'''Addition of natural numbers''' is the most basic arithmetic operation. The operation [[addition]] takes two [[natural number]]s, the augend and addend, and produces a single number, the sum. The set of natural numbers will be denoted by '''N''', and "0" will be used to denote the natural number which is not the successor of any other natural number.

== Notation and terms ==

The operation of '''addition''', commonly written as the [[Infix notation|infix]] [[operator]] "+", is a [[function (mathematics)|function]] +: '''N''' × '''N''' → '''N'''. For [[natural number]]s ''a'', ''b'', and ''c'', we write

:<math>a + b = c</math>

Here, ''a'' is the ''augend'', ''b'' is the ''addend'', and ''c'' is the ''sum''.

== Definition ==

Assume that '''N''' has been defined by the [[Peano postulates]]. We let ''S''(''a'') denote the ''successor of a''.

Addition is defined inductively by fixing the augend. In other words, we let ''a'' be any arbitrary, but fixed natural number, and we then make the following definitions:

* ''a'' + 0 = ''a'' [A1]
* ''a'' + ''S''(''b'') = ''S''(''a'' + ''b'') [A2]

By the recursion theorem, this defines a unique function "''a'' +": '''N''' → '''N'''. In words, it says that adding zero to ''a'' gives back ''a'', and that applying the successor function to the addend has the effect of applying the successor function to the sum.

Since ''a'' was an arbitrary natural number, we can "put together" all these functions into a single binary operation '''N''' &times; '''N''' → '''N'''.

== Properties ==

The following are three immediate and important properties of addition which can be deduced from the definition.

* [[associative|Associativity]]: for all natural numbers ''a'', ''b'', and ''c'', we have
:<math>(a + b) + c = a + (b + c);\,</math> ([[Addition of natural numbers/proofs#Proof of associativity|proof]])
* [[commutative|Commutativity]]: for all natural numbers ''a'' and ''b'', we have
:<math>a + b = b + a;\,</math> ([[Addition of natural numbers/proofs#Proof of commutativity|proof]])
* [[Identity element]]: for all natural numbers ''a'', we have
:<math>a + 0 = 0 + a = a.\,</math> ([[Addition of natural numbers/proofs#Proof of identity element|proof]])

Together, these three properties show that the set of natural numbers '''N''' under addition is a commutative [[monoid]].

==See also==
*[[Addition of natural numbers/Proofs]]
[[Category:Elementary arithmetic]]

[[fr:Addition des entiers naturels]]
[[ja:加法]]