natural number, any number in the set of positive integers {1, 2, 3…} and sometimes zero. The term natural likely refers to humanity’s innate understanding and use of positive discrete values. Sometimes called “counting numbers,” natural numbers predate recorded history, as it is believed that prehistoric humans had some sense of determining differences in quantities. Whether zero is considered a natural number differs across definitions, and there is no general consensus on its inclusion or exclusion from the set.

Definition

Mathematicians have formulated various definitions for natural numbers. One of the most popular is based on the principles of set theory, and it proposes that every natural number can be defined as its own set. The union ∪ of these individual sets then composes the complete set of natural numbers (ℕ), a process aided by successor functions, which establish the successor of a number as S(x) = x∪{x}. The proof begins by establishing 0 as the empty set, a set with no elements, before moving to define 1, its successor.0 = {}1 = 0∪{0} = {{}} = {0}.

The rest of the set of natural numbers ℕ can be defined through the resulting pattern:2 = 1∪{1} = {1, {1}} = {{}, {{}}} = {0, 1}3 = 2∪{2} = {2, {2}} = {{}, {{}}, {{{}}}} = {0, 1, 2}4 = 3∪{3} = {3, {3}} = {{}, {{}}, {{{}}}, {{{{}}}}} = {0, 1, 2, 3}n = (n − 1)∪{n − 1} = {n − 1, {n − 1}} = {{}, {{}},…} = {0, 1, 2,…, n − 2, n − 1}.

A second popular definition is based on the Peano axioms. Introduced by Italian mathematician Giuseppe Peano in 1889, the method provides the foundation for the infinite set of natural numbers through five axioms:

• 1. Zero is a natural number.
• 2. Every natural number has a successor in the natural numbers.
• 3. Zero is not the successor of any natural number.
• 4. If the successor of two natural numbers is the same, then the two numbers are the same.
• 5. If a set contains zero and the successor of every number is in the set, then the set contains the natural numbers.

The fifth axiom, known as the axiom of induction, allows for descriptions of an infinite number of cases without the need for an infinite number of proofs.

History

Natural numbers presumably existed long before recorded history. The earliest representations of counting numbers—records including notches or scratches on sticks, stones, pottery, and other objects—likely came long after humans learned to count. As history evolved and number systems became more sophisticated, specific terminology emerged to describe the set of numbers used for counting. One of the earliest references is from French mathematician Nicolas Chuquet, who described the sequence 1, 2, 3, 4,… as “appellee par les anciens progression naturelle” (“called by the ancients natural progression”) in 1484. English mathematician William Emerson is credited as one of the first to use the phrase natural number in English, in his book The Method of Increments (1763).

In the following centuries mathematicians including Peano, Richard Dedekind, and Gottlob Frege sought to define and examine the properties of natural numbers. Although a seemingly simple concept, definitions vary on whether the set of natural numbers includes zero. Many formal definitions, such as Peano’s updated axioms, include zero in the set of natural numbers. Other definitions leave it out and instead refer to the set as “whole numbers” when zero is included. There is no consensus on whether the digit is considered a natural number; indeed, some advise that its status should be decided on the basis of convenience.

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Properties

Natural numbers possess properties that are fundamental for mathematics—notably, for addition and multiplication. Using the successor function S defined above, addition within the set of natural numbers can be defined using n and m as natural numbers:n + S(m) = S(n + m).

Multiplication within the set of natural numbers can be similarly defined:n × S(m) = (n × m) + nn × 0 = 0.

Both addition and multiplication satisfy the commutativen + m = m + nn × m = m × nand associative properties,l + (m + n) = (l + m) + nl × (m × n) = (l × m) × nas well as the distributive propertyl × (m + n) = (l × m) + (l × n),where l, m, and n are natural numbers. Subtraction and division are not considered properties of natural numbers, since performing those operations with natural numbers may result in values outside the set of natural numbers.

The application of natural numbers extends to providing the basis for counting and ordering objects. Cardinal numbers, which are those used to convey the size of a set, may be interpreted as an extension of natural numbers when used to count the number of objects in a finite set. Ordinal numbers may also be considered an extension of natural numbers. Instead of counting objects, a set of well-ordered natural numbers can be used to signify the order of a group of objects.