Addition of natural numbers is the most basic arithmetic operation. In its simplest form, addition combines two numbers (terms, summands), the augend and addend, into a single number, the sum.

Notation and terms

The operation of addition, commonly written as the infix operator "+", is a function + : N × N → N. For natural numbers a, b, and c, we write Infix has similar meanings in linguistics and mathematics. ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... Partial plot of a function f. ...

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

Definition

We let S(a) denote the successor of a as defined in the Peano postulates. In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ...

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]
• S(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 × N → N.

Properties

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

• Associativity: for all natural numbers a, b, and c, we have

(proof)

• Commutativity: for all natural numbers a and b, we have

(proof)

• Identity element: for all natural numbers a, we have

(proof)

Together, these three properties show that the set of natural numbers N under addition is a commutative monoid. In mathematics, associativity is a property that a binary operation can have. ... Here we will define it from Peanos axioms (see natural number) and prove some simple properties. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... Here we will define it from Peanos axioms (see natural number) and prove some simple properties. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... Here we will define it from Peanos axioms (see natural number) and prove some simple properties. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ...