# Properties of Division of Rational Numbers

We will learn the three properties of division of rational numbers.

Property 1 (Closure Property of division of rational numbers):

If a/b and c/d are any two rational numbers such that c/d ≠ 0 then

(a/b ÷ c/d) is also a rational number.

For example:

(i) 2/3 and 4/9 are any two rational numbers and clearly 4/9 ≠ 0 then

(2/3 ÷ 4/9) = (2/3 × 9/4) = (2 × 9)/(3 × 4) = 18/12 = 3/2 is also a rational number.

(ii) 6/7 and -5/7 are any two rational numbers and clearly -5/7 ≠ 0 then

(6/7 ÷ -5/7) = (6/7 × 7/-6) = (6 × 7)/(7 × -6) = 42/-42 = 1/-1 is also a rational number.

Property 2 (Property of 1 of division of rational numbers):

For every rational number a/b we have:

(a/b ÷ 1) = a/b

For example:

(i) 5/8 ÷ 1 = 5/8

(ii) -4/9 ÷ 1 = -4/9

(iii) 5/-2 ÷ 1 = 5/-2

Property 3:

For every nonzero rational number a/b, we have:

{a/b ÷ a/b} = 1

For example:

(i) 4/7 ÷ 4/7 = 4/7 × 7/4 = (4 × 7)/(7 × 4) = 28/28 = 1

(ii) -3/11 ÷ -3/11 = -3/11 × 11/-3 = (-3 × 11)/(11 × -3) = -33/-33 = 1

(iii) 3/-2 ÷ 3/-2 = 3/-2 × -2/3 = (3 × -2)/(-2 × 3) = -6/-6 = 1

Rational Numbers

What is Rational Numbers?

Is Every Rational Number a Natural Number?

Is Zero a Rational Number?

Is Every Rational Number an Integer?

Is Every Rational Number a Fraction?

Positive Rational Number

Negative Rational Number

Equivalent Rational Numbers

Equivalent form of Rational Numbers

Rational Number in Different Forms

Properties of Rational Numbers

Lowest form of a Rational Number

Standard form of a Rational Number

Equality of Rational Numbers using Standard Form

Equality of Rational Numbers with Common Denominator

Equality of Rational Numbers using Cross Multiplication

Comparison of Rational Numbers

Rational Numbers in Ascending Order

Representation of Rational Numbers on the Number Line

Addition of Rational Number with Same Denominator

Addition of Rational Number with Different Denominator

Properties of Addition of Rational Numbers

Subtraction of Rational Number with Different Denominator

Subtraction of Rational Numbers

Properties of Subtraction of Rational Numbers

Simplify Rational Expressions Involving the Sum or Difference

Multiplication of Rational Numbers

Properties of Multiplication of Rational Numbers

Rational Expressions Involving Addition, Subtraction and Multiplication

Division of Rational Numbers

Rational Expressions Involving Division

Properties of Division of Rational Numbers

To Find Rational Numbers