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The integers (Latin word integer means "untouched") are the natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and zero.

More formally, the integers are the only integral domain whose positive elements are well-ordered, and this order is preserved by addition. Like the natural numbers, the integers form a countably infinite set. The set of all integers is usually denoted by a boldface Z (or blackboard bold , or Unicode ℤ), which stands for Zahlen (German for "numbers").[1]

The term rational integer is used in algebraic number theory to distinguish these "ordinary" integers, embedded in the field of rational numbers, from other "integers" such as the algebraic integers.

Algebraic properties

Like the natural numbers, Z is closed below the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. Also, unlike the natural numbers, it is closed under subtraction. Z is not closed under the operation of division, since the quotient of two integers (e.g., 1 divided by 2), need not be an integer.


addition multiplication
No zero divisors: if ab = 0, then either a = 0 or b = 0 (or both)







Types of integers

  • Positive integers is a set ℤ+ ∈ ℤ which contains only positive numbers (same as natural numbers);
  • Negative integers is a set ℤ- ∈ ℤ which contains only negative numbers;
  • Non-positive integers is a subset of ℤ which contains negative integers and zero (ℤ- ∪ {0});
  • Non-negative integers is a subset of ℤ which contains positive integers and zero (ℤ+ ∪ {0});
  • Almost integer is a number that is very close to an integer;
  • Radical integers is a set of numbers, that is closed under addition, multiplication, subtraction and root extraction.

Order-theoretic properties







Conversion of real numbers to integers

For more information, see: Floor or ceiling functions.

It is possible to convert a real number to an integer. One way is called rounding, when a real number is rounded down if its fractional part less than 5, and up if more than or equal to 5. Also there are two functions called floor and ceiling functions. A floor function of x (denoted as ⌊x⌋) has a value of the biggest number not more than x:

A ceiling function of x (denoted as ⌈x⌉) has a value of the smallest number not less than x:

Unicode entities for floor function symbols are &#8970 and &#8971, for ceiling function are &#8968 and &#8969.


The integers can be constructed from the set of natural numbers by defining them to be the set of equivalence classes of pairs of natural numbers under the equivalence relation


Taking 0 to be a natural number, the natural numbers may be considered to be integers by the embedding that maps to where denotes the equivalence class having as a member.

Addition and multiplication of integers are defined as follows:

It is easily verified that the result is independent of the choice of representatives of the equivalence classes.

Typically, is denoted by



This notation recovers the familiar representation of the integers as

Some examples are:

Integers in computing


