Definition:Addition

Naturally Ordered Semigroup
Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup.

Then the operation $\circ$ in $\left({S, \circ, \preceq}\right)$ is called addition.

Peano Structure
Let $\left({P, 0, s}\right)$ be a Peano structure.

Then we define the binary operation $+$ in $P$ as follows:
 * $\forall x, y \in P: \begin{cases}

0 + y & = y \\ s \left({x}\right) + y & = s \left({x + y}\right) \end{cases}$

This operation is called addition, and it exists and is unique.

Natural Numbers
The addition operation in the domain of natural numbers $\N$ is defined in Definition by Induction of Natural Number Addition and written $+$.

As the Natural Numbers are a Naturally Ordered Semigroup, it is appropriate to write this set as $\left({\N, +, \le}\right)$.

From this, it can be seen that $+$ corresponds with the operation $\circ$ in the structure $\left({S, \circ, \preceq}\right)$.

Integers
The addition operation in the domain of integers $\Z$ is written $+$.

As the set of integers is the Inverse Completion of Natural Numbers, it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times \N$ where two tuples are equivalent if the Unique Minus between the two elements of each tuple is the same.

Thus addition can be formally defined on $\Z$ as the operation induced on those equivalence classes as specified in the definition of integers.

That is, the integers being defined as all the Unique Minus congruence classes, integer addition can be defined directly as the operation induced by natural number addition on these congruence classes:


 * $\forall \left({a, b}\right), \left({c, d}\right) \in \N \times \N: \left[\!\left[{a, b}\right]\!\right]_\boxminus + \left[\!\left[{c, d}\right]\!\right]_\boxminus = \left[\!\left[{a + c, b + d}\right]\!\right]_\boxminus$

Modulo Addition
The addition operation on $\R_z$, the set of set of all residue classes modulo $z$, is defined by the rule:


 * $\left[\!\left[{a}\right]\!\right]_z +_z \left[\!\left[{b}\right]\!\right]_z = \left[\!\left[{a + b}\right]\!\right]_z$

Although the operation of addition modulo $z$ is denoted by the symbol $+_z$, if there is no danger of confusion, the symbol $+$ is often used instead.

More usually, though, the notation $a + b \left({\bmod\, z}\right)$ is used instead of $\left[\!\left[{a}\right]\!\right]_z +_z \left[\!\left[{b}\right]\!\right]_z$.

It means the same thing and, although obscuring the true meaning behind modulo arithmetic, is more streamlined and less unwieldy.

See modulo addition.

Also see

 * Identity Element of Addition on Numbers
 * Inverse Elements for Addition on Numbers


 * Quaternion Addition