Rational Numbers

Rational numbers are a set of numbers that can be expressed as the ratio of two integers, where the denominator is not zero.

Definition

A rational number, commonly denoted as ℚ, can be defined as follows:

ℚ = { p/q | p, q ∈ ℤ, q ≠ 0 }

In this notation, p/q represents the ratio of two integers, p, q ∈ ℤ means that p and q are both integers, and q ≠ 0 means that the denominator cannot be zero.

Properties of Rational Numbers

Rational numbers have several important properties:

1. Closure Property: If you add, subtract, or multiply two rational numbers, or divide one rational number by another (except for division by zero), the result is always a rational number.

2. Associative Property: For all rational numbers a, b, and c, (a + b) + c = a + (b + c), (a - b) - c = a - (b + c), and (a * b) * c = a * (b * c).

3. Commutative Property: For all rational numbers a and b, a + b = b + a and a * b = b * a.

4. Distributive Property: For all rational numbers a, b, and c, a * (b + c) = a * b + a * c.

5. Identity Property: The number 0 is the additive identity in the set of rational numbers, and the number 1 is the multiplicative identity.

6. Additive and Multiplicative Inverses: Every rational number a has an additive inverse -a, such that a + (-a) = 0, and a multiplicative inverse 1/a (if a ≠ 0), such that a * 1/a = 1.

Relationship to Integers

Rational numbers are an extension of integers, which include all positive and negative whole numbers and zero. Every integer can be considered a rational number with a denominator of 1.

Applications

Rational numbers are used in a variety of mathematical contexts, such as measuring quantities that can be divided into parts, performing division operations, and representing proportions and ratios. They also form the basis for the real number system.