Properties of Rational Numbers with examples: There is a need to extend the number system to find answers to problems like \(\frac{1}{3}-\frac{1}{2}\) or \(-5 \div 3\). Just as we extend the whole system to the left of \(0\) to get negative integers, we now extend the number system to include all negative fractions. In this number system, corresponding to every positive fraction to the right of zero, there is a corresponding negative fraction to the left of zero.

Thus, \(-\frac{1}{3}\) corresponds to \(+\frac{1}{3} ;-\frac{5}{9}\) corresponds to \(+\frac{5}{9}\), and so on. We call this new number system the rational number system. This article helps to learn the properties of addition, subtraction, multiplication, and division of rational numbers. The article will also discuss additive properties of rational numbers, properties of rational numbers class 8, etc.

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Definition of Rational Numbers

A rational number is the one that can be written in the form \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0.\)

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Denominator and numerator of rational numbers: In the rational number \(\frac{p}{q}\), the integer \(p\) is the numerator, and the integer \(q\) (should not be equal to zero) is the denominator. So, the numerator in \(\frac{-5}{7}\) is \(-5\) and the denominator is \(7\).

Properties of Addition of Rational Numbers

Note: When we say that the number \(a\) is a rational number, it means that it can be expressed in the form \(\frac{p}{q}\) where \(q \neq 0\).

Closure Property

The sum of two rational numbers is also a rational number, i.e., if \(a\) and \(b\) are rational numbers, \(a+b\) is also a rational number.

Examples:

1. \(\frac{1}{2}+\frac{3}{4}=\frac{2+3}{4}=\frac{5}{4}\)
2. \(\frac{-3}{5}+\frac{-2}{3}=\frac{-9-10}{15}=\frac{-19}{15}\)

Commutative Property:

The sum of two rational numbers does not depend on the order in which they are added, i.e., if \(a\) and \(b\) are rational numbers, \(a+b=b+a\).

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Examples:

1. \(\quad \frac{1}{2}+\frac{1}{3}=\frac{1}{3}+\frac{1}{2}\)
\(\Rightarrow \frac{5}{6}=\frac{5}{6}\)

2. \(\quad \frac{-4}{5}+\frac{2}{3}=\frac{2}{3}+\frac{-4}{5}\)
\(\Rightarrow \frac{-2}{15}=\frac{-2}{15}\)

Associative Property

The sum of three or more rational numbers does not depend on the way the rational numbers are grouped, i.e., if \(a, b, c\) are three rational numbers, then:
\(a+(b+c)=(a+b)+c\)

Example: \(\frac{2}{3}+\left(\frac{-3}{5}+\frac{-3}{4}\right)=\left(\frac{2}{3}+\frac{-3}{5}\right)+\frac{-3}{4}\)
\(\Rightarrow \frac{2}{3}+\left(\frac{-12-15}{20}\right)=\left(\frac{10-9}{15}\right)+\frac{-3}{4}\)
\(\Rightarrow \frac{2}{3}+\left(\frac{-27}{20}\right)=\left(\frac{1}{15}\right)+\frac{-3}{4}\)
\(\Rightarrow \frac{40-81}{60}=\frac{4-45}{60}\)
\(\Rightarrow \frac{-41}{60}=\frac{-41}{60}\)
L.H.S \(=\) R.H.S

Identity Property (Property of 0)

\(0\) added to any rational number leaves the rational number unchanged, i.e., if \(a\) is a rational number, then \(a+0=0+a=a\). \(0\) is called the identity element for the addition of rational numbers.

Examples:

1. \(\frac{5}{6}+0=0+\frac{5}{6}=\frac{5}{6}\)
2. \(\frac{1}{6}+0=0+\frac{1}{6}=\frac{1}{6}\)

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Additive Inverse (Negative)

If the sum of two rational numbers is \(0\), then the two numbers are called additive inverses on negatives, i.e., if \(a\) and \(b\) are rational numbers such that \(a+b=0\), then \(a\) and \(b\) are the additive inverse of each other.

With this we can conclude that for any rational number \(a\),
\(a+(-a)=0 \Longrightarrow-a\) is the additive inverse of \(a\).

Examples:
1. Additive inverse of \(\frac{2}{5}\) is \(\frac{-2}{5}\).
2. Negative of \(\frac{-3}{8}\) is \(\frac{3}{8}\).

Properties of Subtraction of Rational Numbers

Closure Property

The difference between the two rational numbers is also a rational number, i.e., if \(a\) and \(b\) are rational numbers, \(a-b\) is also a rational number.

Examples:

1. \(\frac{1}{2}-\frac{3}{4}=\frac{2-3}{4}=\frac{-1}{4}\)
2. \(\frac{-3}{5}-\frac{-2}{3}=\frac{-9+10}{15}=\frac{1}{15}\)

Commutative Property

The commutative property does not hold good for the subtraction of rational numbers.
Thus, if \(a\) and \(b\) are rational numbers, \(a-b \neq b-a\) (except when \(a=b\))

Example: \(\frac{1}{2}-\frac{1}{3} \neq \frac{1}{3}-\frac{1}{2}\)
\(\Rightarrow \frac{3-2}{6} \neq \frac{2-3}{6}\)
\(\Rightarrow \frac{1}{6} \neq \frac{-1}{6}\)
L.H.S \(\neq\) R.H.S

Associative Property

The associative property does not hold good for the subtraction of rational numbers.
Thus, if \(a, b\) and \(c\) are rational numbers, \(a-(b-c) \neq(a-b)-c\)

Example: \(\frac{2}{5}-\left(\frac{1}{5}-\frac{3}{5}\right)=\left(\frac{2}{5}-\frac{1}{5}\right)-\frac{3}{5}\)
\(\Rightarrow \frac{2}{5}-\left(\frac{1-3}{5}\right)=\left(\frac{2-1}{5}\right)-\frac{3}{5}\)
\(\Rightarrow \frac{2}{5}-\left(\frac{-2}{5}\right)=\left(\frac{1}{5}\right)-\frac{3}{5}\)
\(\Rightarrow \frac{4}{5}=\frac{1-3}{5}\)
\(\Rightarrow \frac{4}{5} \neq \frac{-2}{5}\)
L.H.S \(\neq\) R.H.S

Property of Zero

\(0\) subtracted from any rational number leaves it unchanged, i.e., if \(a\) is \(a\) rational number,
\(a-0=a\)

However, any rational number subtracted from \(0\) gives its additive inverse, i.e. if \(a\) is a rational number \(0-a=-a\)

Examples:

1. \(\frac{4}{5}-0=\frac{4}{5}\)
2. \(0-\frac{4}{5}=-\frac{4}{5}\)

Properties of Multiplication of Rational Numbers

Closure Property

The product of two rational numbers is also a rational number, i.e., if \(a\) and \(b\) are rational numbers, \(a \times b\) is also a rational number.

Examples:

1. \(\frac{1}{2} \times \frac{3}{4}=\frac{3}{8}\)
2. \(\frac{-3}{5} \times \frac{-2}{3}=\frac{6}{15}=\frac{2}{5}\)

Commutative Property

The product of two rational numbers does not depend on the order in which they are added,
i.e., if \(a\) and \(b\) are rational numbers, \(a \times b=b \times a\).

Examples:
1. \(\frac{1}{2} \times \frac{1}{3}=\frac{1}{3} \times \frac{1}{2}\)
\(\Rightarrow \frac{1}{6}=\frac{1}{6}\)

2. \(\quad \frac{-4}{5} \times \frac{2}{3}=\frac{2}{3} \times \frac{-4}{5}\)
\(\Rightarrow \frac{-8}{15}=\frac{-8}{15}\)

Associative Property

The product of three or more rational numbers does not depend on how the rational numbers are grouped, i.e., if \(a, b, c\) are three rational numbers, then:
\(a \times(b \times c)=(a \times b) \times c\)

Example: \(\frac{2}{3} \times\left(\frac{3}{5} \times \frac{3}{4}\right)=\left(\frac{2}{3} \times \frac{3}{5}\right) \times \frac{3}{4}\)
\(\Rightarrow \frac{2}{3} \times\left(\frac{9}{20}\right)=\left(\frac{6}{15}\right) \times \frac{3}{4}\)
\(\Rightarrow \frac{18}{60}=\frac{18}{60}\)
L.H.S \(=\) R.H.S

Identity Property

The product of any rational number with \(1\) is the rational number itself, i.e., if \(a\) is a rational number, \(a \times 1=1 \times a=a\).
Rational number \(1\) is the multiplicative identity for rational numbers.

Example: \(\frac{1}{2} \times 1=1 \times \frac{1}{2}\)
\(\Rightarrow \frac{1}{2}=\frac{1}{2}\)

Zero Property

The product of any rational number and \(0\) is \(0\), i.e., if a is a rational number,
\(a \times 0=0 \times a=0\)

Example: \(\frac{1}{2} \times 0=0 \times \frac{1}{2}\)
\(\Rightarrow 0=0\)

Distributive Property of Multiplication over Addition and Subtraction

If \(a, b\) and \(c\) are rational numbers, then.
\(a \times(b+c)=a \times b+a \times c\)
And \(a \times(b-c)=a \times b-a \times c\)

Example: \(\frac{-2}{3} \times\left(\frac{-3}{4}+\frac{5}{6}\right)=\frac{-2}{3} \times \frac{-3}{4}+\frac{-2}{3} \times \frac{5}{6}\)
\(\Rightarrow \frac{-2}{3} \times\left(\frac{1}{12}\right)=\frac{6}{12}+\frac{-10}{18}\)
\(\Rightarrow \frac{-2}{3} \times\left(\frac{1}{12}\right)=\frac{18-20}{36}\)
\(\Rightarrow \frac{-1}{18}=\frac{-2}{36}\)
\(\Rightarrow \frac{-1}{18}=\frac{-1}{18}\)
L.H.S \(=\) R.H.S

Properties of Division of Rational Numbers

Closure Property

If \(a\) and \(b\) are rational numbers and \(b \neq 0\), then \(a \div b\) is also a rational number, i.e., the division is closed in rational numbers.

Examples:
1. \(\frac{1}{2} \div \frac{3}{4}=\frac{1}{2} \times \frac{4}{3}=\frac{-1}{2}\)
2. \(\frac{-3}{5} \div \frac{-2}{3}=\frac{-3}{5} \times \frac{-3}{2}=\frac{9}{10}\)

Commutative Property

The commutative property does not hold good for the division of rational numbers.
Thus, if \(a\) and \(b\) are rational numbers, \(a \div b \neq b \div a\) (except when \(a=b\))

Example: \(\frac{1}{2} \div \frac{1}{3} \neq \frac{1}{3} \div \frac{1}{2}\)
\(\Rightarrow \frac{1}{2} \times \frac{3}{1} \neq \frac{1}{3} \times \frac{2}{1}\)
\(\Rightarrow \frac{3}{2} \neq \frac{2}{3}\)
L.H.S \(\neq\) R.H.S

Associative Property

The associative property does not hold good for the division of rational numbers.
Thus, if \(a, b\) and \(c\) are rational numbers, \((a \div b) \div c \neq a \div(b \div c)\)

Property of \(1\) and \(-1\): For any rational number a, \(a \div 1=a\) and \(a \div(-1)=-a\).

Examples:
1. \(\frac{1}{2} \div 1=\frac{1}{2} \times \frac{1}{1}=\frac{1}{2}\)
2. \(\frac{-1}{2} \div(-1)=\frac{-1}{2} \times \frac{1}{-1}=\frac{1}{2}\)

Division by Itself

For any non-zero rational number \(a, a \div a=1\) and \(a \div(-a)=-1\).

Examples:
1. \(\quad \frac{5}{7} \div\left(\frac{5}{7}\right)=\frac{5}{7} \times\left(\frac{7}{5}\right)=\frac{35}{35}=1\)
2. \(\frac{3}{2} \div\left(\frac{-3}{2}\right)=\frac{3}{2} \times\left(\frac{-2}{3}\right)=\frac{-6}{6}=-1\)

Properties of Rational Numbers Chart

Operations/Properties	Closure Property	Commutative Property	Associative Property	Distributive Property
Addition	yes	yes	yes	Yes
Subtraction	yes	No	No	Yes
Multiplication	yes	yes	yes	No
Division	yes	No	No	No

Note: Yes \( \to \) Satisfies the condition
No \( \to \) Does not satisfy the condition

Solved Examples – Properties of Rational Numbers

Q.1. Identify the property in the following statement:
\(\frac{1}{2}+\left(\frac{2}{3}+\frac{1}{5}\right)=\left(\frac{1}{2}+\frac{2}{3}\right)+\frac{1}{5}\).
Ans: Given the statement \(\frac{1}{2}+\left(\frac{2}{3}+\frac{1}{5}\right)=\left(\frac{1}{2}+\frac{2}{3}\right)+\frac{1}{5}\). In addition, it is an example of the associative property of rational numbers.

Q.2. Verify the property \(a \times b=b \times a\) by taking:
\(a=2, b=\frac{-3}{5}\)
Ans: From the given, the condition is \(a \times b=b \times a\) and values \(a=2, b=\frac{-3}{5}\).
\(\Rightarrow a \times b=b \times a\)
\(\Rightarrow 2 \times \frac{-3}{5}=\frac{-3}{5} \times 2\)
\(\Rightarrow \frac{-6}{5}=\frac{-6}{5}\)
L.H.S \(=\) R.H.S
Hence, it is verified.

Q.3. Divide the following rational numbers: \(\frac{2}{7} \div \frac{2}{7}\)
Ans: When we divide two given rational numbers, the first rational number remains the same and multiply it with the reciprocal of the second rational number.
\(\Rightarrow \frac{2}{7} \div\left(\frac{2}{7}\right)=\frac{2}{7} \times\left(\frac{7}{2}\right)=\frac{14}{14}=1\)
Hence, \(\frac{2}{7} \div \frac{2}{7}=1\).

Q.4. Solve \(\frac{1}{3}-\frac{5}{6}\)
Ans: We have \(\frac{1}{3}-\frac{5}{6}\)
\(=\frac{2-1}{6}\)
\(=\frac{1}{6}\)
Hence, the obtained value is \(\frac{1}{6}\).

Q.5. Find the additive inverse of \(\frac{1}{9}\).
Ans: If the sum of two rational numbers is \(0\), then the two numbers are called additive inverses or negatives.
So, the additive inverse of \(\frac{1}{9}\) is \(\frac{-1}{9}\).

Q.6. Verify the property \(a \times(b-c)=a \times b-a \times c\) by taking:
\(a=\frac{1}{2}, b=\frac{-3}{4}, c=\frac{-2}{3}\).
Ans: Substitute the values \(a=\frac{1}{2}, b=\frac{-3}{4}, c=\frac{-2}{3}\) in \(a \times(b-c)=a \times b-a \times c\).
\(\Rightarrow a \times(b-c)=a \times b-a \times c\)
\(\Rightarrow \frac{1}{2} \times\left(\frac{-3}{4}-\frac{-2}{3}\right)=\frac{1}{2} \times \frac{-3}{4}-\frac{1}{2} \times \frac{-2}{3}\)
\(\Rightarrow \frac{1}{2} \times\left(\frac{-9+8}{12}\right)=\frac{-3}{8}+\frac{2}{6}\)
\(\Rightarrow \frac{1}{2} \times\left(\frac{-1}{12}\right)=\frac{-3}{8}+\frac{2}{6}\)
\(\Rightarrow \frac{-1}{24}=\frac{-9+8}{24}\)
\(\Rightarrow \frac{-1}{24}=\frac{-1}{24}\)
L.H.S \(=\) R.H.S
Hence, it is verified.

Summary

A rational number can be represented as \(\frac{a}{b}\) with \(a\) and \(b\) being integers and \(b \neq 0\). The basic properties of rational numbers are closure property, commutative property, associative property, distributive property, identity property, inverse properties, and examples in the article.

It helps for a good understanding of the properties of rational numbers. The outcome of this article helps in apply the properties of rational numbers while solving the different problems.

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Frequently Asked Questions (FAQs) – Properties of Rational Numbers

Q.1: What are the properties of rational numbers?
Ans: Basic properties of rational numbers are:
1. Closure Property of rational numbers.
2. Commutative property of rational numbers.
3. Associative property of rational numbers.
4. Distributive property of rational numbers.
5. Identity Property of rational numbers.
6. Inverse Property of rational numbers.

Q.2: What are the four types of rational numbers?
Ans: \(4\) types of rational numbers are,
1. Natural Numbers
2. Whole Numbers
3. Integers
4. Fractions

Q.3: What is the commutative property of a rational number?
Ans: The commutative property of rational numbers states that adding or multiplying any two rational numbers in any order produces the same result. However, if the order of the numbers is changed in subtraction and division, the result will vary.

Q.4: What is the distributive property of rational numbers?
Ans: Any equation containing three rational numbers \(A, B\), and \(C\), given in the form \(A \times(B+C)\), is resolved as \(A \times(B+C)=A \times B+A \times C\). or \(A \times(B-C)=A \times B-A \times C\),
according to the distributive property of rational numbers. This signifies that operand \(A\) is shared by the other two operands, \(B\) and \(C\).

Q.5: What is the closure property of a rational number?
Ans: The closure property states that \(x-y\) is also a rational number for any two rational integers \(x\) and \(y\). As a result, an outcome is a rational number. As a result, while subtracting rational numbers, the rational numbers are closed.

Q.6: What is the closure property of the addition formula?
Ans: The sum of two rational numbers is also a rational number, i.e., if \(a\) and \(b\) are rational numbers, \(a+b\) is also a rational number.
Examples: \(\frac{1}{4}+\frac{3}{4}=\frac{1+3}{4}=\frac{4}{4}=1\)

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