Natural number

A natural number is a number used for counting or ordering. Conventions differ on whether the sequence begins with zero or one.

Peano axioms: Axioms characterizing the natural numbers through zero, succession, and induction. They give a foundational account of natural numbers and their successor operation.
Addition: An arithmetic operation combining quantities into a sum. Repeated successor steps define addition on natural numbers.
Counting: The process of assigning number words or symbols to items to determine how many there are. Counting is the basic use that gives natural numbers their cardinal role.
Whole number: A nonnegative integer, commonly an element of {0, 1, 2, …}. In common usage it includes zero, while some definitions of natural number do not.
Successor function: A function that assigns each natural number its next number. Successor formalizes the step from one natural number to the next.
Multiplication: An arithmetic operation that combines equal groups or scales a quantity. Repeated addition defines multiplication for natural numbers.
Ordinal number: A number describing an item's position in an ordered sequence. Natural numbers can name finite positions as well as quantities.
Rational number: A number expressible as the quotient of two integers with nonzero denominator. Rationals include fractions that are not natural numbers.
Mathematical induction: A proof method establishing a property for all natural numbers from a base case and an inductive step. Induction exploits the successor structure to prove statements across the sequence.
Exponentiation: An operation that raises a base to a whole-number power by repeated multiplication. Natural-number exponents encode repeated multiplication, with a conventional zero-power case.

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