Rational numbers are of the form p/q, where p and q are integers and q ≠ 0. Because of the underlying structure of numbers, or p/q form, most individuals find it difficult to distinguish between fractions and rational numbers. The rational number is represented by Q. The main six properties of rational numbers are:
Closure Property
The closure property of the rational number states that performing an operation on the rational number always results in a rational number, i.e., the operations are closed on the rational number. The operations allowed in this property are addition, subtraction, and multiplication. The division operation is not allowed in the closure property of the rational number, as dividing two rational numbers may or may not result in the rational number.
Suppose we take two rational numbers, X and Y, then.
For Addition: The sum of any two rational numbers always results in the rational number. This can be represented as,
X + Y = Rational Number
For example, 2/3 + 4/3 = 6/3 = 2 (Rational Number)
For Subtraction: The difference between any two rational numbers always results in the rational number. This can be represented as,
X - Y = Rational Number
For example, 2/3 - 4/3 = -2/3 (Rational Number)
For Multiplication: The multiplication of any two rational numbers always results in the rational number. This can be represented as,
X × Y = Rational Number
For example, 2/3 × 4/3 = 8/9 (Rational Number)
For Division: The division of any two rational numbers may or may not always results in the rational number. This can be represented as,
X/Y = May or may not Rational Number
For example,
- 2/3 4/3 = 1/2 (Rational Number)
- 2/3/0 = undefined (Not a rational Number)
Commutative Property
The commutative property of the rational number states that changing the order of the multiplication and the addition of the rational number does not change the result.
Now let's learn about this property in detail below.
Suppose we take two rational numbers, X and Y, then.
For Addition: The sum of any two rational numbers is always commutative, i.e., it does not change with the change of the order. This can be represented as,
X + Y = Y + X
For example, 2/3 + 4/3 = 4/3 + 2/3 = 6/3
For Subtraction: The difference between any two rational numbers is not commutative, i.e., the difference if the order is changed is not equal. This can be represented as,
X - Y ≠ Y - X
For example, 2/3 - 4/3 = -2/3
4/3 - 2/3 = 2/3
2/3 - 4/3 ≠ 4/3 - 2/3
For Multiplication: The multiplication of any two rational numbers is always commutative, i.e., it does not change with the change of the order. This can be represented as,
X × Y = Y × X
For example, 2/3 × 4/3 = 4/3 × 2/3 = 8/9
For Division: The division of two rational numbers is not commutative, i.e., if the order of division is changed, the result is not equal. This can be represented as,
X ÷ Y ≠ Y ÷ X
For example, 2/3 ÷ 4/3 = 1/2
4/3 ÷ 2/3 = 2
2/3 ÷ 4/3 ≠ 4/3 ÷ 2/3
Associative Property
The associative property of the rational number states that if we take three or more rational numbers, then adding or multiplying them in any order does not change the result.
Now let's learn about this property in detail below.
Suppose we take three rational numbers, X, Y, and Z. Then,
For Addition: The sum of three rational numbers is always associative, i.e., it does not change with the change of the order. This can be represented as,
(X + Y) + Z = X + (Y + Z)
For example, (2/3 + 4/3) + 6/3 = 2/3 + (4/3 + 6/3)
LHS = (2/3 + 4/3) + 6/3
= 6/3 + 6/3
= 12/3 = 4
RHS = 2/3 + (4/3 + 6/3)
= 2/3 + 10/3
= 12/3 = 4
Hence, LHS = RHS proved
Thus, the property is verified.
For Subtraction: The difference of three or more rational numbers is not commutative, i.e., the difference of three or more rational numbers if the order is changed is not equal. This can be represented as,
(X - Y) - Z ≠ X - (Y - Z)
For Multiplication: The product of three rational numbers is always associative, i.e., it does not change with the change of the order. This can be represented as,
(X × Y) × Z = X × (Y × Z)
For example, (2/3 × 4/3) × 6/3 = 2/3 × (4/3 × 6/3)
LHS = (2/3 × 4/3) × 6/3
= 8/3 × 6/3
= 48/9 = 16/3
RHS = 2/3 × (4/3 × 6/3)
= 2/3 × 24/3
= 48/9 = 16/3
Hence, LHS = RHS proved
Thus, the property is verified.
For Division: The division of three or more rational numbers is not commutative, i.e., the division of three or more rational numbers if the order is changed is not equal. This can be represented as,
(X ÷ Y) ÷ Z ≠ X ÷ (Y ÷ Z)
Distributive Property
The distributive property of the rational number distributes multiplication over addition and subtraction. Suppose we have three rational numbers,
X, Y, and Z; then the distributive property of the rational number states that,
X × ( Y + Z) = X × Y + Y × Z
This is called the distributive property of rational numbers over addition.
X × ( Y - Z) = X × Y - Y × Z
This is called the distributive property of rational numbers over subtraction.
Example 1: Simplify 3 × (11 + 4)
Solution:
= 3 × (11 + 4)
= 3×11 + 3×4
= 33 + 12
= 45
Example 2: Simplify 3 × (11 - 4)
Solution:
= 3 × (11 - 4)
= 3×11 - 3×4
= 33 - 12
= 21
Additive Property
The additive property of the rational number is classified into categories that include,
1. Additive Identity Property
The additive identity property states that among rational numbers we have an identity element such that adding it to any other rational number results in the same rational number. The additive identity element of the rational number is 0. Thus, for any rational number A
A + 0 = A
2. Additive Inverse Property
The additive inverse property states that among rational numbers we have an inverse element of all the elements such that adding these elements results in the identity element (0). The inverse element of any rational number A is (-A). Thus,
A + (-A) = 0
Identity and Inverse Properties
The additive property of the rational number is classified into categories that include,
Multiplicative Identity Property
The multiplicative identity property states that among rational numbers we have an identity element such that multiplying it by any other rational number results in the same rational number. The multiplicative identity element of the rational number is 1. Thus, for any rational number A
A × 1 = A
Multiplicative Inverse Property
The multiplicative inverses property states that among rational numbers we have an inverse element of all the elements such that multiplying these elements results in the identity element (1). The inverse element of any rational number A is (1/A), which is also called the reciprocal of the number. Thus,
A × (1/A) = 1
Solved Examples
Example 1: Verify the associative property of a rational number if a = 1/2 and b = 3/4 and c = 2/3.
Solution:
Given,
- a = 1/2
- b = 3/4
- c = 2/3
For Associative Property of Addition,
(a + b) + c = a + (b + c)
(1/2 + 3/4 ) + 2/3 = 1/2 + (3/4 + 2/3)
5/4 + 2/3 = 1/2 + 17/12
23/12 = 23/12
Proved.
For Associative Property of Addition,
(a.b).c = a.(b.c)
(1/2 . 3/4 ). 2/3 = 1/2 . (3/4 . 2/3)
2/8 = 2/8
3/8 . 2/3 = 1/2 . 2/4
1/4 = 1/4
Proved
Example 2: Verify the distributive property of a rational number if a = 1/2 and b = 3/4 and c = 2/3
Solution:
Given,
- a = 1/2
- b = 3/4
- c = 2/3
Distributive Property of Multiplication over Addition
a × (b + c) = a × b + a × c
1/2 × ( 3/4 + 2/3 ) = (1/2 × 3/4) + (1/2 × 2/3)
1/2 × 17/12 = 3/8 + 2/6
17/24 = 17/24
Proved
Distributive Property of Multiplication over Subtraction
a × (b - c) = a × b - a × c
1/2 × (3/4 - 2/3) = 1/2 × 3/4 - 1/2 × 2/3
1/2 × 1/12 = 3/8 - 2/6
1/24 = 1/24
Proved