Feedback

What do you think about us?

Your name

Your email

Message

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.

Show More

## Fraction Form

### Definition

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

### Mixed Numbers

Rational numbers that include both a whole number and a fraction

### Repeating Decimals

Rational numbers that have a repeating pattern in their decimal representation

## Continued Fractions

### Definition

A unique way to express rational numbers through a sequence of integers and fractions

### Compact Form

A more concise way to write continued fractions

### Example

[2; 1, 2] is a continued fraction representation of the rational number 2 + 1/(1 + 1/2)

## Egyptian Fractions

### Definition

A way to represent a rational number as a sum of distinct unit fractions

### Example

2 + 1/2 + 1/6 is an Egyptian fraction representation of the rational number 2 2/3

## Prime Factorization

### Definition

Expressing a rational number as a product of prime numbers to various powers

### Example

2^3 × 3^-1 is the prime factorization of the rational number 8/3

## Quote Notation

### Definition

A method of representing rational numbers often used in measurements, such as 3'6"

### Example

3'6" is a quote notation representation of the rational number 3.5

## Formal Construction of Rational Numbers

### Definition

The set of rational numbers is constructed from equivalence classes of ordered pairs of integers

### Equivalence Relation

An established relation between ordered pairs of integers that is congruent with addition and multiplication

### Operations

Addition and multiplication operations defined for ordered pairs of integers

## Canonical Representation and Equivalence Classes

### Definition

Each rational number is represented by a unique simplest form in an equivalence class

### Canonical Representative

The simplest form of a rational number, also known as the reduced fraction or lowest terms

### Integration of Integers

Integers can be included in the rational number system by representing them as n/1

## Ordering Rational Numbers

### Definition

Rational numbers can be ordered by comparing the cross-products of their numerators and denominators

### Consistency with Integers

The ordering of rational numbers extends the natural order of integers

Algorino

Edit available