Comparing and ordering real numbers worksheet answer key

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

An ordered set is one in which we can compare any two elements of the set, 𝑎 and 𝑏, with one of three possible outcomes:

• 𝑎 and 𝑏 are equal,
• the order of 𝑎 is greater than that of 𝑏,
• the order of 𝑏 is greater than that of 𝑎.

Of course, we have not described exactly what is meant by an order since we are working with a general set and there are many different possible interpretations for an ordering. Usually, when working with numbers, we work with the idea of value. So, we can compare the value of two numbers and determine if their values are the same or if one is greater than the other.

For the rest of this explainer, we will only be talking about the set of real numbers ℝ. There are two methods we can use to compare real numbers. First, we can compare their decimal expansions. For example, we can see that the value of 0.6 is greater than 0.2 because 6 is greater than 2. This also extends to negative numbers since we can say that negative numbers are always less than positive numbers, and the further from the origin a negative number is, the less it is.

However, it can be difficult to compare the decimal expansions of some numbers. For example, numbers like 𝜋 have a nonrepeating infinite decimal expansion, and it can be difficult to find the digits of irrational numbers.

A second method we can use is to recall that all real numbers can be represented by points on a number line. For example, let’s say 𝑎,𝑏∈ℝ are represented by points 𝐴 and 𝐵 on the number line as shown.

We can say that 𝐵 is to the right of 𝐴, and since this is the positive direction of the number line, we can also say that 𝑏 is greater than 𝑎. It is worth noting that this is the same as saying that 𝑎 is less than 𝑏. Of course, the points could be the other way around.

In which case we know that 𝑏 is less than 𝑎; or alternatively, 𝑎 is greater than 𝑏. Finally, points 𝐴 and 𝐵 can be coincident. In other words, they can be the same point on the number line.

Then, we can say that 𝑎 is equal to 𝑏. We can now describe this ordering formally as follows.

Definition: Ordering Real Numbers

If 𝑎 and 𝑏 are real numbers represented by points 𝐴 and 𝐵 on a number line, then we know the following:

• If 𝐴 lies to the right of 𝐵, we say that 𝑎 is greater than 𝑏, and this is written as 𝑎>𝑏.
• If 𝐴 lies to the left of 𝐵, we say that 𝑎 is less than 𝑏, and this is written as 𝑎<𝑏.
• If points 𝐴 and 𝐵 are congruent, we say that 𝑎 is equal to 𝑏, and this is written as 𝑎=𝑏.

It is also worth noting that saying 𝑎>𝑏 is the same as saying that 𝑏<𝑎, so we can always switch the numbers and switch the direction of the order.

Let’s now see an example of comparing two real numbers given in different forms.

Example 1: Comparing a Decimal Number to a Given Fraction

Fill in the blank using <, =, or >: 7304.9.

Answer

We start by recalling that we order numbers based on their position on a number line. So, we should start by representing each number on a number line. We can do this by finding decimal expansions of each number.

We note that 4.9 is already a decimal, and since 30>7, we can note that 7 30<1. Adding these to a number line gives us the following.

We can see that 730 is to the left of 4.9, so we say it is less than 4.9.

We write this as 730<4.9.

Let’s now see another example of comparing two real numbers.

Example 2: Comparing Decimals to the Absolute Value of Fractions

Fill in the blank using < , =, or >: 7.2|||−4738|||.

Answer

We start by recalling that we order numbers based on their position on a number line. So, we should start by representing each number on a number line. We can do this by finding decimal expansions of each number.

We note that 7.2 is already a decimal. We know that taking the absolute value removes the sign of the number, so |||−4738|||=4738.

We could now find the value of this fraction by using a calculator; however, this is not necessary. Instead, we can note that 38 goes into 47 with a remainder of 9, so 4738=1+938.

We know that 938 will be between 0 and 1 since the denominator is greater than the numerator. Hence, 4738 is between 1 and 2 on the number line. This gives us the following.

We can see that 7.2 is to the right of |||−4 738|||, so we say it is greater than |||−4738|||.

We write this as 7.2>|||−4738|||.

Before we move onto our next example, we can use this definition of the comparison of real numbers to define some useful subsets of the real numbers.

Definition: Positive and Negative Numbers

The set of positive real numbers ℝ is the set of all real numbers greater than 0. So, ℝ={𝑥∶𝑥>0,𝑥∈ℝ}.

The set of negative real numbers ℝ is the set of all real numbers less than 0. So, ℝ={𝑥∶𝑥<0,𝑥∈ℝ}.

We call the elements of ℝ positive and the elements of ℝ negative.

It is worth noting that 0 is not in either of these sets, so we consider 0 to not be positive nor negative. We can include 0 by considering the nonpositive and nonnegative numbers as follows.

Definition: Nonpositive and Nonnegative Numbers

The set of nonnegative real numbers is the set of all real numbers that are not negative. It is given by ℝ∪{0}.

The set of nonpositive real numbers is the set of all real numbers that are not positive. It is given by ℝ∪{0}.

We can also see that all real numbers are positive, negative, or equal to zero. In other words, ℝ=ℝ∪{0}∪ℝ.

We can see this on a number line.

The negative numbers are those that lie to left of 0, and the positive numbers are those that lie to the right of 0.

Let’s now see an example of determining whether a given real number is positive or negative, also known as determining the sign of the number.

Example 3: Identifying Whether a Number is Positive or Negative

For a real number 𝑥, determine whether 𝑥 is positive or negative in each of the following cases.

1. 𝑥=−7
2. 𝑥>2
3. −3>𝑥

Answer

We recall that positive numbers lie to the right of 0 on a number line, and negative numbers lie to the left of 0. We can therefore determine the signs of 𝑥 in each case by considering the possible positions of 𝑥 on a number line.

Part 1

We know that −7 will lie to the left of 0 on a number line.

Hence, 𝑥 is negative in this case.

Part 2

We are told that 𝑥>2. We recall that this means 𝑥 is to the right of 2 on a number line. We can sketch this on a number line as follows.

Since 𝑥 lies to the right of 2 and 2 lies to the right of 0, we can conclude that 𝑥 lies to the right of 0 and is hence positive.

Part 3

We are told that −3>𝑥. We recall that this means −3 is to the right of 𝑥 on a number line. This is the same as saying 𝑥 lies to the left of −3, or in other words, 𝑥<−3.

We can sketch this on a number line as follows.

We can see that 𝑥 must lie to the left of 0 and hence is negative.

In our next example, we will compare a real number to the square root of its square.

Example 4: Comparing a Real Number with the Square Root of Its Square

Is (𝜋−38) greater than, equal to, or less than 𝜋−38?

Answer

We may be tempted to answer this question as “equal to” since it seems like the square root of a square should always return the same number. However, this is not the case. To see this, let’s consider (− 1). We know that (−1) =1. Hence, (−1)=√1=1=−(−1).

In this case, the square root of the square is greater than the original number. The reason for this is that the square root always gives the nonnegative root. Thus, √𝑥=|𝑥|, and in particular, √𝑥≥𝑥.

We can now consider this method to help us answer the question. First, we note that 38 is much greater than 𝜋, since 𝜋≈3 .14. This means that 𝜋−38<0; in other words, it is negative.

Hence, ( 𝜋−38)=|𝜋−38|= 38−𝜋.

We have shown that (𝜋−38)>𝜋−38. So, the square root of the square of this number is greater than the original number.

Now that we can compare any two real numbers, we can use this to order any list of any real numbers. We can order the list in one of two ways, either from least to greatest, which is called ascending order since the numbers get bigger, or from greatest to least, which is called descending order. We can define these terms formally as follows.

Definition: Ascending and Descending Order

A list of real numbers 𝑎,𝑎 ,…,𝑎 is said to be in ascending order if 𝑎<𝑎<⋯<𝑎. In other words, the numbers are getting larger.

A list of real numbers 𝑎,𝑎,…,𝑎 is said to be in descending order if 𝑎>𝑎>⋯>𝑎. In other words, the numbers are getting smaller.

We can order a list of real numbers into either ascending or descending order just by comparing pairs of numbers in the list. However, in order to compare irrational numbers to rational numbers, we will need to know a few useful properties.

First, if 𝑎>𝑏 and 𝑎,𝑏≥1, then 𝑎>𝑏. Second, if we have two positive numbers such that 𝑎>𝑏, then √𝑎>√𝑏.

Let’s now see an example of using these properties to order a list of real numbers.

Example 5: Arranging Real Numbers from Least to Greatest

By considering square numbers, order √19, √24, √28, 4, √17, 5, and 4.5 from least to greatest.

Answer

To do this, we can start by ordering all of the rational numbers. We can see that 4<4.5 and 4.5<5. We note that none of the radicals are square roots of square numbers, so we cannot easily write these as decimals. Instead, we will use the fact that if we have two positive numbers such that 𝑎>𝑏, then √𝑎>√𝑏. This allows us to write these radicals in ascending order: √17, √19, √24, √28.

We now need to compare the radicals to the rational numbers. To do this, we can rewrite each rational number as a radical. We note that 4=√16, 4.5=√4.5, and 5=√25.

This will allow us to order all of the numbers; we just need to evaluate 4.5 . One way of doing this is to note that 4.5=92, so 4.5=92=814=20.25. 

We can now order the radical expressions: √16,√17,√19,√20.25,√24,√25,√28.

Then, replacing the three radical expressions with their original rational equivalents gives us 4,√17,√19,4.5,√24,5,√28.

In our final example, we will need to order a list of real numbers involving the irrational number 𝜋 into ascending order.

Example 6: Arranging Real Numbers Involving 𝝅 in Ascending Order

Without using a calculator, order the numbers 3𝜋, 3√2, 𝜋, 9, and 3√ 3 from least to greatest.

Answer

Since the real numbers in this question involve both 𝜋 and radical expressions, we are going to need to compare the values of the radicals with 𝜋. We can do this by recalling a few facts:

• 𝜋>3,
• If 𝑎>𝑏≥0, then 𝑎>𝑏.

This allows us to note that 𝜋>3, and we know that 3=9. The remaining three expressions are all of the form 3𝑥 for some positive real value 𝑥. We note that if 𝑥>𝑦 for positive real numbers 𝑥 and 𝑦, then 3𝑥 >3𝑦. This means we need to compare √2, √3 , and 𝜋.

We recall that if 𝑎>1, then √𝑎<𝑎, so √3 <3; and we know that 3<𝜋, so √3<𝜋. We also recall that if we have two positive numbers such that 𝑏<𝑎, then √𝑏<√𝑎. Hence, √2<√3. Thus, 3√2<3√3<3𝜋.

We now need to compare these two lists. We can do this by comparing 3𝜋 to 9. Since 9= 3×3 and 𝜋>3, we know that 9<3𝜋. However, we cannot add it onto the list without comparing 9 to 3√3. Since 9=3×3 and √3<3, we know that 3√3<9, giving us 3√2<3√3<9<3𝜋.

Finally, since we already know that 9<𝜋, we just need to compare 𝜋 to 3𝜋. We know that 3<𝜋, so 3𝜋<𝜋, giving us 3√2,3√3,9,3𝜋,𝜋.

Let’s finish by recapping some of the important points from this explainer.

Key Points

• If 𝑎 and 𝑏 are real numbers represented by points 𝐴 and 𝐵 on a number line, then we know the following:
  • If 𝐴 lies to the right of 𝐵, we say that 𝑎 is greater than 𝑏, and this is written as 𝑎>𝑏.
  • If 𝐴 lies to the left of 𝐵, we say that 𝑎 is less than 𝑏, and this is written as 𝑎<𝑏.
  • If points 𝐴 and 𝐵 are coincident, we say that 𝑎 is equal to 𝑏, and this is written as 𝑎=𝑏.
• Saying 𝑎>𝑏 is the same as saying that 𝑏<𝑎, so we can always switch the numbers and switch the direction of the order.
• The set of positive real numbers ℝ is the set of all real numbers greater than 0.
• The set of negative real numbers ℝ is the set of all real numbers less than 0.
• The set of nonnegative real numbers is the set of all real numbers that are not negative. It is given by ℝ∪{0}.
• The set of nonpositive real numbers is the set of all real numbers that are not positive. It is given by ℝ∪{0}.
• A list of real numbers 𝑎,𝑎,…,𝑎 is said to be in ascending order if 𝑎<𝑎<⋯ <𝑎. In other words, the numbers are getting larger.
• A list of real numbers 𝑎,𝑎,…,𝑎 is said to be in descending order if 𝑎>𝑎>⋯>𝑎. In other words, the numbers are getting smaller.
• For real numbers 𝑎,𝑏≥0, if 𝑎>𝑏, then 𝑎>𝑏.
• For real numbers 𝑎,𝑏≥0, if 𝑎>𝑏, then √𝑎>√𝑏.
• For any real number 𝑥, we have √𝑥=|𝑥|. In general, we can say that √𝑥≥𝑥, with equality only when 𝑥 ≥0.