# The Integers

## Limitations of the Natural Numbers

The natural numbers are sufficient for counting and addition, but they are not sufficient for subtraction. For example,

$$
7-3=4
$$

is a natural number, while

$$
3-7
$$

is not contained in $\mathbb{N}$.

To make subtraction universally possible, mathematics enlarges the number system by introducing the integers.

The set of integers is denoted by

$$
\mathbb{Z}=\{\ldots,-3,-2,-1,0,1,2,3,\ldots\}.
$$

The symbol $\mathbb{Z}$ comes from the German word *Zahlen*, meaning numbers.

The integers contain the positive numbers, the negative numbers, and zero.

## Positive and Negative Numbers

Every positive integer $a$ has a corresponding negative integer $-a$. These numbers satisfy

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

The number $-a$ is called the additive inverse of $a$.

The integer $0$ plays a distinguished role. It satisfies

$$
a+0=a
$$

for every integer $a$. Thus $0$ is called the additive identity.

The integers may therefore be viewed as an extension of the natural numbers in which subtraction becomes an internal operation.

## Arithmetic Operations on Integers

Addition, subtraction, and multiplication extend naturally from the arithmetic of natural numbers.

For addition, integers with the same sign are added by adding their absolute values and preserving the sign:

$$
(-3)+(-5)=-8.
$$

If the signs differ, subtraction of absolute values occurs:

$$
7+(-4)=3.
$$

Subtraction is defined through additive inverses:

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

Multiplication obeys the familiar sign rules:

$$
(+)(+)=+,
\qquad
(+)(-)=-,
\qquad
(-)(+)=-,
\qquad
(-)(-)=+.
$$

Thus,

$$
(-3)(5)=-15
$$

while

$$
(-3)(-5)=15.
$$

These rules preserve the distributive law:

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

## Algebraic Structure

The integers satisfy important algebraic properties.

### Closure

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

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

and

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

### Commutative Laws

$$
a+b=b+a
$$

and

$$
ab=ba.
$$

### Associative Laws

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

and

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

### Identity Elements

$$
a+0=a,
\qquad
a\cdot1=a.
$$

### Additive Inverses

For every integer $a$, there exists an integer $-a$ such that

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

These properties make the integers one of the fundamental algebraic systems in mathematics.

## Order on the Integers

The integers inherit a natural order extending the order of the natural numbers:

$$
\cdots<-3<-2<-1<0<1<2<3<\cdots
$$

Negative integers lie to the left of zero, while positive integers lie to the right.

The order relation is compatible with arithmetic. If

$$
a<b,
$$

then

$$
a+c<b+c.
$$

If $c>0$, then

$$
ac<bc.
$$

Multiplication by a negative integer reverses inequalities:

$$
a<b
\quad\Longrightarrow\quad
ac>bc
$$

whenever $c<0$.

## Absolute Value

The absolute value of an integer measures its distance from zero on the number line. It is defined by

$$
|a|=
\begin{cases}
a, & a\ge0,\\
-a, & a<0.
\end{cases}
$$

For example,

$$
|5|=5,
\qquad
|-5|=5.
$$

Absolute value satisfies several important properties:

$$
|ab|=|a||b|
$$

and

$$
|a+b|\le |a|+|b|.
$$

The latter inequality is called the triangle inequality.

## Integers in Number Theory

The integers form the central domain of classical number theory. Questions concerning divisibility, primes, congruences, and Diophantine equations are fundamentally questions about integers.

For example, the equation

$$
x^2+y^2=z^2
$$

asks for integer solutions, while divisibility questions such as

$$
a\mid b
$$

depend entirely on integer arithmetic.

The passage from the natural numbers to the integers therefore marks the beginning of arithmetic as a complete algebraic theory.