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

$a\; +\; b\; =\; c.\backslash ,$

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.

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 ^*
• S(a) + S(b) = S(a + b) ^*

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

$(a\; +\; b)\; +\; c\; =\; a\; +\; (b\; +\; c);\backslash ,$ (proof)

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

$a\; +\; b\; =\; b\; +\; a;\backslash ,$ (proof)

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

$a\; +\; 0\; =\; 0\; +\; a\; =\; a.\backslash ,$ (proof)

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

Elementary arithmetic

Addition des entiers naturels | 加法