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Addition of natural numbers is the most basic arithmetic operation. The operation addition takes two natural numbers, 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.

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

a + b = c

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

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 × N → N.

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.

• Addition of natural numbers/Proofs