# Absolute Value and Distance

## Magnitude and Sign

The order relation distinguishes positive and negative integers, but in many situations the sign of a number is less important than its magnitude. For example, the integers

$$
5
\quad\text{and}\quad
-5
$$

have opposite signs, yet both lie at the same distance from zero on the number line.

This observation leads to the concept of absolute value.

The absolute value of an integer $a$ is denoted by

$$
|a|.
$$

It is defined by

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

Thus absolute value removes the sign while preserving magnitude.

For example,

$$
|7|=7,
\qquad
|-7|=7,
\qquad
|0|=0.
$$

Absolute value therefore measures size independently of direction.

## Distance on the Number Line

The integers may be represented geometrically on a number line:

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

The absolute value of an integer equals its distance from zero.

For example,

$$
|-4|=4
$$

because the point $-4$ lies four units from the origin.

More generally, the distance between two integers $a$ and $b$ is

$$
|a-b|.
$$

For example, the distance between $3$ and $-5$ is

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

Similarly, the distance between $7$ and $2$ is

$$
|7-2|=5.
$$

Distance is always nonnegative.

## Basic Properties

Absolute value satisfies several important algebraic properties.

### Nonnegativity

For every integer $a$,

$$
|a|\ge0.
$$

Moreover,

$$
|a|=0
$$

if and only if

$$
a=0.
$$

### Symmetry

For every integer $a$,

$$
|-a|=|a|.
$$

Changing the sign does not change magnitude.

### Multiplicative Property

For all integers $a$ and $b$,

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

For example,

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

and

$$
|-3||5|=3\cdot5=15.
$$

This property plays an important role in divisibility theory and algebra.

## Triangle Inequality

One of the most fundamental inequalities in mathematics is the triangle inequality:

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

Geometrically, this states that the direct distance between two points is never greater than the length of a path passing through an intermediate point.

For example,

$$
|3+(-5)|=|-2|=2,
$$

while

$$
|3|+|-5|=3+5=8.
$$

Hence

$$
2\le8.
$$

Equality does not always occur. However, if $a$ and $b$ have the same sign, then

$$
|a+b|=|a|+|b|.
$$

The triangle inequality is fundamental in analysis, geometry, and number theory.

## Reverse Triangle Inequality

Another useful relation is

$$
\bigl||a|-|b|\bigr|\le |a-b|.
$$

This inequality compares the magnitudes of two integers with the distance between them.

For example,

$$
\bigl||7|-|3|\bigr|=|7-3|=4,
$$

and

$$
|7-3|=4.
$$

Thus equality may occur.

## Bounds Using Absolute Value

Absolute value provides a convenient way to express inequalities involving magnitude.

The inequality

$$
|a|\le n
$$

means that

$$
-n\le a\le n.
$$

Similarly,

$$
|a|<n
$$

means

$$
-n<a<n.
$$

For example,

$$
|x|\le3
$$

is equivalent to

$$
-3\le x\le3.
$$

This notation becomes increasingly useful in higher mathematics.

## Absolute Value and Divisibility

Absolute value interacts naturally with divisibility.

If

$$
a\mid b,
$$

then

$$
|a|\mid |b|.
$$

Furthermore, divisibility depends only on magnitude and not on sign. For example,

$$
3\mid12,
\qquad
-3\mid12,
\qquad
3\mid(-12).
$$

Thus positive and negative divisors are treated symmetrically.

In many contexts, one therefore restricts attention to positive divisors.

## Absolute Value in Number Theory

Absolute value appears throughout number theory. It measures the size of integers, bounds solutions of equations, and controls error terms in analytic estimates.

For example, Diophantine equations often ask for integer solutions satisfying inequalities such as

$$
|x|<100.
$$

Analytic number theory studies estimates of the form

$$
|f(n)|\le Cn^{1/2}.
$$

Even elementary arguments involving divisibility frequently depend on size considerations expressed through absolute value.

Thus absolute value connects arithmetic with geometry and analysis. It transforms the integers from a purely algebraic system into a metric structure in which notions of distance and approximation become meaningful.