Properties of Rational Numbers (Part - 1): Explanation and Examples

Rational numbers are those numbers that can be expressed in the form of p/q, where p and q are integers and q is not equal to zero. In this article, we will discuss some of the important properties of rational numbers and their examples.

Closure property under addition and multiplication

The addition and multiplication of two rational numbers always result in a rational number. This property is called closure property.

Example:

1/2 + 3/4 = 2/4 + 3/4 = 5/4 (a rational number)

2/3 x 3/5 = 6/15 (a rational number)

Commutative property under addition and multiplication

The order of addition or multiplication of two rational numbers does not affect the result. This property is called commutative property.

Example:

4/5 + 3/7 = 3/7 + 4/5 = 47/35 (a rational number)

2/3 x 4/5 = 4/5 x 2/3 = 8/15 (a rational number)

Associative property under addition and multiplication

The grouping of rational numbers while addition or multiplication does not affect the result. This property is called associative property.

Example:

(2/3 + 4/5) + 1/6 = 2/3 + (4/5 + 1/6) = 79/30 (a rational number)

(2/3 x 4/5) x 3/4 = 2/3 x (4/5 x 3/4) = 2/5 (a rational number)

Existence of additive identity and multiplicative identity

There exist two special rational numbers, 0 and 1, that do not change the value of a rational number when added or multiplied with it. These two numbers are called additive identity and multiplicative identity, respectively.

Example:

1/2 + 0 = 1/2 (a rational number)

2/3 x 1 = 2/3 (a rational number)

Existence of additive inverse and multiplicative inverse

For every rational number, there exists a unique rational number such that their sum is equal to the additive identity (0) and their product is equal to the multiplicative identity (1). These two numbers are called additive inverse and multiplicative inverse, respectively.

Example:

1/2 + (-1/2) = 0 (additive identity)

2/3 x (3/2) = 1 (multiplicative identity)

Conclusion

Properties of rational numbers are important to understand for various mathematical operations. The above-mentioned properties with examples help in the better understanding of rational numbers.