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Lesson Video: Comparing and Ordering Real Numbers Mathematics • 8th Grade

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

16:49

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‘Ž and 𝑏 with one of three possible outcomes. Either π‘Ž and 𝑏 are equal, the order of π‘Ž is greater than that of 𝑏, or the order of 𝑏 is greater than that of π‘Ž.

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

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. Let’s 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.

Let’s 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 β€œgreater than.” 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‘Ž sub one, π‘Ž sub two, and so on, up to π‘Ž sub 𝑛 is said to be in ascending order if π‘Ž sub one is less than π‘Ž sub two, and so on, which is less than π‘Ž sub 𝑛. In other words, the numbers are getting larger. In the same way, a list of real numbers π‘Ž sub one, π‘Ž sub two, and so on, up to π‘Ž sub 𝑛 is said to be in descending order if π‘Ž sub one is greater than π‘Ž sub two, and so on, which is greater than π‘Ž sub 𝑛. 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‘Ž is greater than 𝑏 and π‘Ž and 𝑏 are both greater than or equal to one, then π‘Ž squared is greater than 𝑏 squared. Secondly, if π‘Ž and 𝑏 are two positive numbers such that π‘Ž is greater than 𝑏, then the square root of π‘Ž is greater than the square root of 𝑏. 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‘Ž is less than 𝑏, then the square root of π‘Ž is less than the square root of 𝑏. 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‘Ž and 𝑏 are real numbers represented by the points capital 𝐴 and capital 𝐡 on a number line, then we know the following. If 𝐴 lies to the right of 𝐡, then π‘Ž is greater than 𝑏. If 𝐴 lies to the left of 𝐡, then π‘Ž is less than 𝑏. And if 𝐴 and 𝐡 are coincident, then π‘Ž is equal to 𝑏. We noted that saying that π‘Ž is greater than 𝑏 is the same as saying that 𝑏 is less than π‘Ž. This means that we’re 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‘Ž sub one, π‘Ž sub two, up to π‘Ž sub 𝑛 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‘Ž and 𝑏 that are greater than or equal to zero, then if π‘Ž is greater than 𝑏, then π‘Ž squared is greater than 𝑏 squared and root π‘Ž is greater than root 𝑏.

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.

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