entier

An integer, or entier, is a whole number that can be positive, negative, or zero. In mathematics the set of all integers is denoted by Z, derived from the German word Zahlen. In French usage, nombres entiers relatifs refer to the integers, while nombres entiers naturels denote the nonnegative subset, depending on convention.

Integers are closed under addition, subtraction, and multiplication; they form a ring with identity. Division is not closed: the quotient of two integers need not be an integer. The integers include zero and all sign variations of the natural numbers.

Every integer is even or odd. The positive integers factor uniquely into primes, up to order and sign (Fundamental Theorem of Arithmetic). The greatest common divisor of two integers a and b can be expressed as ax + by for some integers x and y (Bezout's identity).

The division algorithm states that for any integers a and b with b not equal to zero, there exist unique q and r such that a = bq + r and 0 ≤ r < |b|. This underpins modular arithmetic, where a ≡ b (mod n) means n divides a − b.

In computing and applications, integers are used for exact counting, indexing, and discrete calculations. Integer types may be signed or unsigned and vary in range and precision, influencing behavior such as overflow.