Absolute Values

Measuring Size in Arithmetic

The ordinary absolute value on the real numbers measures magnitude:

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

This function allows us to define distance,

d(x,y)=|x-y|,

and therefore notions such as convergence, continuity, and completeness.

In number theory, however, there are many different ways to measure the size of numbers. The ordinary absolute value is only one possibility.

The theory of absolute values provides a unified framework for studying these different notions of size. From this viewpoint arise the $p$ -adic numbers, local fields, and modern local-global principles.

Definition of an Absolute Value

Let $K$ be a field.

Definition. An absolute value on $K$ is a function

|\cdot|:K\to\mathbb{R}_{\ge0}

satisfying the following properties for all $x,y\in K$ :

Nonnegativity:

|x|\ge0, \qquad |x|=0 \iff x=0.

Multiplicativity:

|xy|=|x||y|.

Triangle inequality:

|x+y|\le |x|+|y|.

The pair $(K,|\cdot|)$ becomes a metric space with metric

d(x,y)=|x-y|.

Thus an absolute value imposes geometric structure on a field.

The Standard Absolute Value

The familiar absolute value on

\mathbb{Q},\quad \mathbb{R},\quad \mathbb{C}

is called the archimedean absolute value.

For rational numbers,

\left|\frac{a}{b}\right| = \frac{|a|}{|b|}.

This absolute value reflects ordinary Euclidean geometry.

It is called archimedean because repeated addition eventually dominates every fixed quantity. More precisely, for any positive numbers $x$ and $y$ , there exists $n\in\mathbb{N}$ such that

nx>y.

This property fails for $p$ -adic absolute values.

Non-Archimedean Absolute Values

Number theory reveals another family of absolute values associated with prime numbers.

Fix a prime $p$ . Every nonzero rational number can be written uniquely as

x=p^k\frac{a}{b},

where $a$ and $b$ are integers not divisible by $p$ .

Definition. The $p$ -adic absolute value of $x$ is

|x|_p=p^{-k}.

Additionally,

|0|_p=0.

Thus divisibility by large powers of $p$ makes a number small in the $p$ -adic sense.

For example, in the $2$ -adic absolute value,

|8|_2=2^{-3}=\frac18,

while

|1/8|_2=2^3=8.

Hence powers of $2$ become increasingly small.

This behavior is opposite to the ordinary absolute value.

The Ultrametric Inequality

The $p$ -adic absolute value satisfies a stronger version of the triangle inequality.

Theorem.

|x+y|_p \le \max(|x|_p,|y|_p).

This is called the ultrametric inequality or non-archimedean triangle inequality.

An immediate consequence is that if

|x|_p\ne |y|_p,

then

|x+y|_p = \max(|x|_p,|y|_p).

Thus cancellation behaves very differently in $p$ -adic geometry.

For example, in ordinary geometry a triangle may have three distinct side lengths. In ultrametric geometry, every triangle is isosceles with the two larger sides equal.

The geometry induced by non-archimedean absolute values is therefore highly rigid.

Equivalent Absolute Values

Different absolute values may define the same notion of convergence.

Definition. Two absolute values $|\cdot|_1$ and $|\cdot|_2$ on a field $K$ are equivalent if there exists $c>0$ such that

|x|_1=|x|_2^c

for all $x\in K$ .

Equivalent absolute values induce the same topology on the field.

For the rational numbers, Ostrowski’s theorem gives a complete classification.

Ostrowski’s Theorem

One of the foundational results in local number theory is the following theorem.

Theorem (Ostrowski). Every nontrivial absolute value on $\mathbb{Q}$ is equivalent either to:

the ordinary absolute value;
the $p$ -adic absolute value for some prime $p$ .

Thus the rational numbers possess exactly two kinds of geometry:

archimedean geometry;
$p$ -adic geometry.

This classification explains why prime numbers play such a fundamental role in arithmetic.

Completions

Absolute values allow one to define Cauchy sequences and completeness.

The rational numbers are not complete with respect to the ordinary absolute value because sequences such as decimal approximations to $\sqrt2$ converge to numbers outside $\mathbb{Q}$ .

Completing $\mathbb{Q}$ with respect to the ordinary absolute value produces

\mathbb{R}.

Similarly, completing $\mathbb{Q}$ with respect to $|\cdot|_p$ produces the field of $p$ -adic numbers:

\mathbb{Q}_p.

Thus every absolute value naturally leads to a completed field.

The fields

\mathbb{R},\quad \mathbb{C},\quad \mathbb{Q}_p

form the basic local fields of number theory.

Product Formula

The ordinary and $p$ -adic absolute values are linked by a remarkable global relation.

For every nonzero rational number $x$ ,

|x|_\infty \prod_{p}|x|_p = 1,

where $|x|_\infty$ denotes the ordinary absolute value.

For example, if

x=12=2^2\cdot3,

then

|12|_\infty=12,

|12|_2=\frac14, \qquad |12|_3=\frac13,

and all other $p$ -adic absolute values equal $1$ . Therefore

12\cdot\frac14\cdot\frac13=1.

This formula expresses a deep balance between all primes simultaneously.

It foreshadows the adelic viewpoint in modern number theory.

Topological Consequences

Absolute values convert algebra into topology.

One may define:

convergence;
continuity;
compactness;
analytic functions;
integration.

The resulting theories differ radically between archimedean and non-archimedean settings.

For example, a series may converge $p$ -adically even when it diverges over the real numbers.

The geometric intuition of calculus therefore changes substantially in local arithmetic.

Absolute Values in Modern Number Theory

Absolute values form the foundation of local methods in arithmetic.

They appear in:

local fields;
Diophantine equations;
Hasse principles;
Galois representations;
automorphic forms;
arithmetic geometry.

Modern number theory studies arithmetic simultaneously over all completions of a field. The local information from each absolute value is then assembled into global arithmetic statements.

Thus absolute values provide the bridge between algebraic structure and analytic geometry, making them one of the fundamental organizing principles of modern number theory.