Addition of natural numbers

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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.\,]

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 [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

[(a + b) + c = a + (b + c);\,] (proof)

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

[a + b = b + a;\,] (proof)

Identity element: for all natural numbers a, we have

[a + 0 = 0 + a = a.\,] (proof)

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

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