Rational Numbers – Definition & Properties

The word ‘rational’ originated from the word ‘ratio’. And the numbers that are represented in terms of ratio $\frac{p}{q}$, where $p$ and $q$ both are integers and $q \ne 0$ are called rational numbers.

In fact, the numbers in the sets of natural numbers, whole numbers, and integers are also rational numbers, as the numbers like $5$ or $-8$ can also be expressed in the form $\frac {5}{1}$ or $-\frac {8}{1}$. Even the number $0 = \frac {0}{1}$ is a rational number.

What are Rational Numbers?

A rational number is a number that is of the form $\frac {p}{q}$ where $p$ and $q$ are integers and $q \ne 0$. The set of rational numbers is denoted by $Q$. 

A set of rational numbers is a superset of sets of natural numbers, whole numbers, and integers i.e, all natural numbers, whole numbers, and integers are rational numbers.

Rational Numbers Examples

If any number that can be expressed in the form of $\frac {p}{q}$ where $p$ and $q$ are integers and $q \ne 0$, then that number is called a rational number. Some of the examples of rational numbers are

• $23 \left( \text{can be written as } \frac {23}{1}\right)$. It’s also a natural number and a whole number.
• $-15 \left( \text{can be written as } -\frac {15}{1}\right)$. It’s also an integer.
• $\frac {3}{4}$. It’s a proper fraction.
• $-2\frac {5}{7}$. It’s an improper fraction.
• $2.96$. It’s a decimal number.

Is 0 a Rational Number?

As discussed above any number that can be expressed in the form $\frac {p}{q}$, where $p$ and $q$ are integers and $q \ne 0$ is called a rational number.

The number $0$ can be written as $\frac {0}{1}$, where $0$ and $1$ are both integers and obviously, $1 \ne 0$, therefore, $0$ is a rational number.

Note: $\frac {1}{0}$ is not a rational number. In fact, it’s undefined.

Properties of Rational Numbers

You can perform any of the following four basic operations on rational numbers. 

• Addition
• Subtraction
• Multiplication 
• Division

Each of these operations shows one or more of the following properties:

• Closure Property
• Commutative Property
• Associative Property
• Distributive Property
• Additive Inverse Property
• Multiplicative Inverse Property
• Identity Property

Let’s understand these properties of rational numbers in detail.

Closure Property of Rational Numbers

The closure property states that if any two numbers from a set are operated by an arithmetic operation then their result also lies in the same set.

The closure property is exhibited by the operations of addition, subtraction, multiplication, and division in the set of rational numbers.

Closure Property of Addition of Rational Numbers

It states that when two rational numbers are added, then their sum is also a rational number.

Mathematically, it is represented as if $a, b \in Q, \text {then  } a + b \in Q$.

For example, $12$ and $14$ are rational numbers, then their sum $12 + 14 = 26$ is also a rational number.

Similarly, $\frac {1}{6}$ and $5$ are rational numbers, then their sum $\frac {1}{6} + 5 = \frac {31}{6}$ is also a rational number.

Also, $2.8$ and $6.78$ are rational numbers, then their sum $2.8 + 6.78 = 9.58$ is also a rational number.

Closure Property of Subtraction of Rational Numbers

It states that when two rational numbers are subtracted, then their difference is also a rational number.

Mathematically, it is represented as if $a, b \in Q, \text {then  } a – b \in Q$.

For example, $12$ and $17$ are rational numbers, then their difference $12 – 17 = -5$ as well as $17 – 12 = 5$ are also  rational numbers.

Similarly, $\frac {5}{9}$ and $\frac {2}{3}$ are rational numbers, then their difference $\frac {5}{9} – \frac {2}{3} = -\frac {1}{9}$ as well as $\frac {2}{3} – \frac {5}{9} = \frac {1}{9}$ are also rational numbers.

Also, $4.96$ and $2.35$ are rational numbers, then their difference $4.96 – 2.35 = 2.61$ as well as $2.35 – 4.96 = -2.61$ are also rational numbers.

Closure Property of Multiplication of Rational Numbers

It states that when two rational numbers are multiplied, then their product is also a rational number.

Mathematically, it is represented as if $a, b \in Q, \text {then  } a \times b \in Q$.

For example, $7$ and $12$ are rational numbers, then their product $7 \times 12 = 84$ is also a rational number.

Similarly, $-14$ and $16$ are rational numbers, then their product $-14 \times 16 = -224$ is also a rational number.

Also, $\frac {2}{3}$ and $\frac {4}{5}$ are rational numbers, then their product $\frac {2}{3} \times \frac {4}{5} = \frac {8}{15}$ is also a rational number.

Closure Property of Division of Rational Numbers

It states that when two rational numbers are divided, then their quotient is also a rational number.

Mathematically, it is represented as if $a, b \in Q, \text {then  } a \div b \in Q$.

For example, $7$ and $12$ are rational numbers, then their quotient $7 \div 12 = \frac {7}{12}$ or $0.58333…$ is also a rational number.

Similarly, $-14$ and $16$ are rational numbers, then their quotient $-14 \div 16 = -\frac {7}{8}$  or $-0.875$ is also a rational number.

Commutative Property of Rational Numbers

The commutative property deals with the ordering of numbers in an operation. It states that the result remains the same even if the order of numbers in the operation is changed or swapped.

The commutative property is exhibited by the operations of addition and multiplication in the set of rational numbers. The operations subtraction and division do not show exhibit commutative property in the set of rational numbers.

Commutative Property of Addition of Rational Numbers

It states that for any two rational numbers their sum remains the same even if the positions of the numbers are interchanged or swapped.

Mathematically, it is represented as if $a, b \in Q, \text {then  } a + b = b + a$.

For example, $4$ and $11$ are two rational numbers. $4 + 11 = 8$ and also $11 + 4 = 15$.

Similarly, for two rational numbers, $3.87$ and $12.92$, $3.87 + 12.92 = 16.79$ and $12.92 + 3.87 = 16.79$.

Or, for two rational numbers $\frac {2}{5}$ and $\frac {1}{7}$, $\frac {2}{5} + \frac {1}{7} = \frac {19}{35}$.

Commutative Property of Multiplication of Rational Numbers

It states that for any two rational numbers their product remains the same even if the positions of the numbers are interchanged or swapped.

Mathematically, it is represented as if $a, b \in Q, \text {then  } a \times b = b \times a$.

For example, $17$ and $14$ are two rational numbers. $17 \times 14 = 238$ and also $14 \times 17 = 238$.

Similarly, for two rational numbers, $\frac {5}{9}$ and $\frac {7}{11}$, $\frac {5}{9} \times \frac {7}{11} = \frac {7}{11} \times \frac {5}{9} = \frac {35}{99}$.

Associative Property of Rational Numbers

The associative property deals with the grouping of numbers in an operation. It states that the result remains the same even if the grouping of numbers is changed while performing the operation.

The associative property is exhibited by the operations addition and multiplication in the set of rational numbers. The operations subtraction and division do not show exhibit associative property in the set of rational numbers.

Associative Property of Addition of Rational Numbers

It states that the sum of any three rational numbers remains the same even if the grouping of the numbers is changed. 

Mathematically, it is represented as if $a, b, c \in Q, \text {then  } \left (a + b \right) + c = a + \left(b + c \right)$. 

For example, for three rational numbers $7$, $9$ and $\frac {5}{6}$, $\left( 7 + 9 \right) + \frac {5}{6} = 16 + \frac {5}{6} = \frac {101}{6}$ and $7 + \left(9 + \frac {5}{6} \right) = 7 + \frac {59}{6} = \frac {101}{6}$.

Similarly, for three rational numbers $-10$, $\frac {2}{7}$ and $12$, $\left(-10 + \frac {2}{7} \right) + 12 = -\frac {68}{7} + 12 = \frac {16}{7}$ and $-10 + \left(\frac {2}{7} + 12 \right) = -10 + \frac {86}{7} = \frac {16}{7}$.

Associative Property of Multiplication of Rational Numbers

It states that the product of any three rational numbers remains the same even if the grouping of the numbers is changed. 

Mathematically, it is represented as if $a, b, c \in Q, \text {then  } \left (a \times b \right) \times c = a \times \left(b \times c \right)$. 

For example, for three rational numbers $\frac {1}{2}$, $5$ and $12$, $\left( \frac {1}{2} \times 5 \right) \times 12 = \frac {5}{2} \times 12 = 30$ and $\frac {1}{2} \times \left(5 \times 12 \right) = \frac {1}{2} \times 60 = 30$.

Similarly, for three rational numbers $\frac {4}{5}$, $\frac {2}{3}$ and $-7$, $\left(\frac {4}{5} \times \frac {2}{3} \right) \times \left(-7 \right) = \frac {8}{15} \times \left(-7 \right) = -\frac {56}{15}$ and $\frac {4}{5} \times \left(\frac {2}{3} \times \left(-7 \right) \right) = \frac {4}{5} \times \left(-\frac {14}{3} \right) = -\frac {56}{15}$.

Distributive Property of Rational Numbers

The distributive property of rational numbers deals with the splitting of the distribution of rational numbers through addition and subtraction while performing the multiplication operation.

There are two forms of distributive property of rational numbers.

• Distributive property of multiplication over addition
• Distributive property of multiplication over subtraction

Distributive Property of Multiplication Over Addition of Rational Numbers

It states that for any three rational numbers the expression of the form $\left(a + b \right) \times c$ can be solved as $a \times b + a \times c$.

For example, $\left(\frac {1}{2} + \frac {2}{3} \right) \times 6$ can be solved as $\frac {1}{2} \times 6 + \frac {2}{3} \times 6 = 3 + 4 = 7$.

This also $\left(\frac {1}{2} + \frac {2}{3} \right) \times 6$ on solving gives $\frac {7}{6} \times 6 = 7$. 

Similarly, $\left(15 + \frac {2}{3} \right) \times \frac {4}{5}$ can be solved as $15 \times \frac {4}{5} + \frac {2}{3} \times \frac {4}{5} = 12 + \frac {8}{15} = \frac {188}{15}$.

This also $\left(15 + \frac {2}{3} \right) \times \frac {4}{5} = \frac {47}{3} \times \frac {4}{5} = \frac {188}{15}$.

Distributive Property of Multiplication Over Subtraction of Rational Numbers

It states that for any three rational numbers the expression of the form $\left(a – b \right) \times c$ can be solved as $a \times b – a \times c$.

For example, $\left(14 – 6 \right) \times \frac {3}{8}$ can be solved as $14 \times \frac {3}{8} – 6 \times \frac {3}{8} = \frac {21}{4} – \frac {9}{4} = \frac {12}{4} = 3$.

This also $\left(14 – 6 \right) \times \frac {3}{8}$ on solving gives $8 \times \frac {3}{8} = 3$. 

Note: The distributive property does not hold for division in the case of rational numbers.

Additive Identity Property of Rational Numbers

The additive identity property of rational numbers is also known as the identity property of rational numbers of addition, which states that adding $0$ to any rational number, results in the number itself. This is due to the fact that when we add $0$ to any rational number, it does not change the number and keeps its identity.

Mathematically, it is expressed as for any rational number, $a$ there exists a rational number $0$ such that $a + 0 = 0 + a = a$.

For example $19 + 0 = 0 + 19 = 19$, or $-\frac {3}{7} + 0 = 0 + \left(-\frac {3}{7} \right) = -\frac {3}{7}$.

Note: $0$ is called the additive identity of rational numbers.

Multiplicative Identity Property of Rational Numbers

The multiplicative identity property of rational numbers is also known as the identity property of rational numbers of multiplication, which states that multiplying $1$ to any rational number, results in the number itself. This is due to the fact that when we multiply $1$ to any rational number, it does not change the number and keeps its identity.

Mathematically, it is expressed as for any rational number, $a$ there exists a rational number $1$ such that $a \times 1 = 1 \times a = a$.

For example $16 \times 1 = 1 \times 16 = 16$, or $\frac {5}{7} \times 1 = 1 \times \frac {5}{7}  = \frac {5}{7}$.

Note: $1$ is called the multiplicative identity of rational numbers.

Additive Inverse Property of Rational Numbers

The additive inverse of a rational number is its opposite number. If a rational number is added to its additive inverse, the sum of both the numbers becomes zero ($0$). 

Mathematically, it is expressed as for every rational number $a$, there exists a rational number $-a$, such that $a + \left(-a \right) = -a + a = 0$. $-a$ is called the additive inverse of rational number $a$.

The simple rule is to change the positive rational number to a negative rational number and vice versa. 

For example, additive inverse of $17$ is $-17$, since, $17 + \left(-17 \right) = -17 + 17 = 0$.

Or, additive inverse of $\frac {5}{6}$ is $-\frac {5}{6}$, since, $\frac {5}{6} + \left(-\frac {5}{6}\right) = -\frac {5}{6} + \frac {5}{6} = 0$.

Multiplicative Inverse Property of Rational Numbers

The multiplicative inverse of a rational number is its opposite number. If a rational number is multiplied by its multiplicative inverse, the product of both the numbers becomes one ($1). 

Mathematically, it is expressed as for every rational number $a$, there exists a rational number $\frac {1}{a}$, such that $a \times \frac {1}{a} = \frac {1}{a} \times a = 1$. $\frac {1}{a}$ is called the multiplicative inverse of rational number $a$.

The simple rule is to change the rational number to its multiplicative inverse is to take its reciprocal.

For example, multiplicative inverse of $6$ is $\frac {1}{6}$, since, $6 \times \frac {1}{6} = 1$.

Or, multiplicative inverse of $\frac {7}{9}$ is $\frac {9}{7}$, since $\frac {7}{9} \times \frac {9}{7} = 1$.

Conclusion

Rational numbers are the numbers which can be expressed in the form $\frac {p}{q}$, such that $p$ and $q$ are integers and $q \ne 0$. The rational numbers exhibit the properties – closure property, commutative property, associative property, distributive property, additive identity property, multiplicative identity property, additive inverse property, and multiplicative inverse property.

Practice Problems

1. Fill in the blanks
   • Rational numbers are the numbers which can be expressed in the form $\frac {p}{q}$, where $p$ and $q$ are __________.(natural numbers, whole numbers, integers)
   • In a rational number $\frac {p}{q}$, $q$ cannot be ___________ ($0$, $1$, $-1$)
   • Additive identity of rational numbers is ___________ ($0$, $1$, $-1$)
   • Multiplicative identity of rational numbers is ___________ ($0$, $1$, $-1$)
   • Additive inverse of a rational number $a$ is ___________ $\left(-a, \frac {1}{a} \right)$
   • Multiplicative inverse of a rational number $a$ is ____________ $\left(-a, \frac {1}{a} \right)$
2. State True or False
   • The sum of a rational number and its additive inverse is $0$
   • The sum of a rational number and its additive inverse is $1$
   • The sum of a rational number and its additive inverse is $-1$
   • The product of a rational number and its multiplicative inverse is $0$
   • The product of a rational number and its multiplicative inverse is $1$
   • The product of a rational number and its multiplicative inverse is $-1$

Recommended Reading

• Associative Property – Meaning & Examples
• Distributive Property – Meaning & Examples
• Commutative Property – Definition & Examples
• Closure Property(Definition & Examples)
• Additive Identity of Decimal Numbers(Definition & Examples)
• Multiplicative Identity of Decimal Numbers(Definition & Examples)
• Natural Numbers – Definition & Properties
• Whole Numbers – Definition & Properties
• What is an Integer – Definition & Properties

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