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Rational numbers are numbers that can be expressed as a fraction of two integers, with a non-zero denominator. They include integers, whole numbers, and their opposites, and are denoted by ℚ. These numbers are either terminating or repeating decimals and form a field in algebra. Simplification, arithmetic operations, ordering, and the concept of density are key aspects of rational numbers. They also have a unique representation through continued fractions and play a crucial role in advancing mathematical concepts.

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## Definition and Characteristics

### Definition of Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers with a non-zero denominator

### Representation and Equivalence

Equivalence Classes

Rational numbers are represented as equivalence classes of ordered pairs of integers based on the principle of equality through cross-multiplication

Field Structure

Rational numbers form a field, a fundamental algebraic structure with the properties of arithmetic operations

### Simplification and Arithmetic

Rational numbers can be simplified to their simplest form and arithmetic operations follow specific rules to ensure the result is in its simplest form

## Ordering and Density

### Ordering Rational Numbers

Rational numbers can be ordered by comparing their sizes when written as fractions with positive denominators

### Density of Rational Numbers

The density of rational numbers states that between any two real numbers, there exists a rational number, which is crucial in mathematical analysis

### Connections to Real Numbers

The density of rational numbers is fundamental in constructing the real number system from the rationals

## Continued Fractions

### Representation of Rational Numbers

Rational numbers can be represented by finite continued fractions, providing insight into their structure and interrelationships

### Euclidean Algorithm

The coefficients of continued fractions can be determined using the Euclidean algorithm

### Connections to Number Theory

Continued fractions offer a unique perspective on the connections between rational numbers and number theory

## Role of Rational Numbers in Mathematics

### Foundational Elements

Rational numbers are foundational elements in the study of number systems, with clear and consistent arithmetic properties

### Progression to Complex Number Systems

The field of rational numbers is essential for understanding more complex number systems, such as algebraic numbers and the real numbers

### Applications in the Physical World

Rational numbers are used to model and interpret the physical world, making them crucial in advanced mathematical concepts and applications

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