Wholenumbers

Whole numbers are the non-negative integers, typically written as 0, 1, 2, 3, and so on. Depending on the context, the term may include or exclude zero, but most mathematical references treat zero as a member, making the whole numbers the non-negative part of the integers. They are often denoted as Z≥0 to distinguish them from negative integers. In many contexts, natural numbers and whole numbers overlap, with some definitions of natural numbers starting at 1 and others including 0.

The set of whole numbers is closed under addition and multiplication: the sum or product of any two whole numbers is another whole number. Subtraction and division, however, are not generally closed within this set; for example, 7 − 9 equals −2, which is not a whole number, and dividing two whole numbers may yield fractions or decimals.

Whole numbers form a discrete, countably infinite, ordered set with a first element, 0, that extends without bound as numbers increase. They provide a foundation for counting objects, defining arithmetic in number theory and combinatorics, and they sit as a subset of the integers and a superset of the natural numbers in contexts where zero is included.

In common usage, natural numbers N may be defined as {1, 2, 3, ...} or as {0, 1, 2, ...}, so the two sets often coincide with the whole numbers. In computer science, zero-based indexing uses whole numbers to label positions in arrays and data structures.

Applications include counting, arithmetic, combinatorics (such as binomial coefficients with nonnegative integers), and various mathematical constructions like factorials and sequences. Whole numbers thus provide a simple, well-ordered basis for arithmetic and are studied in elementary number theory and discrete mathematics.