Mathematics# Real numbers

## What are real numbers?

## Which are the real numbers?

## What are real numbers for?

## Characteristics

## History

## Classification

## Real numbers properties

## Operations

## Examples

**Real numbers** are all those numbers that are included within **rational** numbers. They can be **positive**, **negative** and include the number **zero**, as in the case of **irrational** numbers. These numbers can be written in different ways, some of them very simple, generally used in **simple** mathematical **operations**, and in more **complex** forms. Also included in this group of numbers are whole **number** **fractions** that have in their denominator numbers that are not null.

**Related topics**

Algebraic numbers, complex numbers, integer numbers, irrational numbers, prime numbers, rational numbers

Real numbers are all those numbers found on the **number line** that make up the group of **rational** and **irrational** numbers, negative and positive, including the number zero.

The system of real numbers is formed mainly by two large groups, the **rational numbers**, which are all those numbers that can be expressed as the division of two whole numbers, and the system of **irrational numbers** whose decimal representation is expansive, infinite and aperiodic. Rational numbers can also be divided into **subgroups**, among which we can mention: **non whole fractions** with their negative notations; **whole numbers** including negative numbers and positive integers; the latter in turn include **natural numbers** and **zero**.

Real numbers are digits that allow us to carry out all the **mathematical operations** necessary to solve a **situation** or problem. They also have as function, to designate the **quantity of elements** that have a certain group. They can be used to **identify** places or objects and to **order** and **prioritize** categories.

Among the main characteristics of real numbers we can mention the following:

- They are formed by the
**union**of**rational**and**irrational** - They are a
**complete set**. - This type of numbers and the
**number line**have a close relationship. - For each real number there is a
**point**that represents it within the number line. - Natural numbers are
**complete**, and it is an**ordered** - They are numbers that have
**associativity**and**commutativity**. - They all have an
**order**and are written**consecutively**. - When they are used for
**counting**, then we mean that they have a**cardinal**

They appeared as a need for the people of the **primitive era** to tell and solve problems that were constantly presented to them. It is recorded that the **Babylonians** used them to count, for example, their animals. The **Egyptians** used the fractions for the first time through mathematicians such as **Pythagoras**. During the 19th century, two important mathematicians, **Georg Cantor** and **Richard** **Dedekind**, made **constructions** and **systemizations** of real numbers. These two historical figures in Mathematics managed to systematize real numbers using a series of advances invented by them.

Real numbers are classified or divided into the following groups:

**Natural numbers**: These are the numbers we normally use to**count**. They can start with 0 or with 1. They serve as a basis for**making**larger numbers, and they are numbers that have**divisibility**and number**distribution**. With them, you can add, subtract, multiply and divide.**Integers**: These are numbers that**can be written without**using a**fraction**. They are complete numbers and are used to express**quantities**,**depths**,**temperatures**. Together, they form the smallest group of real numbers.**Rational numbers**: These are the numbers that can be expressed as a**fraction**of two whole numbers, which have a**numerator**and a**denominator**. It is represented by the letter They can also be defined as types of**equivalences**of whole pairs.**Irrational numbers**: These are real numbers that are not rational numbers either. These numbers**cannot be**expressed as**fractions**. Among them we can mention the**radius**of a circumference, the golden number and the**square root**.

The property of the real numbers can change depending on the **type of operation**, being this the case, then we have the following ones:

**Properties of the Sum****Internal property**: the result of adding two real numbers will result in another real number.**Associative property:**if there are more than two summands, it doesn’t matter which sum is done first, if the numbers are all real.**Commutative property**: the order of the digits will not alter the sum.**Existence of the neutral element**: any number that is added with 0 will result in the same number.

**Subtraction properties**- If the minuend or subtracting are positive, the subtraction will give a
**positive result**, otherwise the result will be negative. - If the minuend is negative and the subtracting positive, the addition is done, and the result has a
**minus sign**. - Subtracting a positive number is the
**same**as adding a negative number. - Subtracting a negative number is the
**same**as adding a positive number.

- If the minuend or subtracting are positive, the subtraction will give a
**Properties of multiplication****Internal****Associative****Commutative****Distributive****Inverse**or opposite element.**Common factor**.

The mathematical operations that can be performed using real numbers are: **addition**, **multiplication**, **subtraction**, **division** and **potentiation** or enhancement.

- 120
- 1234512345…
- – ½
- √5

Written by Gabriela Briceño V.

Briceño V., Gabriela. (2019). *Real numbers*. Recovered on 24 February, 2024, de Euston96: https://www.euston96.com/en/real-numbers/