Integer numbers

In mathematics there are many types of numbers depending on 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 whole numbers was established to refer to indivisible units such as a person or a country.

What are whole numbers?

The integers are elements of a numerical set that groups the natural numbers, their additive inverses and zero . There are positive, negative, and zero integers. 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 it and zero to the left, the progression of negative numbers is given and from zero to the right, the progression of positive numbers.

  • What are the whole numbers
  • What are whole numbers for?
  • features
  • History
  • How they are represented
  • Properties
  • Integer operations
  • Examples

What are the whole numbers

The whole numbers are: the natural numbers, the zero and the negative numbers:

  • The natural numbers are those that are used to count the elements of a set and to carry out elementary calculation operations.
  • The zero is a number of null value representing that there is no item number or count.
  • Negative numbers are those that result from subtracting a natural number from a greater one. Negative numbers are less than zero and represent losses, decreases, among other things.

What are whole numbers for?

Whole numbers are used to express a countable 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 whole numbers, the following can be mentioned:

  • They are made up of positive, negative, and zero integers.
  • They have no decimals .
  • The symbol represents the letter Z .
  • Zero is a valueless number that divides positive numbers from negative ones. All numbers greater than zero are positive and all numbers less than zero are negative.
  • The zero is considered a neutral number .
  • They cannot be divided unless the division is exact.
  • Positive numbers that are farther from zero will represent more.
  • Negative numbers are located on a number line on the left , zero in the center, and positive numbers on the right .


The history of whole numbers has a beginning in the time of prehistoric man , who was developing with his intellect his ability to count elements that were in nature or in his mind. At first, he made use of natural numbers but over time these were not enough because he needed an element opposite to positive to express debts, or lack of integer elements (without decimals or incomplete). All this evolution of the uses and functions of positive and negative integers allowed man to generate various mathematical operations . In ancient times, specifically in the 3rd century BC, the zero appears in Mesopotamia to refer to a null quantity or something that lacks value and that also marks the limit of negative numbers and the beginning of positive numbers.

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 , music, among others. In the modern era, they were located by their characteristics as part of the rational numbers , which in turn are part of the 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 seek accuracy or more complex equations than in other times. They are often used as meteorological indicators to indicate temperatures that are below or above zero. In this sense, it can be said that in the evolution of man, integers have played a fundamental role in his logical-mathematical development.

How they are represented

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

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


The properties of integers are as follows:

Properties of whole numbers as a set

The whole numbers are an extension of the natural numbers and they are also a subset of the rational numbers . These numbers are an indeterminate set since neither beginning nor end is known and whose origin is zero.

They have their value determined in the position they occupy on the number line

Properties of whole numbers in multiplication operations

The multiplication properties are:

  • Associative (when the factors are multiplied together).
  • 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 additions and multiplications).

Properties in addition operations

The addition properties are:

  • Associative (association of addends).
  • Commutative (the addends can vary their order without altering the result).
  • The neutral element is zero (any number added to zero does not change the result).

Integer operations

With integers 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 a whole number .


Here are several examples of whole numbers.

Examples of positive integers


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

Examples of negative integers

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

Example of integer 0

Zero is the neutral element in the sum. Every number added to zero is equal to the number that is added. Example. 5 + 0 = 5

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