# Properties of Rational Numbers

1. Closure Property

This shows that the operation of any two same types of numbers is also the same type or not.

a. Whole Numbers

If p and q are two whole numbers then

b. Integers

If p and q are two integers then

c. Rational Numbers

If p and q are two rational numbers then

2. Commutative Property

This shows that the position of numbers does not matter.

a. Whole Numbers

If p and q are two whole numbers then

b. Integers

If p and q are two integers then

c. Rational Numbers

If p and q are two rational numbers then

3. Associative Property

This shows that the grouping of numbers does not matter i.e. we can use operations on any two numbers first and the result will be the same.

a. Whole Numbers

If p, q and r are three whole numbers then

b. Integers

If p, q and r are three integers then

c. Rational Numbers

If p, q and r are three rational numbers then

Zero is the additive identity for whole numbers, integers and rational numbers.

Multiplicative Identity

One is the multiplicative identity for whole numbers, integers and rational numbers.

Additive inverse of a number is the number that when added to yields zero.

Multiplicative Inverse

The multiplicative inverse of any rational number

Example

The reciprocal of is .

Distributivity of Multiplication over Addition and Subtraction for Rational Numbers

This shows that for all rational numbers p, q and r

1. p(q + r) = pq + pr

2. p(q – r) = pq – pr

Example

Check the distributive property of the three rational numbers and .

Solution

Let’s find the value of

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