Properties of Rational Numbers with FAQs

The major properties of rational numbers are:(1) Closure, (2) Commutativity, (3) Associativity, (4) Distributive, (5) Identity Property and(6) Inverse Property.

Properties of Rational Numbers

Rational numbers are numbers that can be expressed in the fractional form i.e., p/q, where both p and q are integers and
q ≠ 0. The rational numbers include integers, whole numbers and natural numbers. Rational numbers can be also described as terminating decimal numbers, or as non-terminating but repeating decimal numbers.  

Properties of rational numbers are: 

(1) Closure property

(2) Associative property

(3) Commutative property

(4) Distributive property

(5) Identity property

(6) Inverse Property

1. Closure Property of Rational Numbers

This property states that when mathematical operations like addition, subtraction, and multiplication are applied on any two rational numbers then the result is also a rational number. Hence, rational numbers are closed under addition, subtraction and multiplication.

For Two Rational numbers\frac{5}{2} \text { and } \frac{3}{2}

Closure under Addition

\frac{5}{2}+\frac{3}{2}=\frac{8}{2}=4, 4 is a rational number

Hence we say that rational numbers are closed in addition

Therefore, If a and b are rational numbers then (a+b) is also a rational number

closure under subraction

\frac{5}{2}-\frac{3}{2}=\frac{2}{2}=1, 1 is  a rational number

Hence we say that rational numbers are closed in subraction

Therefore, If a and b are rational numbers then (a-b) is also a rational number

Closure under Multiplication

\frac{5}{2} \times \frac{3}{2}=\frac{15}{4}, \frac{15}{4}is a rational number

Hence we say that rational numbers are closed in subraction

Therefore, If a and b are rational numbers then \Rightarrow(\mathbf{a} \times \mathbf{b})is also a rational number

The division is not under closure property because division by zero is undefined.

Hence, we can also say that except ‘0’ all numbers are closed under division.

Closure under division (only for non zero denominator)

\left(\frac{5}{2} \div \frac{3}{2}\right)=\frac{5}{2} \times \frac{2}{3}=\frac{5}{3}, \frac{5}{3}

Hence, we say that ratonal numbers are closed under division.

Therefore, If a and b are rational numbers then \Rightarrow(a \div b) for \mathbf{b} \neq 0

2. Commutative Property 

This property states that when two rational numbers are added or multiplied in any order the outcome of the operation is equal. But in the case of subtraction and division, the outcome values will not remain equal if the order of the numbers is changed.

For Two Rational numbers\frac{5}{2} \text { and } \frac{3}{2}

Commutative under addition

\frac{5}{2}+\frac{3}{2}=\frac{8}{2}=4

Hence we say that rational numbers are commutative under the addition

Therefore, If a and b are rational numbers \Rightarrow(\mathbf{a}+\mathbf{b})=(\mathbf{b}+\mathbf{a})

Noncommutative under subtraction

\frac{5}{2}-\frac{3}{2}=\frac{2}{2}=1

\frac{3}{2}-\frac{5}{2}=\frac{-2}{2}=-1

Hence we say that rational numbers are not commutative under the subraction

Therefore, If a and b are rational numbers\Rightarrow \mathbf{a}-\mathbf{b} \neq \mathbf{b}-\mathbf{a}

commutative under multiplication

\frac{5}{2} \times \frac{3}{2}=\frac{15}{4}

\frac{3}{2} \times \frac{5}{2}=\frac{15}{4}

Hence we say that the rational numbers are commutative under multiplication

Therefore a + b are rational numbers \Rightarrow(\mathbf{a} \times \mathbf{b})=(\mathbf{b} \times \mathbf{a})

Noncommutative under Division

\frac{5}{2} \div \frac{3}{2}=\frac{5}{3}

\frac{3}{2} \div \frac{5}{2}=\frac{3}{5}

Hence we say that rational numbers are not commutative under the division

Therefore, If a and b are rational numbers\Rightarrow \mathbf{a} \div \mathbf{b} \neq \mathbf{b} \div \mathbf{a}

3. Associative Property

This property states that when any three rational numbers are added or multiplied by grouping in any manner the outcome is equal. But in the case of subtraction and division, the outcome value will not be equal when the order of the numbers are reversed or grouped differently.

For three rational numbers  \frac{7}{2}, \frac{5}{2} \text { and } \frac{3}{2}

Associative under addition

Hence we say that rational numbers are associative under addition

Therefore a b and c are rational numbers\Rightarrow(\mathbf{a}+\mathbf{b})+\mathbf{c}=\mathbf{a}+(\mathbf{b}+\mathbf{c})

nonassociative under subractionl

Hence we say that rational numbers are not associative under subraction

Therefore if a b and c are rational numbers\Rightarrow(\mathbf{a}-\mathbf{b})-\mathbf{c} \neq \mathbf{a}-(\mathbf{b}-\mathbf{c})

Associative under multiplication

Hence  Hence we say that rational numbers are associative under multiplication

therfore a b and c are rational numbers \Rightarrow(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}=\mathbf{a} \times(\mathbf{b} \times \mathbf{c})

Nonassociative under Division

\left(\frac{7}{2} \div \frac{5}{2}\right) \div \frac{3}{2}=\frac{7}{5} \div \frac{3}{2}=\frac{14}{15}

Hence we say that rational numbers are not associative under division

therfore a b and c are rational numbers\Rightarrow(\mathbf{a} \div \mathbf{b}) \div \mathbf{c} \neq \mathbf{a} \div(\mathbf{b} \div \mathbf{c})

4. Distributive Property

This property states that

If a b and c are rational numbers then

a \times(b+c)=(a \times b)+(a \times c)

a \times(b-c)=(a \times b)-(a \times c)

For three rational numbers  \frac{7}{2}, \frac{5}{2} \text { and } \frac{3}{2}

\frac{7}{2} \times\left(\frac{5}{2}+\frac{3}{2}\right)=\frac{7}{2} \times \frac{8}{2}=\frac{7}{2} \times 4=14

\left(\frac{7}{2} \times \frac{5}{2}\right)+\left(\frac{7}{2} \times \frac{3}{2}\right)=\frac{35}{4}+\frac{21}{4}=\frac{56}{4}=14

\therefore \frac{7}{2} \times\left(\frac{5}{2}+\frac{3}{2}\right)=\left(\frac{7}{2} \times \frac{5}{2}\right)+\left(\frac{7}{2} \times \frac{3}{2}\right)

similiarly 

\frac{7}{2} \times\left(\frac{5}{2}-\frac{3}{2}\right)=\frac{7}{2} \times \frac{2}{2}=\frac{7}{2}

\left(\frac{7}{2} \times \frac{5}{2}\right)-\left(\frac{7}{2} \times \frac{3}{2}\right)=\frac{35}{4}-\frac{21}{4}=\frac{14}{4}=\frac{7}{2}

5. Identity Property

For any rational number, the additive identity is ‘0’ and the multiplicative identity is ‘1’.

For the rational number  \frac{3}{4}

\frac{3}{4}+0=\frac{3}{4} \Rightarrow 0is the additive identity

\frac{3}{4} \times 1=\frac{3}{4} \Rightarrow 1is the multiplicative identity

6. Inverse Property

For any rational number, the additive inverse is the negative of that number and the multiplicative inverse is its reciprocal value.

For the rational number \frac{3}{4}

\frac{3}{4}+\left(-\frac{3}{4}\right)=0 \Rightarrow-\frac{3}{4} is the additive inverse of \frac{3}{4}

\frac{3}{4} \times \frac{4}{3}=1 \Rightarrow \frac{4}{3}is the multiplicative inverse of \frac{3}{4}