Rational Numbers

Properties

Definition

Role Of 0

Role Of 1

Multiplicative Inverse

Numbers which can be written in the form p/q where q ≠ 0 are called as rational numbers

Examples :- 5/3 , 35/1, 12/45

Commutative property

Distributive Property

Closure Property

Associative property

The closure property means that a set is closed for some mathematical operation.

The commutative property states that the numbers on which we operate can be moved or swapped from their position without making any difference to the answer.

According to the distributive property, multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together.

Closure Property of Rational Numbers

Subtraction

Multiplication

Addition

Division

Example: -5 / 7 - 2 / 3 = - 29 / 21

Example: - 2 / 3 x 4 / 5 = - 8 / 15

a ÷ 0 is not defined for rational numbers a and 0. Thus, rational numbers are not closed under division.

For an two rational Numbers a and b, a + b is also a rational number

Example: - 3 / 8 + (- 5 / 7) = -19 / 56

For an two rational Numbers a and b, a - b is also a rational number

For an two rational Numbers a and b, a x b is also a rational number

Subtraction

Multiplication

Addition

Division

For any Two rational numbers a and b, a / b = b / a So, Division is not commutative for rational numbers

For any Two rational numbers a and b, a x b = b x a So, Multiplication is commutative for rational numbers

For any two rational numbers a and b, a - b ≠ b - a. So, Subtraction is not commutative for rational numbers

For any Two rational numbers a and b, a + b = b + a So, Addition is commutative for rational numbers

Example:- 2 / 3 + 5 / 7 = 5 / 7 + 2 / 3 = 29 / 21 or 1 8 / 21

Example:- 2 / 3 - 5 / 7 ≠ 5 / 7 - 2 / 3

Example:- 2 / 3 x 5 / 7 = 5 / 7 x 2 / 3 = 10 / 21

Example:- 2 / 3 ÷ 5 / 7 ≠ 5 / 7 ÷ 2 / 3

Distributivity of Addition over Multiplication

Distributivity of Subtraction over Multiplication

For Any Three Rational Numbers a, b and c a (b + c) = ab + ac

For Any Three Rational Numbers a, b and c a (b - c) = ab - ac

Example:- 3 / 4 x ( 2 / 3 + 5 / 6 ) = (3 / 4 x 2 / 3) + (3 / 4 x 5 / 6)

Example:- 3 / 4 x ( 2 / 3 - 5 / 6 ) = ( 3 / 4 x 2 / 3 ) - ( 3 / 4 x 5 / 6 )

The associative property is a math rule that says that the way in which factors are grouped in a multiplication problem does not change the product.

Addition

Multiplication

Subtraction

Division

According to the associative property of addition, the sum of three or more numbers remains the same regardless of how the numbers are grouped.

Example:- ( 2 / 4 + 3 / 4 ) + 4 / 4 = 2 / 4 + ( 3 / 4 + 4 / 4 )

For any three Rational Numbers a , b and c, a - (b - c) ≠ ( a - b ) - c

Example:- ( 2 / 4 - 3 / 4 ) - 4 / 4 ≠ 2 / 4 - ( 3 / 4 - 4 / 4 )

For any three Rational Numbers a , b and c, a x (b x c) = ( a x b ) x c

Example:- ( 2 / 4 x 3 / 4 ) x 4 / 4 = 2 / 4 x ( 3 / 4 x 4 / 4 )

For any three Rational Numbers a , b and c, a ÷ (b / c) ≠ ( a / b ) ÷ c

Example:- ( 2 / 4 ÷ 3 / 4 ) ÷ 4 / 4 ≠ 2 / 4 ÷ ( 3 / 4 ÷ 4 / 4 )

1 is the Multiplicative Identity for Rational Numbers because for a given Rational Number a, a x 1 = a

Example:- 3 / 2 x 1 = 3 / 2

Example:- 3 / 2 + 0 = 3 / 2

0 is the Additive Identity For Rational Numbers because for a given rational number a, a + 0 = a

For a given Rational Number p / q , its reciprocal or multiplicative inverse is q / p because p / q x q / p = 1

Example:- 21 / 32 x 32 / 21 = 1

Additive Inverse

For a given Rational Number p / q, its additive inverse is - p / q because p / q = - p / q = 0

Example:- 2 / 1 + (- 2 / 1) = 0

By - Om Mahendra Ahire