# Arithmetic Operations

## Operations as Rules

An arithmetic operation is a rule that combines numbers to produce another number. The most basic operations on integers are addition, subtraction, multiplication, and division.

Addition and multiplication are always possible inside the integers. If

$$
a,b\in\mathbb{Z},
$$

then

$$
a+b\in\mathbb{Z}
$$

and

$$
ab\in\mathbb{Z}.
$$

Subtraction is also always possible because the integers contain additive inverses:

$$
a-b=a+(-b).
$$

Division is different. The quotient $a/b$ need not be an integer. For example,

$$
6/3=2
$$

is an integer, but

$$
7/3
$$

is not. This failure of division inside $\mathbb{Z}$ is one reason divisibility becomes a central topic in number theory.

## Addition

Addition combines quantities. It is written

$$
a+b.
$$

The integer $0$ is the identity element for addition:

$$
a+0=a.
$$

Every integer has an additive inverse:

$$
a+(-a)=0.
$$

Addition is commutative:

$$
a+b=b+a,
$$

and associative:

$$
(a+b)+c=a+(b+c).
$$

Associativity means that a sum of several integers may be written without ambiguity:

$$
a+b+c.
$$

The order of addition does not matter, and the grouping of terms does not matter.

## Subtraction

Subtraction is defined using addition and additive inverses:

$$
a-b=a+(-b).
$$

Thus subtraction does not require a separate theory. It is addition with the opposite number.

For example,

$$
8-11=8+(-11)=-3.
$$

Subtraction is not commutative. In general,

$$
a-b\ne b-a.
$$

For instance,

$$
5-2=3,
\qquad
2-5=-3.
$$

Subtraction is also not associative. Usually,

$$
(a-b)-c\ne a-(b-c).
$$

For example,

$$
(10-4)-3=3,
$$

while

$$
10-(4-3)=9.
$$

These failures explain why addition is structurally simpler than subtraction.

## Multiplication

Multiplication represents repeated addition. For positive integers,

$$
ab=\underbrace{b+b+\cdots+b}_{a\text{ times}}.
$$

For all integers, multiplication is extended using the sign rules:

$$
(-a)b=-(ab),
\qquad
a(-b)=-(ab),
\qquad
(-a)(-b)=ab.
$$

The integer $1$ is the identity element for multiplication:

$$
a\cdot 1=a.
$$

Multiplication is commutative:

$$
ab=ba,
$$

and associative:

$$
(ab)c=a(bc).
$$

It also distributes over addition:

$$
a(b+c)=ab+ac.
$$

The distributive law is one of the most important laws in arithmetic. It connects addition and multiplication and allows expressions to be expanded and factored.

## Division

Division asks whether one integer is an exact multiple of another. The expression

$$
a/b
$$

means a number $q$ such that

$$
bq=a.
$$

Such a number $q$ may fail to be an integer. Therefore division is not always an operation inside $\mathbb{Z}$.

When $b\ne0$ and there exists an integer $q$ such that

$$
a=bq,
$$

we say that $b$ divides $a$, and write

$$
b\mid a.
$$

For example,

$$
3\mid 12
$$

because

$$
12=3\cdot4.
$$

But

$$
3\nmid 14
$$

because no integer $q$ satisfies

$$
14=3q.
$$

This distinction between exact and non-exact division is the first appearance of divisibility.

## Closure and Failure of Closure

A number system is said to be closed under an operation if applying that operation to numbers in the system always produces another number in the same system.

The integers are closed under addition:

$$
a,b\in\mathbb{Z}
\quad\Longrightarrow\quad
a+b\in\mathbb{Z}.
$$

They are closed under subtraction:

$$
a,b\in\mathbb{Z}
\quad\Longrightarrow\quad
a-b\in\mathbb{Z}.
$$

They are closed under multiplication:

$$
a,b\in\mathbb{Z}
\quad\Longrightarrow\quad
ab\in\mathbb{Z}.
$$

They are not closed under division. Since

$$
1/2\notin\mathbb{Z},
$$

division leads outside the integers.

This failure is not a defect. It is the source of many central problems in number theory.

## Arithmetic Expressions

An arithmetic expression is built from numbers, variables, and operations. For example,

$$
3a^2-5b+7
$$

is an arithmetic expression involving addition, subtraction, multiplication, and powers.

The usual order of operations applies. Multiplication is performed before addition, unless parentheses specify otherwise:

$$
2+3\cdot4=14,
$$

while

$$
(2+3)\cdot4=20.
$$

Parentheses make the structure of an expression explicit. Much of algebra and number theory consists of rewriting expressions while preserving their values.

## Role in Number Theory

Arithmetic operations provide the language in which number theory is written. Divisibility, prime factorization, congruences, Diophantine equations, and modular arithmetic all depend on the interaction between addition, multiplication, and exact division.

The integers are simple enough to define early, but rich enough to support deep questions. Even the elementary expression

$$
a^2+b^2=c^2
$$

leads to the theory of Pythagorean triples, quadratic forms, elliptic curves, and modern arithmetic geometry.