How to Use the Properties of Rational Numbers in the Questions?


Question: How to use the properties of rational numbers in the questions?

A rational number is formed when any two integers, such as p and q, are expressed in the form of p/q, where q ≠ 0.

Answer: To use the different properties of rational numbers, we use the definition of the properties involved with different mathematical operations.

Let us discuss the properties of rational numbers in details

Explanation:

Let's list the properties of rational numbers with examples.

  • Closure Property

Rational numbers are closed under addition, subtraction and multiplication. Let us take a few examples:
1. 2/3 + 3/5 = 10/15 + 9/15 = 19/15
2. 7/9 - 1/2 = 14/18 - 9/18 = 5/18
3. 2/5 × 1/7 = 2/35

Therefore, the mathematical operations addition, subtraction, and multiplication when performed on rational numbers always results in a rational number.
Only division of rational numbers is not closed, as:

7/11 ÷ 0 is not defined

  • Commutative Property

Rational numbers are commutative under addition and multiplication, but are not commutative under subtraction and division. Let us take a few examples:
1. 2/3 + 3/5 = 3/5 + 2/3
2. 7/9 - 1/2 ≠ 1/2 - 7/9
3. 2/5 × 1/7 = 1/7 × 2/5
4.  5/4 ÷ 1/9 ≠ 1/9 ÷ 5/4

  • Associative Property

Rational numbers are associative under addition and multiplication, but are not associative under subtraction and division. Let us take a few examples:
1. (2/3 + 3/5) + 3/8 = 2/3 + (3/5 + 3/8)
2. (7/9 - 1/2) - 2/5 ≠ 7/9 - (1/2 - 2/5)
3. (2/5 × 1/7) × 5/7 = 2/5 × (1/7 × 5/7)
4.  (5/4 ÷ 1/9) ÷ 3/19  ≠ 5/4 ÷ (1/9 ÷ 3/19)

  • Identity Property

1. Rational numbers follow the additive identity as, when any rational number 'a' is added to 0 it results in 'a'. Therefore, 0 is an additive identity.
2. Rational numbers follow the multiplicative identity as, when any rational number 'a' is multiplied to 1 it results in 'a'. Therefore, 1 is a multiplicative identity.

  • Inverse Property

1. Rational numbers follow the additive inverse as, when any rational number 'a' is added to '-a' it results in 0. Therefore, -a is an additive inverse.
2. Rational numbers follow the multiplicative inverse as, when any rational number 'a' is multiplied to '1/a' it results in 1. Therefore, 1/a is a multiplicative inverse.

Therefore, we can use the different properties of rational numbers in different questions.