Lesson Explainer: The Set of Rational Numbers Mathematics • 6th Grade

In this explainer, we will learn how to identify rational numbers and find the position of a rational number on a number line.

Rational numbers are often called fractions. However, they are a specific type of fraction where both the numerator and denominator must be integers and the denominator cannot be equal to zero.

One way of application of rational numbers is to consider proportions of a whole. For example, if we split a pizza into 8 equal slices, then each slice represents 18 of the whole pizza.

Similarly, we can think about these numbers by considering number lines. We can ask which number lies halfway between 0 and 1. Since double this number must be 1, we can see that this is 12.

We can define the set of all rational numbers more formally as follows.

Definition: The Set of Rational Numbers

The set of rational numbers, written β„š, is the set of all quotients of integers. Therefore, β„š contains all elements of the form π‘Žπ‘ where π‘Ž and 𝑏 are integers and 𝑏 is nonzero. In set builder notation, we have β„š=ο«π‘Žπ‘βˆΆπ‘Ž,π‘βˆˆβ„€π‘β‰ 0.and

Using this definition, we can see some interesting properties of the set of rational numbers. First, we can note that all integers are rational numbers, since if π‘βˆˆβ„€, then 𝑐=𝑐1, so π‘βˆˆβ„š. This means that the set of integers is a subset of the set of rational numbers. We can then recall that the set of natural numbers is a subset of the integers, giving us β„•βŠ‚β„€βŠ‚β„š.

Although it is beyond the scope of this explainer to prove this result, some numbers such as √2 or πœ‹ are not rational and are called irrational. It is worth noting that a number cannot be rational and irrational at the same time.

We can also use this definition to find some examples of rational numbers. For example, 1 and 2 are integers, so 12βˆˆβ„š. Similarly, 53 and 127 are rational numbers. One way of conceptualizing rational numbers like these is to consider them as multiples of simpler fractions. For example, we can think about 53 as 5 lots of 13.

This is not the only way of representing rational numbers; we have also seen that 12=0.5, so we can also represent rational numbers as decimals. It is worth noting that any decimal expansion with a finite number of digits or a repeating expansion is rational. We can represent numbers like this using a line over the repeating digits, so 0.7Μ‡1Μ‡2=0.7121212… and 0.7Μ‡1Μ‡2βˆˆβ„š. Similarly, we can represent fractions as mixed numbers; for example, 53=123, which is also a rational number.

We can represent this information in the following Venn diagram.

In our first example, we will determine if a given number is a rational number.

Example 1: Determining the Rationality of a Number

Is 1256 a rational number?

Answer

We begin by recalling that the set of rational numbers, written β„š, is the set of all quotients of integers. Therefore, β„š=ο«π‘Žπ‘βˆΆπ‘Ž,π‘βˆˆβ„€π‘β‰ 0and.

Thus, to determine if 1256 is rational, we need to check if we can write this number in the form π‘Žπ‘ for integers π‘Ž and 𝑏 with 𝑏≠0. We can do this by recalling that 12 can be written as 12Γ—66=726. Hence, 1256=726+56=776.

Thus, 1256 can be written as a quotient of integers and so the answer is yes, it is a rational number.

In our next example, we will consider if all rational numbers are integers.

Example 2: Comparing Integers and Rational Numbers

Is every rational number an integer?

Answer

We begin by recalling that the set of rational numbers, written β„š, is the set of all quotients of integers. Therefore, β„š=ο«π‘Žπ‘βˆΆπ‘Ž,π‘βˆˆβ„€π‘β‰ 0and.

Therefore, numbers such as 12 are rational numbers. We know that 12 is between 0 and 1 and not equal to either integer, so 12βˆ‰β„€. Hence, 12 is a rational number that is not an integer, and so the answer is no, not all rational numbers are integers.

In our next example, we will check if a given integer is a rational number.

Example 3: Determining If a Number Belongs or Does Not Belong in the Set of Rational Numbers

Which of the following is true?

  1. 1βˆˆβ„š
  2. 1βˆ‰β„š

Answer

We begin by recalling that the set of rational numbers β„š is the set of all quotients of integers. In particular, we can note that 1=11, so it is a quotient of integer values.

It is worth noting that any π‘βˆˆβ„€ is a rational number, since 𝑐=𝑐1. By setting 𝑐=1, we have that 1βˆˆβ„š.

Hence, 1βˆˆβ„š, which is option A.

Thus far, we have focused on the formal definition of a rational number. However, there is a lot of use in considering the visualization of rational numbers. In particular, we can represent any rational number on a number line by using distances.

For example, consider the following number line with markers representing the integers.

Let’s say we wanted to find the point on the number line representing βˆ’53. We could start by noting that this number is negative, so it lies on the negative side of the number line, and that 3 goes into 5 once with a remainder of 2, so βˆ’53 must lie between βˆ’1 and βˆ’2. However, we want the exact point that represents this number, and to do this, we note that the denominator is 3 and so we will need to split the number line into increments of 13. In particular, βˆ’53 will be 5 increments of 13 on the negative side, as shown.

We note that βˆ’2=βˆ’63 and that βˆ’1=βˆ’33 and that each increment of 13 will increase or decrease this value by 13. This allows us to see that the marked points between βˆ’1 and βˆ’2 will represent βˆ’43 and βˆ’53 respectively.

Using this same method, we can represent any rational number on a number line; we just need to determine the integers it lies between and then split the line into equal increments based on the denominator of the fraction.

This also allows us to note that the sign of π‘Žπ‘ is determined by the signs of π‘Ž and 𝑏. In general, if π‘Ž and 𝑏 have the same sign, then π‘Žπ‘ is positive, if π‘Ž and 𝑏 have different signs, then π‘Žπ‘ is negative, and if π‘Ž=0, then π‘Žπ‘=0. Also, when π‘Žπ‘ is negative, the negative sign is written at the front of the fraction, as it does not matter if this factor of βˆ’1 is in the numerator or denominator.

Let’s now see an example of determining the point that represents a given rational number.

Example 4: Identifying the Positions of Rational Numbers on a Number Line

Which of the numbers 𝑙,π‘š,𝑛, and π‘œ is 410?

Answer

To determine the position of 410 on the number line, we note that since the denominator of the fraction is 10, we want to divide the number line into increments of 110. If we look at the number line given, we can see that there are 5 increments for each integer pair. So, we will need to split each of these in half to make increments of 110. We also note that 410 is positive, so we want the point 4 increments of 110 in the positive direction, as shown.

We can then see that this is marked 𝑛 on the number line.

In our next example, we will use the idea of finding points representing rational numbers on a number line to determine the rational number that lies between two given rational numbers.

Example 5: Solving Problems Involving Rational Numbers

Find the rational number lying halfway between βˆ’27 and 435.

Answer

Since we are asked to find the rational number lying halfway between βˆ’27 and 435, we should start by representing these numbers on a number line. To do this, we note that βˆ’27 will be 2 increments of 17 in the negative direction and that 435 will be 4 increments of 135 in the positive direction. We can also note that 35=7Γ—5, so we can find the increments of 135 by splitting the increments of 17 into 5 equal sections. This gives us the following.

We want the number that lies halfway between βˆ’27 and 435. This means it must lie an equal number of increments from both of these numbers. Since there are 14 increments between the numbers, we want the number that is 7 increments either side of these end points. We can see on the number line that this is the following point.

Since this point is 3 increments of 135 to the left of 0, we can write this number as βˆ’335.

In our final example, we will determine which of four given expressions is rational given the values of the variables.

Example 6: Identifying the Rational Expression from a List of Given Expressions

Which of the following expressions is rational given π‘Ž=1 and 𝑏=34?

  1. 39π‘π‘βˆ’34
  2. βˆ’39π‘Žβˆ’1
  3. π‘π‘Ž
  4. 39π‘π‘Žβˆ’1

Answer

We begin by recalling that the set of rational numbers, written β„š, is the set of all quotients of integers. Therefore, β„š=ο«π‘Žπ‘βˆΆπ‘Ž,π‘βˆˆβ„€π‘β‰ 0and.

We can determine which of these expressions is rational by substituting the values of π‘Ž and 𝑏 into each expression separately.

In expression A, we get 39π‘π‘βˆ’34=39(34)34βˆ’34.

Evaluating the denominator, we get 39(34)34βˆ’34=39(34)0.

Since we cannot divide by 0, this is not a rational number.

We have a similar story in expressions B and D. For expression B, we have βˆ’39π‘Žβˆ’1=βˆ’391βˆ’1=βˆ’390, and for expression D, we have 39π‘π‘Žβˆ’1=39(34)1βˆ’1=39(34)0.

Thus, neither of these represents rational numbers.

Finally, in expression C, we have π‘π‘Ž=341, which is the quotient of two integer values where the denominator is nonzero.

Hence, only option C is a rational number.

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

Key Points

  • The set of rational numbers, written β„š, is the set of all quotients of integers.
  • Every integer is a rational number and β„•βŠ‚β„€βŠ‚β„š, but not all rational numbers are integers.
  • We can represent the rational number π‘Žπ‘ on a number line by splitting the number line into increments of 1𝑏 and choosing the point that is π‘Ž increments from 0, where the sign of π‘Ž tells us the direction.
  • The sign of π‘Žπ‘ is determined by the signs of π‘Ž and 𝑏. In general, if π‘Ž and 𝑏 have the same sign, then π‘Žπ‘ is positive, if π‘Ž and 𝑏 have different signs, then π‘Žπ‘ is negative, and if π‘Ž=0, then π‘Žπ‘=0.

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