π is said to be in
descending order if π sub one is greater than π sub two, and so on, which is
greater than π sub π. In this case, the numbers are
getting smaller.

In our next example, we need to
order a set of rational and irrational numbers. To help us order any irrational
numbers, we recall two key properties. Firstly, if π is greater than π
and π and π are both greater than or equal to one, then π squared is greater than
π squared. Secondly, if π and π are two
positive numbers such that π is greater than π, then the square root of π is
greater than the square root of π. We will now see an example where we
need to use these properties to order a list of real numbers.

By considering square numbers,
order the square root of 19, the square root of 24, the square root of 28, four, the
square root of 17, five, and 4.5 from least to greatest.

In this question, we have a mixture
of rational and irrational numbers that we need to order from least to greatest. This is known as ascending
order. We begin by ordering the three
rational numbers. Four is less than 4.5, which is
less than five. Since 17, 19, 24, and 28 are not
square numbers, the square roots of these numbers would give us irrational
numbers.

Next, we recall that if we have two
positive numbers such that π is less than π, then the square root of π is less
than the square root of π. As such, we can order the four
irrational numbers. The square root of 17 is less than
the square root of 19, which is less than the square root of 24, which is less than
the square root of 28.

We now need to compare the
irrational numbers or radicals to the three rational numbers. One way to do this is to rewrite
each rational number as a radical. Since four squared is equal to 16,
we know that four is equal to the square root of 16. Likewise, since five squared is 25,
the square root of 25 is equal to five. We are now left with 4.5, which is
equal to the square root of 4.5 squared. Noting that 4.5 is equal to the
improper or top-heavy fraction nine over two, we can calculate 4.5 squared by
squaring nine over two. We square the numerator and
denominator separately, giving us 81 over four, which is equal to the decimal
20.25. 4.5 is therefore equal to the
square root of 20.25.

We can now order the seven radical
expressions. The smallest is the square root of
16. This is followed by the square root
of 17, the square root of 19, the square root of 20.25, the square root of 24, the
square root of 25, and finally the square root of 28. Replacing the three radical
expressions with their original rational equivalents gives us the set of seven
numbers from least to greatest. In ascending order, we have four,
the square root of 17, the square root of 19, 4.5, the square root of 24, five, and
the square root of 38.

We will now finish this video by
summarizing the key points. We began this video by noting that
if π and π are real numbers represented by the points capital π΄ and capital π΅ on
a number line, then we know the following. If π΄ lies to the right of π΅, then
π is greater than π. If π΄ lies to the left of π΅, then
π is less than π. And if π΄ and π΅ are coincident,
then π is equal to π. We noted that saying that π is
greater than π is the same as saying that π is less than π. This means that weβre able to
switch the numbers and switch the direction of the order.

We saw the notation for the set of
positive and negative real numbers, where the set of positive real numbers is all
real numbers greater than zero and the negative real numbers are all real numbers
less than zero. We extended this to consider the
set of nonnegative and nonpositive real numbers. We saw that a set of real numbers
π sub one, π sub two, up to π sub π is said to be in ascending or descending
order if the inequality shown holds. When the numbers are in ascending
order, they are getting larger. And when the numbers are in
descending order, they are getting smaller. For two real numbers π and π that
are greater than or equal to zero, then if π is greater than π, then π squared is
greater than π squared and root π is greater than root π.

Finally, for any real number π₯, we
have the square root of π₯ squared is equal to the absolute value of π₯. In general, we can say that the
square root of π₯ squared is greater than or equal to π₯ with equality only when π₯
is greater than or equal to zero.