- Addition of natural numbers
**Addition of natural numbers**is the most basic arithmetic binary operation. The operationaddition takes twonatural number s, 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 infixoperator "+", is a function +:**N**×**N**→**N**. Fornatural number s "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 $mathbb\{N\}$, the set of Natural Numbers, has been defined by the

Peano postulates as follows:* $0\; in\; mathbb\{N\}$

* $n\; in\; mathbb\{N\}\; arr\; S(n)\; in\; mathbb\{N\}$Where $S(a)$ is the successor of a defined as

* $S\; :\; mathbb\{N\}\; arr\; mathbb\{N\}$

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]

* $a\; +\; S(b)\; =\; S(a\; +\; b)$ [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 $+\; :\; mathbb\{N\}\; arr\; mathbb\{N\}$, in other words a unique function "+" that maps $mathbb\{N\}$ back onto $mathbb\{N\}$.

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

: Let $a,\; b,\; c\; in\; mathbb\{N\}$

* Associativity: (proof):$(a\; +\; b)\; +\; c\; =\; a\; +\; (b\; +\; c);,$

* Commutativity: (proof):$a\; +\; b\; =\; b\; +\; a;,$

*Identity element : (proof):$a\; +\; 0\; =\; 0\; +\; a\; =\; a.,$ :where 0 is known as the additive identity under $mathbb\{N\}$. Strictly it is defined as a right identity, but by commutativity, is becomes the general identity in the group $(mathbb\{N\},\; +)$Together, these three properties show that the set of natural numbers $mathbb\{N\}$ under addition is a commutative

monoid .**ee also***

Addition of natural numbers/Proofs *Method to Derive Polynomial Representations that Sum Natural Numbers with Exponents [

*http://upload.wikimedia.org/wikipedia/commons/6/62/Sum_of_i.pdf*]

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2010.*