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Rational Numbers

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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.

Exploring the World of Rational Numbers

Rational numbers are an integral part of the number system in mathematics, defined as numbers that can be expressed as the quotient of two integers, with a non-zero denominator. These numbers encompass all integers, including whole numbers and their opposites, and are denoted by the symbol ℚ. Rational numbers are characterized by their ability to be written as either terminating or repeating decimals. They are a subset of the real numbers, which also include irrational numbers—those that cannot be expressed as a fraction of two integers.
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Defining Features and Mathematical Structure of Rational Numbers

The defining features of rational numbers include their representation as equivalence classes of ordered pairs of integers. This equivalence is based on the principle that two fractions are considered equal if the product of the numerator of one and the denominator of the other is the same as the product of the associated terms in the reverse order. Rational numbers form a field, an algebraic structure where addition, subtraction, multiplication, and division (except by zero) are possible and follow the usual properties of arithmetic. This field is a fundamental building block in algebra and is known as a prime field because it does not contain any smaller field within it and has characteristic zero, indicating an infinite set.

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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.

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