SafekipediaInteger
Adapted from Wikipedia · Discoverer experience
An integer is a type of number used in mathematics. It includes zero, like (/wiki/0), positive numbers such as 1, 2, and 3, and their opposites called negative numbers, like −1, −2, and −3. These numbers are important because they help us count, measure, and solve problems without using fractions or decimals.
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, but numbers like 9.75 or the square root of 2 are not.
Integers are also important in algebra and number theory. They form the smallest group and ring that include natural numbers, making them a key part of many mathematical ideas and calculations.
History
The word integer comes from the Latin word meaning "whole" or "untouched." Originally, only positive integers were considered, like 1, 2, and 3. Over time, mathematicians recognized the usefulness of including negative numbers, such as −1 and −2, in this group.
By the late 19th century, the idea of a set of integers was introduced. Today, we commonly use the letter Z to represent all integers, thanks to a suggestion from a German word for "numbers." This notation became standard in the middle of the 20th century.
Algebraic properties
Integers, which include zero, positive numbers like 1 and 2, and negative numbers like -1 and -2, have special rules when you add, subtract, or multiply them. For example, adding or multiplying any two integers always gives you another integer. However, dividing two integers doesn’t always work out to 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 straightforward, although not every integer can be reversed under multiplication (for example, 2 doesn’t have a whole number that multiplies back to 1). These properties help mathematicians understand how numbers behave in various situations.
| 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.
This article is a child-friendly adaptation of the Wikipedia article on Integer, available under CC BY-SA 4.0.