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Rational numbers are a fundamental concept in mathematics, represented as ratios of integers. This overview explores their various forms, including fractions, mixed numbers, repeating decimals, continued fractions, Egyptian fractions, and prime factorization. It delves into the formal construction of rational numbers through equivalence classes, their canonical representation, and the integration and ordering of integers within the rational number system.
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Rational numbers can be expressed as a ratio of two integers, where the denominator is not zero
Rational numbers that include both a whole number and a fraction
Rational numbers that have a repeating pattern in their decimal representation
A unique way to express rational numbers through a sequence of integers and fractions
A more concise way to write continued fractions
[2; 1, 2] is a continued fraction representation of the rational number 2 + 1/(1 + 1/2)
A way to represent a rational number as a sum of distinct unit fractions
2 + 1/2 + 1/6 is an Egyptian fraction representation of the rational number 2 2/3
Expressing a rational number as a product of prime numbers to various powers
2^3 × 3^-1 is the prime factorization of the rational number 8/3
A method of representing rational numbers often used in measurements, such as 3'6"
3'6" is a quote notation representation of the rational number 3.5
The set of rational numbers is constructed from equivalence classes of ordered pairs of integers
An established relation between ordered pairs of integers that is congruent with addition and multiplication
Addition and multiplication operations defined for ordered pairs of integers
Each rational number is represented by a unique simplest form in an equivalence class
The simplest form of a rational number, also known as the reduced fraction or lowest terms
Integers can be included in the rational number system by representing them as n/1
Rational numbers can be ordered by comparing the cross-products of their numerators and denominators
The ordering of rational numbers extends the natural order of integers
00
A ______ number can be shown as the quotient of two integers, with the condition that the bottom number isn't zero.
rational
01
The fraction 8/3 is an example of a ______ number in its most commonly seen form.
rational
02
A mixed number, such as 2 2/3, combines a whole number with a ______, and is another form of a rational number.
fraction
