Integer — Safekipedia Adventurer

An integer is a type of number used in mathematics. It includes zero, like (/wiki/0), and whole numbers such as 1, 2, and 3. It also includes their opposites, called negative numbers, like −1, −2, and −3. These numbers help us count, measure, and solve problems.

Integers are part of a larger group of numbers called the real numbers. They are special because they can be written without any parts after a decimal point. For example, numbers like 21, 4, and −2048 are integers. Numbers like 9.75 or the square root of 2 are not.

Integers are important in algebra and number theory. They are a key part of many mathematical ideas and calculations.

History

The word integer comes from the Latin word meaning "whole" or "untouched." At first, only positive integers like 1, 2, and 3 were used. Later, mathematicians saw that negative numbers such as −1 and −2 were also useful.

In the late 19th century, the idea of a set of integers was introduced. Today, we use the letter Z to stand for all integers. This idea came from a German word for "numbers" and became common in the middle of the 20th century.

Algebraic properties

Integers are numbers that include zero, positive numbers like 1 and 2, and negative numbers like -1 and -2. They have special rules when you add, subtract, or multiply them. For example, adding or multiplying any two integers always gives you another integer. But dividing two integers doesn’t always give you another integer—for instance, dividing 1 by 2 gives 0.5, which isn’t an integer.

Integers follow certain patterns that make them useful in math. Under addition, they act like a group where every integer has an opposite (like 5 and -5). They also follow rules that make multiplication easier, although not every integer can be reversed under multiplication. These properties help mathematicians understand how numbers behave in different situations.

Properties of addition and multiplication on integers
	Addition	Multiplication
Closure:	a + b is an integer	a × b is an integer
Associativity:	a + (b + c) = (a + b) + c	a × (b × c) = (a × b) × c
Commutativity:	a + b = b + a	a × b = b × a
Existence of an identity element:	a + 0 = a	a × 1 = a
Existence of inverse elements:	a + (−a) = 0	The only invertible integers (called units) are −1 and 1.
Distributivity:	a × (b + c) = (a × b) + (a × c) and (a + b) × c = (a × c) + (b × c)
No zero divisors:		If a × b = 0, then a = 0 or b = 0 (or both)

Cardinality

The set of integers is countably infinite. This means we can pair each integer with a unique natural number. For example: (0, 1), (1, 2), (−1, 3), (2, 4), and so on.

More formally, the cardinality of the set of integers (written as ⁠ Z {\displaystyle \mathbb {Z} } !{\displaystyle \mathbb {Z} } ⁠) is said to equal ℵ₀ (aleph-null). This pairing is called a bijection.