Natural numbers

Natural numbers (also called whole numbers or counting numbers) are 0, 1, 2, 3, 4 and so on, used for counting objects and ordering positions. They are the earliest type of numbers that humans learned, and form the foundation on which other number systems are built.

The set of natural numbers is infinite: every number has a successor that is one greater. This set cannot be enumerated in a complete list, but it can be counted—meaning each number can be assigned a position.

Is 0 a natural number?

There is no universal agreement among mathematicians. Some—particularly in logic and set theory—begin with 0. Others, notably in number theory, begin with 1. Because of this disagreement, most textbooks state their convention explicitly at the outset.

Operations

Natural numbers can be added and multiplied, and the result is always another natural number. Subtraction and division, however, often fall outside the set: 3 minus 5 and 1 divided by 2 are not natural numbers. For this reason, integers and fractions were introduced.

Logical foundations

In the 19th century, Giuseppe Peano presented fundamental axioms that define natural numbers: there exists a starting number; every number has a successor; no two distinct numbers share the same successor; and the principle of induction—a method for proving that a given property holds for every natural number. These axioms made natural numbers a rigorously defined subject, not merely an intuitive notion of counting.

Applications

Natural numbers serve two purposes: answering "how many?" and answering "what position?" In computer science, they are used for measuring data size, loop iteration, and array indexing. Number theory studies properties of natural numbers such as primality, divisibility, and multiplication.