Definitions

natural numbers

Addition of natural numbers is the most basic arithmetic binary 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.

Repeated addition of natural numbers is generalized as multiplication of natural numbers

Notation and terms

The operation of addition, commonly written as the infix operator "+", is a function +: N × NN. 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

Assume that $mathbb\left\{N\right\}$, the set of Natural Numbers, has been defined by the Peano postulates as follows:

• $0 in mathbb\left\{N\right\}$
• $n in mathbb\left\{N\right\} rarr S\left(n\right) in mathbb\left\{N\right\}$

Where $S\left(a\right)$ is the successor of a defined as

• $S : mathbb\left\{N\right\} rarr mathbb\left\{N\right\}$

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 A1 and A2:

• $a + 0 = a$ [A1]
• $a + S\left(b\right) = S\left(a + b\right)$ [A2]

In words, this says that adding zero, which is the additive-identity, 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.

By the recursion theorem, this defines a unique function $+ : mathbb\left\{N\right\} rarr mathbb\left\{N\right\}$, in other words a unique function "+" that maps $mathbb\left\{N\right\}$ back onto $mathbb\left\{N\right\}$.

Properties

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

Let $a, b, c in mathbb\left\{N\right\}$

$\left(a + b\right) + c = a + \left(b + c\right);,$

$a + b = b + a;,$

$a + 0 = 0 + a = a.,$
where 0 is known as the additive identity under $mathbb\left\{N\right\}$. Strictly it is defined as a right identity, but by commutativity, is becomes the general identity in the group $\left(mathbb\left\{N\right\}, +\right)$

Together, these three properties show that the set of natural numbers $mathbb\left\{N\right\}$ under addition is a commutative monoid.