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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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Rational numbers are numbers that can be expressed as the quotient of two integers with a non-zero denominator
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
Rational numbers can be simplified to their simplest form and arithmetic operations follow specific rules to ensure the result is in its simplest form
Rational numbers can be ordered by comparing their sizes when written as fractions with positive denominators
The density of rational numbers states that between any two real numbers, there exists a rational number, which is crucial in mathematical analysis
The density of rational numbers is fundamental in constructing the real number system from the rationals
Rational numbers can be represented by finite continued fractions, providing insight into their structure and interrelationships
The coefficients of continued fractions can be determined using the Euclidean algorithm
Continued fractions offer a unique perspective on the connections between rational numbers and number theory
Rational numbers are foundational elements in the study of number systems, with clear and consistent arithmetic properties
The field of rational numbers is essential for understanding more complex number systems, such as algebraic numbers and the real numbers
Rational numbers are used to model and interpret the physical world, making them crucial in advanced mathematical concepts and applications
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Definition of Rational Numbers
Numbers expressible as a quotient of two integers, with a non-zero denominator.
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Decimal Representation of Rational Numbers
Rational numbers can be written as terminating or repeating decimals.
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Rational Numbers in the Number System
Rational numbers, denoted by ℚ, are a subset of real numbers excluding irrationals.
