In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. The terms positive integers, non-negative integers, whole numbers, and counting numbers are also used. The set of the natural numbers is commonly denoted with a bold N or a blackboard bold ⁠${\displaystyle \mathbb {N} }$⁠.

The natural numbers are used for counting, and for labeling the result of a count, like "there are seven days in a week", in which case they are called cardinal numbers. They are also used to label places in an ordered series, like "the third day of the month", in which case they are called ordinal numbers. Natural numbers may also be used to label, like the jersey numbers of a sports team; in this case, they have no specific mathematical properties and are called nominal numbers.

Two natural operations are defined on natural numbers, addition and multiplication. Arithmetic is the study of the ways to perform these operations. Number theory is the study of the properties of these operations and their generalizations. Much of combinatorics involves counting mathematical objects, patterns and structures that are defined using natural numbers.

Many number systems are built from the natural numbers and contain them. For example, the integers are made by including 0 and negative numbers. The rational numbers add fractions, and the real numbers add all infinite decimals. Complex numbers add the square root of −1. This makes up natural numbers as foundational for all mathematics.

The term natural numbers has two common definitions: either 0, 1, 2, ... or 1, 2, 3, .... Because there is no universal convention, the definition can be chosen to suit the context of use. To eliminate ambiguity, the sequences 1, 2, 3, ... and 0, 1, 2, ... are often called the positive integers and the non-negative integers, respectively.

The phrase whole numbers is frequently used for the natural numbers that include 0, although it may also mean all integers, positive and negative. In primary education, counting numbers usually refer to the natural numbers starting at 1, though this definition can vary.

The set of all natural numbers is typically denoted N or in blackboard bold as ${\displaystyle \mathbb {N} .}$ Whether 0 is included is often determined by the context but may also be specified by using ${\displaystyle \mathbb {N} }$ or ${\displaystyle \mathbb {Z} }$ (the set of all integers) with a subscript or superscript. Examples include ${\displaystyle \mathbb {N} _{1}}$, or ${\displaystyle \mathbb {Z} ^{+}}$ (for the set starting at 1) and ${\displaystyle \mathbb {N} _{0}}$ or ${\displaystyle \mathbb {Z} ^{0+}}$ (for the set including 0).

An intuitive and implicit understanding of natural numbers is developed naturally through using numbers for counting, ordering and basic arithmetic. Within this are two closely related aspects of what a natural number is: the size of a collection; and a position in a sequence.