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Mathematics# Integer numbers

## What are integer numbers?

## Which are the integer numbers?

## What are integer numbers for?

## Characteristics

## History

## How the integer numbers are represented?

## Properties

#### Properties of integers as a set

#### Properties of integers in multiplication operations

#### Properties in addition operations

## Operations of integers

## Examples

#### Examples of positive integers

#### Examples of negative integers

#### Example of integer 0

In mathematics, there are many types of numbers according to the characteristics they have. In this numerical typology, **integers** can be mentioned, which are those that **do not have a decimal part and include natural numbers, zero and negative numbers**. The notion of integers was established to refer to indivisible units such as a person or a country.

**Related topics**

Algebraic numbers, complex numbers, irrational numbers, prime numbers, rational numbers, real numbers

Integer numbers are elements of a numerical set that **groups natural numbers, their additive inverses, and zero. There are positive integers, negative integers, and zero**. **Indivisible units** are **represented** and therefore, do not accept decimals. They are **symbolized by the letter Z**. They are represented on the number line by placing zero in the center of the line and from zero to the left, there is the progression of negative numbers and from zero to the right, the progression of positive numbers.

Integers are: natural numbers, zero and negative numbers:

**Natural numbers**are those used to count the elements of a set and to perform elementary calculation operations.**Zero**is a null value number that represents that there is no number or element to count.- Negative numbers are those that result from subtracting a natural number with a greater one. Negative numbers are less than zero and represent losses, decreases, among other things.

Integers are used **to express an accounting quantity, the absence of quantity and a negative quantity** that can be the opposite of an amount or a debt.

Among the most outstanding characteristics of integer numbers, the following can be cited:

- They are made up of positive, negative and zero integers.
**They do not have decimals**.- The symbol that represents it is the
**letter Z.** - The zero is a number without value that divides the positive numbers from the negative ones. All numbers greater than zero are positive and all numbers less than zero are negative.
**Zero**is considered a**neutral number**.- They cannot be divided unless the division is exact.
- Positive numbers further from zero will represent more.
- On a number line,
**negative numbers are placed on the left**,**zero is placed in the center**, and**positive numbers are placed on the right**.

The **history** of the integer numbers has a **beginning in the epoch of the prehistoric man**, who with his intellect was developing his capacity to count elements that were in the nature or in his mind. In the beginning, he made use of natural numbers, but with the passing of time these were not enough because he needed an element contrary to positive to express debts, or lack of whole elements (without decimals or incomplete). All this evolution of the uses and functions of positive and negative integers allowed man to **generate several mathematical operations**. In antiquity, specifically in the 3rd century B.C., in **Mesopotamia**, **zero appears** to refer to a null quantity or to something that has no value and that also marks the limit of negative numbers and the beginning of positive ones.

In the Middle Ages and modern times, the knowledge of these numbers and the operations that could be generated with them, served to develop many sciences that required mathematics such as architecture, **medicine**, **economics** and **music** among others. In the modern era, they were considered by their characteristics as part of rational numbers, which in turn are part of real numbers.

Nowadays, **they are used in different simple and complex mathematical operations** as in the commercial operations of our daily life, chemical formulas that look for accuracy or more complex equations than in other times. They are often used as meteorological indicators to indicate temperatures below or above zero. In this sense, it can be said that in man’s evolution, whole numbers have played a fundamental role in his logical-mathematical development.

**They are symbolized by the letter Z **which is the initial letter of the German word Zahien which means number.

The set of integers is usually represented by a **number line** divided by zero. This line starts from zero to the left with the negative numbers and from zero to the right with the positive numbers.

The properties of integer numbers are as follows:

Integers **are an extension of natural numbers and are also a subset of rational numbers**. These numbers are an indeterminate set since it is not known either beginning or end and its origin is zero.

They have their value determined in the position that they occupy in the numerical line.

The multiplication properties are:

- Associative (when factors multiply each other).
- Commutative (when the order of the factors does not alter the product).
- The neutral element that is the unit (any number multiplied by one does not alter the result).
- Distributive (when the factors are distributed in an equation, there are sums and multiplications).

The addition properties are:

- Associative (summands association).
- Commutative (the summands can vary their order without altering the result).
- The neutral element is zero (any number added with zero does not change the result).

With integer numbers, we can perform operations such as **addition, subtraction, multiplication **and **powers** between these numbers, but it can only be used in **division when the quotient is also an integer**.

Here are some examples of integer numbers.

1,2,3,4,5,6,7,8,9

The number 1 is a neutral element in multiplication. Every number multiplied by 1 is equal to the number by which it is multiplied. Example 2×1=2

-1,-2,-3,-4,-5,-6,-7,-8,-9

Zero is the neutral element in the sum. Any number added with zero is equal to the number being added. Example 5+0= 5

Written by Gabriela Briceño V.

Briceño V., Gabriela. (2019). *Integer numbers*. Recovered on 9 March, 2023, de Euston96: https://www.euston96.com/en/integer-numbers/