Properties of Rational Numbers

The major properties of rational numbers are:

Closure Property

Commutativity Property

Associative Property

Distributive Property

Let us now study these properties in detail.

Step-by-step explanation:

Addition of Rational Numbers

The closure property states that for any two rational numbers a and b, a + b is also a rational number.

The result is a rational number. So we say that rational numbers are closed under addition.

2) Subtraction of Rational Numbers

The closure property states that for any two rational numbers a and b, a – b is also a rational number.

The result is a rational number. So the rational numbers are closed under subtraction.

3) MULTIPLICATION of Rational Numbers

The closure property states that for any two rational numbers a and b, a × b is also a rational number.

The result is a rational number. So rational numbers are closed under multiplication.

4) Division of Rational Numbers

The closure property states that for any two rational numbers a and b, a ÷ b is also a rational number.

The result is a rational number. But we know that any rational number a, a ÷ 0 is not defined. So rational numbers are not closed under division. But if we exclude 0, then all the rational numbers are closed under division.

Commutative Property

1. Addition

For any two rational numbers a and b, a + b = b+ a

We see that the two rational numbers can be added in any order. So addition is commutative for rational numbers.

2. Subtraction

For any two rational numbers a and b, a – b ≠ b –

So subtraction is not commutative for ratioanl numbers.

3. Multiplication

For any two rational numbers a and b, a × b = b × a

We see that the two ratrional numbers can be multiplied in any order. So multiplication is commutative for ratioanl numbers.

4. Division

For any two rational numbers a and b, a ÷ b ≠ b

We see that the expressions on both the sides are not equal. So divsion is not commutative for ratioanal numbers.

Associative Property

Take any three rational numbers a, b and c. Firstly add a and b and then add c to the SUM. (a + b) + c. Now again add b and c and then a to the sum, a + (b + c). Is (a + b) + c and a + (b + c) same? Yes and this is how associative property works. It states that you can add or multiply numbers regardless of how they are grouped

Addition and multiplication are associative for rational numbers.

Subtraction and division are not associative for rational numbers.

Distributive Property

Distributive property states that for any three numbers x, y and Z we have

x × ( y + z ) = (x × y) +( x × z)

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