# Definition:Addition

## Definition

**Addition** is the basic operation $+$ everyone is familiar with.

For example:

- $2 + 3 = 5$
- $47 \cdotp 3 + 191\cdotp 4 = 238 \cdotp 7$

### Natural Numbers

Let $\N$ be the natural numbers.

When $\N$ is constructed as a Peano structure $\struct {P, 0, s}$, the **addition operation** $+$ can be defined as:

Let $\struct {P, 0, s}$ be a Peano structure.

The binary operation $+$ is defined on $P$ as follows:

- $\forall m, n \in P: \begin{cases} m + 0 & = m \\ m + \map s n & = \map s {m + n} \end{cases}$

This operation is called **addition**.

### Integers

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

We have that the set of integers is the Inverse Completion of Natural Numbers.

Thus 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 difference 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 difference congruence classes, integer addition can be defined directly as the operation induced by natural number addition on these congruence classes:

- $\forall \tuple {a, b}, \tuple {c, d} \in \N \times \N: \eqclass {a, b} \boxminus + \eqclass {c, d} \boxminus = \eqclass {a + c, b + d} \boxminus$

### Modulo

Let $m \in \Z$ be an integer.

Let $\Z_m$ be the set of integers modulo $m$:

- $\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \ldots, \eqclass {m - 1} m}$

where $\eqclass x m$ is the residue class of $x$ modulo $m$.

The operation of **addition modulo $m$** is defined on $\Z_m$ as:

- $\eqclass a m +_m \eqclass b m = \eqclass {a + b} m$

### Real Modulo

Let $z \in \R$.

Let $\R_z$ be the set of residue classes modulo $z$ of $\R$.

The addition operation is defined on $\R_z$ as follows:

- $\eqclass a z +_z \eqclass b z = \eqclass {a + b} z$

This operation is called **addition modulo $z$**.

### Rational Numbers

The addition operation in the domain of rational numbers $\Q$ is written $+$.

Let:

- $a = \dfrac p q, b = \dfrac r s$

where:

- $p, q \in \Z$
- $r, s \in \Z_{\ne 0}$

Then $a + b$ is defined as:

- $\dfrac p q + \dfrac r s = \dfrac {p s + r q} {q s}$

This definition follows from the definition of and proof of existence of the field of quotients of any integral domain, of which the set of integers is an example.

### Real Numbers

The addition operation in the domain of real numbers $\R$ is written $+$.

From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.

Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}$, where $\eqclass {\sequence {x_n} } {}$ and $\eqclass {\sequence {y_n} } {}$ are such equivalence classes.

Then $x + y$ is defined as:

- $\eqclass {\sequence {x_n} } {} + \eqclass {\sequence {y_n} } {} = \eqclass {\sequence {x_n + y_n} } {}$

### Complex Numbers

The **addition operation** in the domain of complex numbers $\C$ is written $+$.

Let $z = a + i b, w = c + i d$ where $a, b, c, d \in \R, i^2 = -1$.

Then $z + w$ is defined as:

- $\paren {a + i b} + \paren {c + i d} = \paren {a + c} + i \paren {b + d}$

### Multiindices

The addition operation in the domain of multiindices $Z$ is written $+$.

Let $k = \sequence {k_j}_{j \mathop \in J}$ and $l = \sequence {l_j}_{j \mathop \in J}$ be multiindices.

Then $k + l$ is defined as:

- $\paren {\forall j \in J} \paren {\paren {k + l}_j = k_j + l_j}$

## Abstract Algebra

In the context of abstract algebra, the concept of **addition** is defined as follows:

### Ring Addition

The distributand $*$ of a ring $\struct {R, *, \circ}$ is referred to as **ring addition**, or just **addition**.

The conventional symbol for this operation is $+$, and thus a general ring is usually denoted $\struct {R, +, \circ}$.

### Field Addition

The distributand $+$ of a field $\struct {F, +, \times}$ is referred to as **field addition**, or just **addition**.

## Additive Inverse

Let $\Bbb F$ be one of the standard number systems: $\N$, $\Z$, $\Q$, $\R$, $\C$.

Let $a \in \Bbb F$ be any arbitrary number.

The **additive inverse** of $a$ is its inverse under addition, denoted $-a$:

- $a + \paren {-a} = 0$

## Terminology

The symbol $+$ is known as the **plus sign**.

Hence:

- $3 + 5$

is usually read:

*$3$***plus**$5$

### Summand

Let $a + b$ denote the operation of addition on two objects.

The objects $a$ and $b$ are known as the **summands** of $a + b$.

### Sum

Let $a + b$ denote the operation of addition on two objects $a$ and $b$.

Then the result $a + b$ is referred to as the **sum** of $a$ and $b$.

## Commutativity and Associativity

Let $\mathbb F$ be one of the standard number sets: $\N, \Z, \Q, \R$ and $\C$.

### Commutative Law of Addition

- $\forall x, y \in \mathbb F: x + y = y + x$

That is, the operation of addition on the standard number sets is commutative.

### Associative Law of Addition

- $\forall x, y, z \in \mathbb F: x + \paren {y + z} = \paren {x + y} + z$

That is, the operation of addition on the standard number sets is associative.

## Also see

## Historical Note

The symbol $+$ for addition originated in commerce, along with the symbol $-$ for subtraction, where they were used by German merchants to distinguish overweight and underweight items.

These symbols first appeared in print in $1481$.

However, Regiomontanus was the first to use it in its current shape, in an unpublished manuscript from $1456$.

Previous instances of the symbol had appeared in print from $1417$, but the downward stroke was not quite vertical.

## Sources

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*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Example $2.1$ - 1973: C.R.J. Clapham:
*Introduction to Mathematical Analysis*... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $2$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
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