**Short Notes - Rational Numbers**

Rational numbers are an essential concept in mathematics, particularly in the field of number theory. They are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. In other words, rational numbers can be written in the form p/q, where p and q are integers and q is not equal to zero. Here, we will discuss the properties of rational numbers in detail.

**Properties of Rational Numbers:**

1. Closure Property: The sum, difference, product, or quotient of any two rational numbers is always a rational number. For example, if we add or subtract two rational numbers, the result will be a rational number.

2. Commutative Property: The addition and multiplication of rational numbers follow the commutative property. It means that changing the order of rational numbers does not affect the result. For instance, a + b = b + a and a * b = b * a.

3. Associative Property: The addition and multiplication of rational numbers are associative. It means that changing the grouping of rational numbers does not change the result. For example, (a + b) + c = a + (b + c) and (a * b) * c = a * (b * c).

4. Identity Property: The rational number 0 is the additive identity, as adding 0 to any rational number gives the same number. Similarly, 1 is the multiplicative identity, as multiplying any rational number by 1 gives the same number.

5. Inverse Property: Every rational number has an additive inverse, which means that for every rational number a, there exists -a such that a + (-a) = 0. Similarly, every non-zero rational number has a multiplicative inverse, which means that for every non-zero rational number a, there exists 1/a such that a * (1/a) = 1.

6. Distributive Property: The distributive property holds for rational numbers, which states that for any rational numbers a, b, and c, a * (b + c) = a * b + a * c.

7. Unique Representation: Rational numbers have a unique representation. It means that any rational number can be represented in its simplest form, where the numerator and denominator have no common factors other than 1.

Understanding the properties of rational numbers is crucial in various mathematical operations and applications. These properties help in simplifying calculations, solving equations, and analyzing mathematical relationships.