Integer

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Relative numbers representation

Integer is a fundamental concept in mathematics, specifically within the area of number theory. An integer is a whole number that can be positive, negative, or zero. This set of numbers is denoted by the symbol Z, which comes from the German word "Zahlen" (meaning "numbers"). Integers include the counting numbers (1, 2, 3, ...), their negatives (-1, -2, -3, ...), and the number zero. The set of all integers is often represented as {..., -3, -2, -1, 0, 1, 2, 3, ...}.

Properties

Integers possess several important properties that make them a fundamental building block in various areas of mathematics and its applications. Some of these properties include:

• Closure: The sum or product of any two integers is also an integer.
• Associativity: When adding or multiplying integers, the way in which the integers are grouped does not change the result.
• Identity elements: The number 0 acts as an additive identity (adding 0 to any integer does not change the integer), and the number 1 acts as a multiplicative identity (multiplying any integer by 1 does not change the integer).
• Additive inverses: For every integer, there exists another integer that, when added together, equals zero. This second integer is known as the additive inverse.
• Commutativity: The order in which two integers are added or multiplied does not change the result.

Division and Rational Numbers

While integers are closed under addition, subtraction, and multiplication, they are not closed under division. Dividing two integers does not always result in another integer. For example, 1 divided by 2 does not yield an integer but a rational number. A rational number is defined as the quotient of two integers, where the denominator is not zero.

Applications

Integers are used in a wide range of fields beyond pure mathematics. In computer science, integers are used to count, to identify objects, and to perform calculations. In engineering, integers can represent discrete values such as the number of objects or the number of operations. In economics, integers are used to count items such as people, companies, transactions, and more.

Number Theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. It deals with the properties and relationships of integers without the use of geometrical methods. Some of the questions explored in number theory include the distribution of prime numbers, the solution of integer equations, and the exploration of integer sequences.

See Also

• Rational number
• Prime number
• Number theory
• Computer science
• Engineering
• Economics