Integer

In mathematics, an integer is any whole number that can be positive, negative, or zero, but not a fraction or decimal. The set of integers is represented by the symbol ℤ (from the German word "Zahlen" meaning numbers) and includes: ..., -3, -2, -1, 0, 1, 2, 3, ... extending infinitely in both positive and negative directions. Integers are fundamental to number theory and form the building blocks for more complex number systems including rational numbers (fractions), real numbers (including decimals and irrational numbers), and complex numbers.

Integers have several important properties that make them essential in mathematics and everyday applications. They are closed under addition, subtraction, and multiplication (meaning these operations always produce another integer), though division doesn't always yield an integer result (5 ÷ 2 = 2.5, which is not an integer). Every integer has an additive inverse: for any integer n, there exists -n such that n + (-n) = 0. Integers can be classified as positive (greater than zero, also called natural numbers when including zero or not depending on convention), negative (less than zero), or zero itself. The concept of integers is used in countless practical applications: counting objects, representing gains and losses in finance (profit as positive integers, loss as negative integers), measuring temperature (degrees above and below zero), recording elevations (above and below sea level), and computer programming where integer data types are fundamental. Understanding integers and operations with them—particularly rules for adding, subtracting, multiplying, and dividing numbers with different signs—is essential for algebra and higher mathematics. The simplicity yet completeness of integers makes them a cornerstone of mathematical thinking from elementary arithmetic through advanced mathematical theories.