entiers

Entiers, in mathematics, refer to the set of whole numbers including their negatives and zero. In English, they are called integers and are usually denoted by Z. An integer is any number such as -3, 0, 7, or 42. The set of all integers can be written as {..., -2, -1, 0, 1, 2, ...}.

The integers have several important algebraic properties. They are closed under addition, subtraction, and multiplication: adding, subtracting, or multiplying two integers always yields another integer. Division, however, is not closed in the integers, since the quotient of two integers need not be an integer. The integers form a ring with unity (1), meaning they support addition and multiplication with the usual distributive, associative, and identity laws. Every integer has an additive inverse (for example, the inverse of 5 is -5), and the usual total order extends to Z, making it a totally ordered ring. For many questions, divisibility and modular arithmetic (working “mod n”) are central topics within the integers.

The set of integers is infinite but countable, meaning there exists a one-to-one correspondence with the natural numbers. It contains the natural numbers as a subset, and every integer is also a rational number. This places Z between the natural numbers and the rationals in the usual hierarchy of number systems.

Applications of the integers include number theory, cryptography, computer algorithms, and error-detection methods. They underpin concepts such as prime factorization, greatest common divisor, and modular arithmetic, which are foundational in both theoretical and applied disciplines. The term entiers is the French equivalent of this concept.