In this explainer, we will learn how to identify the relationships between the subsets of the real numbers and how to represent real numbers on number lines.β
We recall that the set of rational numbers is the set of all quotients of integers. In other words, it contains all numbers of the form where , and is nonzero. We have also seen that there are numbers such as that cannot be written as the quotients of integers; we call numbers like this irrational numbers. Since it is not rational, it must be an element of the complement of . We call this the set of irrational numbers. We can use this set to construct a new set of numbers called the real numbers.
Definition: Real Numbers and the Set of Real Numbers
The set of real numbers, denoted , is given by
A real number is any element in the set ; this includes all numbers that can be represented on the number line such as the integers, rational numbers, and irrational numbers.
To explore this concept in more detail, letβs move on to the first example.
Example 1: Finding the Union of the Rational and Irrational Numbers
What is ?
Answer
We recall that . This is known as the set of rational numbers; it is the set of all quotients of integers. We can take the complement of this set to represent all numbers that are not rational. In other words, is the set of all numbers that cannot be represented as the quotients of two integers. We call these the irrational numbers.
Finally, we can take the union of these sets to construct the set of real numbers. We have
In our next example, we consider whether nonintegers are all rational numbers.
Example 2: Verifying a Statement about the Set of Integers
Is every noninteger number a rational number?
Answer
We recall that a rational number is one that can be written as the quotient of integers that is equivalent to saying that it has either a terminating decimal expansion or a repeating decimal expansion. We can then recall that numbers like do not have a repeating or finite decimal expansion and are, hence, not rational.
This is enough to answer the question. However, it may be useful to see a formal proof of this fact. Letβs assume that is a rational number. This means that for some integers and where is nonzero. We can note that is not an integer, since there is no integer whose square is 2. If and share any factors, we can cancel them, so we can assume their highest common factor is 1. We now square both sides of the equation to get
If we multiply through by , we have
The left-hand side of the equation is even, so the right-hand side must also be even. However, is an integer, so must be even. Letβs say ; substituting this in gives
Now, the right-hand side of the equation is even, so the left-hand side must also be even. Since is an integer, we must have that is even. But we assumed that and have no common factors. This shows that our original assumption that is a rational number cannot be true. Therefore, is irrational.
Hence, we were able to show that it is not true that every noninteger number is a rational number, so the answer is no.
There are many different subsets of the real numbers: we have the irrational numbers , the rational numbers , the integers , the natural numbers , and many more. Often, we will need to categorize given numbers into each of these sets. However, this can be difficult since some numbers appear in multiple sets. For example, 2 is a natural number, an integer, and a rational number.
To get around this problem we can consider a Venn diagram containing all of these sets. Our outermost set would be . Next, we know that , and we note that there is no number that is both rational and not rational. So, is composed of the disjoint sets and . Finally, . This gives us the following Venn diagram.
We can now use this to classify any real number. For example, we already showed that , , and . We can represent this on the Venn diagram by writing 2 inside all of these sets as follows.
Letβs now see an example of using a Venn diagram to classify a list of given real numbers.
Example 3: Placing Real Numbers in a Venn Diagram
For the numbers , 7, 1.25, , , , , and , which of the following Venn diagrams is true?
Answer
To determine which of the Venn diagrams is correct, we first recall that we need to write each number inside the correct subsets. If a number is encircled by the set, this indicates that it is a member of the set.
The easiest way to answer this question is to eliminate options. We can do this by determining which subset each of the given numbers is a member of and which subsets they are not members of.
Letβs start by recalling what is meant by each set. We will go from the innermost subset outward:
First, the set of natural numbers, , is the set of counting numbers; it is given by . We can immediately see that . We might stop here; however, 9 is a perfect square. So, . Hence, . Similarly, 8 is a perfect cube, so . Hence, . We can use this to eliminate option C, since it does not include and options D and E, since they state that is a natural number.
Second, we recall that is the set of integers that is any whole number including negatives. We can see that . However, the remaining numbers are not integers, so we move on to the next subset.
Third, we recall that is the set of rational numbers: that is the set of all quotients of integer values. Thus, . We also know that any decimal with a finite number of digits is a rational number. In particular, we have that , so . This allows us to eliminate option B, which states that these are not rational numbers.
We could stop here; however, letβs verify that option A is correct. The remaining two numbers, and , cannot be written as the quotient of two integers. To see why this is the case, we first recall that the square root of a nonsquare number is irrational. Since 2 is not a square number, is not rational. This means it is an element of , which is the set of irrational numbers. We also note that , so . Similarly, we can recall that the cube root of a nonperfect cube is irrational. Thus, and .
This gives us the following Venn diagram.
We can see that it exactly matches option A.
Before we move on to our next example, letβs look at classifying numbers given in different forms into the correct subsets of the real numbers. We can do this by writing these numbers in a table with the possible subsets as columns.
| Number | Natural Number | Integer | Rational Number | Irrational Number | Real Number |
|---|---|---|---|---|---|
The first step in classifying each of the given numbers is to recall the definitions of each of the sets:
- The set of natural numbers is the set of counting numbers; it is given by .
- The set of integers is the set of whole numbers including negatives: .
- The set of rational numbers is the set of quotients of integer values: .
- The set of irrational numbers is the complement of the set of rational numbers. It includes all numbers that cannot be written as the quotients of integers.
- The set of real numbers includes the sets of the rational and irrational numbers: .
We can then recall that all of these sets are subsets of the real numbers. In particular,
We can use this to simplify the process of filling in the table.
To determine which set each number is in, we will want to simplify the given expression for each number.
First, we note that , so . We can see that this is not a counting number; however, it is an integer. We then note that , so it is also a rational and a real number. It is not an irrational number since it is rational; this gives us the following.
| Number | Natural Number | Integer | Rational Number | Irrational Number | Real Number |
|---|---|---|---|---|---|
Second, we note that is a decimal with a repeating decimal expansion. In particular, we can note that this is equivalent to , so this is a rational number. It is not a counting number nor is it an integer. Finally, since it is rational it is not irrational and it is a real number. This gives us the following entries in the table.
| Number | Natural Number | Integer | Rational Number | Irrational Number | Real Number |
|---|---|---|---|---|---|
Third, we want to simplify the expression . We can do this by noting that , so . This is a natural number, and by using the fact that , we can see that it is also an integer, a rational number, and a real number. We have shown the following entries.
| Number | Natural Number | Integer | Rational Number | Irrational Number | Real Number |
|---|---|---|---|---|---|
Fourth, we note that 3 is not a square number. We then recall that the square root of a nonsquare number is irrational. Since an irrational number cannot be a natural number, an integer, or a rational number and all irrational numbers are real numbers, we get the following entries.
| Number | Natural Number | Integer | Rational Number | Irrational Number | Real Number |
|---|---|---|---|---|---|
Finally, we can rewrite as 0. This is a natural number, so we get the following entries.
| Number | Natural Number | Integer | Rational Number | Irrational Number | Real Number |
|---|---|---|---|---|---|
This gives us the following final table.
| Number | Natural Number | Integer | Rational Number | Irrational Number | Real Number |
|---|---|---|---|---|---|
In our next example, we will consider what happens when we add numbers from different subsets of the real numbers.
Example 4: Understanding Real Numbers
If , , and , then .
Answer
To answer this question, letβs first try some possible values of , , and . We can do this by recalling the following:
- The set of integers is the set of whole numbers including negatives: .
- The set of rational numbers is the set of quotients of integer values: .
- The set of irrational numbers is the complement of the set of rational numbers. It includes all numbers that cannot be written as the quotients of integers such as .
- The set of real numbers includes the sets of the rational and irrational numbers: .
Thus, we could have , , and . Then, we have . However, we want to know what happens in general, so letβs try another triplet of values.
Letβs say , , . Now we have . We see in this that the sum of these numbers is irrational.
We can use any real value of along side and , so our set must contain all real values. Hence, the answer is .
In general, the sum of numbers in some of the subsets of the real numbers is a closed operation. For example, the sum of two integers is always an integer, the sum of two rational numbers is a rational number, and the sum of two real numbers is a real number.
The same is not true for all of these subsets of . For example, and , but
This means, in general, we need to treat , , and as elements of , since this is the largest subset containing all three numbers. Therefore, summing three elements of gives us a real number.
Hence, the answer is .
There is one final way of visualizing the set of real numbers, and this involves using a number line. To represent a real number on a number line, we start with a point for zero and then denote the right direction as positive and the left direction as negative. Then, we say that a point on the number line represents the real number if its displacement from 0 is units.
For example, we can represent some real numbers on a number line as shown.
We say that two real numbers are equal if the points that represent them on the number line are the same. It is also worth noting that, since these points are based on a displacement, we can determine the displacement from point to point representing real numbers and by calculating . We can determine the distance between the points by using the formula .
Example 5: Placing Real Numbers on a Number Line
If point on the number line represents and point
represents , which of the following line segments has a greater length?
Answer
We can start by evaluating the two given expressions. We note that , so
Similarly, we note that , so
We can use these values to mark and on the number line.
There are now many different ways to determine which line segment is the longest. We can see in the diagram that and are the furthest distance apart, so must be the longest.
It can also be worth recalling how to find the length of the line segment between two points on a number line in general. The distance between values and on a number line is given by . That is the absolute value of the displacement from point to point .
If we label the values represented by , , and as , , and , we could calculate , , and to determine the lengths of the line segments.
However, this is not necessary in this case, since we can see that is the longest of the line segments, which is option C.
Letβs finish by recapping some of the important points from this explainer.
Key Points
- The set of real numbers, denoted , is given by .
- There are many subsets of the real numbers. In particular, .
- Every real number can be represented by a point on a number line.