1 Definition and notation
Natural numbers are the integers used to count objects and to label positions in an ordered sequence. They are the earliest numbers encountered in arithmetic and serve as the primary building blocks for later extensions such as the integers, rational numbers, and real numbers.
1.1 Common conventions
Different mathematical communities adopt slightly different starting points for the symbol set that they call “natural numbers,” particularly regarding whether 0 is included.
1.1.1 Inclusion of zero
In many elementary and computer-science contexts, the natural numbers are written as 0, 1, 2, 3, and so forth. This choice simplifies certain formulas and indexing schemes, because counting and indexing often begin at 0.
1.1.2 Positive integers convention
In other settings, especially in some older or more traditional school curricula and portions of pure mathematics, natural numbers are taken to be 1, 2, 3, and so on. Under this convention, the set of natural numbers is exactly the positive integers.
1.2 Set-theoretic definitions
Natural numbers can be defined rigorously using axiomatic or set-theoretic constructions, ensuring that the meaning of “natural number” does not depend on intuition.
1.2.1 Von Neumann ordinals
One common set-theoretic approach represents each natural number as the set of all smaller natural numbers. In this framework, 0 is represented by the empty set, 1 by the set containing the empty set, 2 by the set containing 0 and 1, and so on. The membership relation then encodes the usual ordering of natural numbers.
1.2.2 Peano axioms
Another classical approach defines natural numbers as the terms in a structure generated from an initial element using a successor operation, subject to axioms that capture the basic behavior of counting. This method emphasizes the logical properties needed to develop arithmetic, rather than the underlying set representation.
1.3 Basic notation
Natural numbers are often denoted by symbols such as ℕ. Depending on convention, ℕ may include 0 or start at 1; mathematicians usually specify which convention is being used in a given work. Variables like *m*, *n*, and *k* typically range over natural numbers in formulas.
2 Arithmetic properties
Arithmetic on natural numbers is governed by operations such as addition and multiplication, along with an order relation that allows comparison.
2.1 Addition
Addition models combining counts: for natural numbers *m* and *n*, the sum *m + n* represents “adding n more.”
2.1.1 Closure under addition
The closure property states that the sum of two natural numbers is again a natural number. Under either the 0-included or positive-integer convention, adding two permitted values yields another permitted value.
2.1.2 Associativity and commutativity
Addition satisfies associativity: *(m + n) + k = m + (n + k)*. It also satisfies commutativity: *m + n = n + m*. These algebraic laws justify rearranging addends when computing totals.
2.2 Multiplication
Multiplication models repeated addition and scaling of counts.
2.2.1 Closure under multiplication
If *m* and *n* are natural numbers, then *m·n* is also a natural number. This ensures multiplication does not leave the number system when applied repeatedly.
2.2.2 Distributive law
Multiplication distributes over addition: *m·(n + k) = m·n + m·k*. This law is essential in algebraic manipulation and in deriving many results in number theory.
2.3 Ordering
Natural numbers carry a natural notion of “less than” and “greater than,” compatible with addition and multiplication.
2.3.1 Less than and greater than
The relation *m < n* means that *m* precedes *n* in the standard order. Equivalently, for appropriate conventions, *m < n* if and only if there exists a nonzero natural number *k* such that *n = m + k*.
2.3.2 Well-ordering property
The well-ordering property asserts that every nonempty set of natural numbers has a least element. This feature underlies proofs by minimal counterexample and many inductive arguments.
3 Number-theoretic structure
Number theory studies fine-grained properties of natural numbers, particularly through divisibility and prime factorization.
3.1 Divisibility
Divisibility formalizes the idea that one count can be partitioned into equal blocks of another.
3.1.1 Factors and multiples
A natural number *a* is a factor (divisor) of *b* if there exists a natural number *c* such that *b = a·c*. In this case, *b* is a multiple of *a*.
3.1.2 Common divisors
If two natural numbers share divisors, those divisors are called common divisors. The set of common divisors is finite and can be organized using ordering, enabling algorithmic computations.
3.2 Prime numbers
Primes are the basic building blocks for multiplication in the sense that they cannot be decomposed into smaller factors (within the natural numbers).
3.2.1 Composite numbers
A natural number greater than the relevant unit value (typically greater than 1, under the positive-integer convention) is composite if it has a nontrivial factorization, meaning it can be written as a product of two smaller natural numbers.
3.2.2 Fundamental theorem of arithmetic
The fundamental theorem of arithmetic states that every integer greater than the unit value (again typically greater than 1) can be expressed as a product of primes, and this factorization is unique up to the order of the prime factors. This theorem provides a structural explanation for many divisibility properties.
3.3 Greatest common divisor and least common multiple
Two key invariants attached to a pair of numbers are the greatest common divisor (gcd) and least common multiple (lcm).
The gcd of *m* and *n* is the largest natural number that divides both. The lcm is the smallest natural number that is divisible by both. Using prime factorizations, the gcd and lcm can be determined by comparing exponents of primes in the factored forms.
4 Subsets and special classes
Natural numbers can be grouped into subsets according to properties preserved by arithmetic operations.
4.1 Even and odd numbers
Even and odd numbers partition natural numbers into two complementary classes.
4.1.1 Parity
A number is even if it is divisible by 2; otherwise it is odd. Parity is central because it interacts predictably with addition and multiplication: for example, adding two even numbers gives an even result, while adding two odd numbers yields an even number.
4.1.2 Alternating patterns
Parity creates alternating sequences in counting: 1, 2, 3, 4, … alternates odd and even. Such alternation is exploited in proofs and in algorithm design where two-case logic is efficient.
4.2 Square numbers
Square numbers are values of the form *n²* for natural numbers *n*.
4.2.1 Cubes and higher powers
Similarly, cubes (*n³*) and higher powers form structured sets. Many properties of these sequences—such as growth rates and spacing—can be studied using modular arithmetic and inequalities.
4.3 Perfect numbers and other special forms
Some natural numbers satisfy additional internal relationships between their factors and sums. For instance, perfect numbers are defined using the sum of their positive divisors (excluding the number itself), and they appear rarely. Other special forms include abundant and deficient numbers, which classify numbers by comparing the sum of their proper divisors to the number.
5 Representation and notation systems
Natural numbers are written in multiple ways depending on the base and the symbolic conventions used.
5.1 Base-10 notation
Base-10 is the standard decimal system used in everyday arithmetic.
5.1.1 Place value
In base-10, each digit’s position indicates a power of 10. A numeral such as *3 4 5* represents *3·10² + 4·10¹ + 5·10⁰*, reflecting the place-value weighting.
5.1.2 Decimal expansions
Every natural number has a finite base-10 expansion. These expansions are central for exact computation and for understanding how arithmetic algorithms work digit by digit.
5.2 Other numeral systems
Changing the base yields different digit sets and place values while representing the same underlying numbers.
5.2.1 Binary
Binary uses base 2, employing digits 0 and 1. Each position represents a power of 2, making binary representation particularly efficient for digital circuitry.
5.2.2 Hexadecimal
Hexadecimal uses base 16 and commonly employs digits 0–9 and letters A–F to represent values 10 through 15. It is frequently used in computing because it provides a compact way to write binary data.
5.3 Written and symbolic representation
Natural numbers can be represented symbolically in expressions and equations using standard arithmetic signs and notation. For example, expressions like *n!*, *2^n*, and *C(n,k)* denote factorial, exponentiation, and combinations respectively, each relying on natural-number inputs.
6 Construction of the number system
Beyond definitions as sets, natural numbers can be constructed and manipulated through logical principles that generate arithmetic operations.
6.1 Axiomatic foundations
Axiomatic systems specify which properties hold and how arithmetic concepts must behave.
6.1.1 Peano arithmetic
Peano arithmetic is an axiomatic framework designed to capture the arithmetic of natural numbers. It includes an initial element, a successor function, and rules that characterize how successor-generated terms behave, allowing formal proofs about arithmetic statements.
6.1.2 Induction
Induction is a core proof method: to show that a property holds for all natural numbers, it is enough to verify it for an initial value and then show that if it holds for an arbitrary number, it also holds for its successor. This technique is fundamental for establishing results about recursively defined functions and for proving general arithmetic laws.
6.2 Recursive definitions
Many arithmetic functions on natural numbers are defined in terms of smaller cases, emphasizing computability and structure.
6.2.1 Successor function
The successor function maps each natural number to the next one in sequence. Many constructions build operations by repeatedly applying successor-based rules.
6.2.2 Recurrence relations
Recurrence relations define a function by specifying its values at small inputs and relating later values to earlier ones. Such relations appear in combinatorics, discrete dynamical systems, and algorithm analysis.
7 Applications and extensions
Natural numbers underpin a wide range of theoretical and practical contexts, from basic counting to algorithmic design.
7.1 Counting and enumeration
The most direct application of natural numbers is counting: determining the size of finite sets and producing labels for ordered collections. Enumeration uses natural numbers to index elements and track positions.
7.2 Combinatorics
Combinatorics uses natural numbers to count possibilities such as selections, arrangements, and partitions. Quantities like binomial coefficients and factorials express how many structures exist under given constraints.
7.3 Computer science uses
Natural numbers are pervasive in computing because data sizes, loop counters, and indexing are inherently discrete.
7.3.1 Natural numbers in algorithms
Algorithm analysis often measures time or steps in terms of natural numbers. Loop invariants, termination conditions, and recurrence-based running time estimates rely on discrete iteration counts.
7.3.2 Data representation
In digital systems, numeric data typically corresponds to natural-number encodings using fixed-width binary representations. Even when the system stores signed or real numbers, conversion and scaling frequently involve operations grounded in natural-number arithmetic.
7.4 Extensions to integers and beyond
Natural numbers extend naturally to the integers by allowing negative values, and further to rationals and reals by permitting division and limits. These extensions preserve arithmetic structure while expanding the range of solvable equations, enabling a broader mathematical toolkit.