# Real numbers

The **real numbers** are all those numbers that are included within the **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 . This group of numbers also includes **fractions of** whole **numbers** that have non- **zero** numbers in their denominator .

## What are real numbers?

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

- What are the real numbers
- What are real numbers for?
- features
- History
- Classification
- Properties of real numbers
- Operations
- Examples

## What are the real numbers

The system of real numbers is formed mainly by two large groups, that of **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-integer fractions** with their negative notations; the **integers** including negative numbers and positive integers; the latter in turn include the **natural numbers** and **zero** .

## What are real numbers for?

Real numbers are digits that help us to carry out all the **mathematical ****operations** necessary to solve a **situation** or problem. They also have the function of designating the **number of elements** that have a certain set. They are used to **identify** places or objects and to **order** and **rank** categories.

## features

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

- It is formed by the
**union**of**rational**and**irrational**numbers . - They are a
**complete set**. - These types of numbers and the
**number line**are closely related. - For each real number there is a
**point**that represents it on the number line. - The natural numbers are
**complete**and it is an**ordered**set . - 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**function .

## History

They arose as a need that the settlers of the **primitive ****era had** to count and solve problems that were constantly presented to them. There is a record that the people of **Babylon** used them to count, for example, their animals. The **Egyptians** first used **fractions** by means of mathematicians like **Pythagoras** . During the 19th centuries, **constructions** and **systematizations** of real numbers were carried out by means of two important mathematicians, **Georg Cantor** and **Richard Dedekind.**. These two historical figures in the world of mathematics managed to systematize real numbers using a series of advances invented by them.

## Classification

Real numbers are classified or divided into the following groups:

**Natural**numbers: These are the numbers that we normally use to**count**. They can start with 0 or 1. They serve as a base to**form**larger numbers, and are numbers that have**divisibility**and**distribution**of numbers. With them you can add, subtract, multiply and divide.**Whole**numbers: They are the numbers that can be**written****without**using a**fraction**. They are complete numbers and are used to express**quantities**,**depths**,**temperatures**. Together, they make the smallest group of real numbers.**Rational**numbers: They 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**Q.**They can also be defined as types of integer pair**equivalences**.**Irrational**numbers: They are the real numbers that are not rational numbers either. These numbers**cannot**be expressed as**fractions**. Among them we can mention the**radius**of a circle, the**golden number**and the**square root**.

## Properties of real numbers

The properties of real numbers vary depending on the **type of operation** , this being the case, then we have the following:

**Properties of Sum****Internal property:**the result of adding two real numbers will result in another real number.**Associative property :**if there are more than two addends, it does not 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 subtrahend are positive, the subtraction will give
**a positive result**, otherwise the result will be negative. - If the minuend is negative and the subtrahend positive, the sum 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 one.

- If the minuend or subtrahend are positive, the subtraction will give
**Multiplication properties****Internal**property**Associative**property**Commutative**property**Distributive**property**Inverse**or opposite element**Common factor**

## Operations

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

## Examples

- 120
- 0.1234512345…
- – ½
- √5