Integers

An integer is a number with no fractional part. It comprises the natural numbers (1, 2, 3, ...), their negatives (−1, −2, −3, ...), and zero. In mathematics it is denoted by the letter Z, derived from the German word 'Zahlen' (numbers).

Integers are an extension of the natural numbers, created to allow subtraction to always yield a result. For example, 3 − 5 has no answer among natural numbers alone, but the integers give −2.

Arithmetic operations

Addition, subtraction, and multiplication of any two integers always produce an integer. This property is called closure. Division is not closed: 7 divided by 2 is not an integer. For this reason the integers form a structure called a ring (halka), an important example in modern mathematics.

Ordering

Integers can be ordered: each one has a predecessor and a successor. They can therefore be represented on a number line extending in both directions. There is no greatest or least integer, but any finite subset has a maximum and minimum.

Divisibility and prime numbers

Study of integers underlies the branch of mathematics called number theory. There one examines divisors, prime numbers, and divisibility properties. The fundamental theorem of arithmetic states that every integer greater than 1 can be broken down into a product of prime numbers in exactly one way, disregarding order.

Applications

In computers, integers are stored in fixed formats such as 32-bit or 64-bit, which creates upper and lower limits. When a result exceeds these bounds, overflow occurs. Integers are also used in counting, graph theory, and cryptography.