Absolute Values on Q (Part I)

Real numbers, Q, can be constructed from the rational numbers Q={a/b where a,b ϵ the integers Z}.  Construction of p-adic numbers, Qp, is done exactly the same way.

Let’s now consider absolute values.  An absolute value is a map from Q to [0,∞) that has the following properties for any x,y ϵ Q:

  1. |x| ≥ 0, and |x| = 0 if and only if x = 0,
  2. |x∙y| = |x|∙|y|,
  3. |x+y| ≤ |x|+|y| (The triangle inequality).

Now that we have some properties, we can look at types of absolute values that can be put on the rationals.  Alexander Ostrowski showed in 1935 that there are only three types.

Definition 1:  Let x be a rational number.

The trivial absolute value of x, denoted |x|0, is defined by

  \[ |x|_0 = \left\{
\begin{array}{l l}
1 & \quad \text{if $x\ne0$ }\\
0 & \quad \text{if $x=0$}
\end{array} \right.\]

The usual absolute value of x, denoted |x|, is defined by

\[ |x|_{\infty} = \left\{
\begin{array}{l l}
x & \quad \text{if $x\ge0$ }\\
-x & \quad \text{if $x\le0$}
\end{array} \right.\]

The p-adic absolute value of x, denoted |x|p, is defined for a given prime p by

\[ |x|_p = \left\{
\begin{array}{l l}
1/p^{ord_p(x)} & \quad \text{if $x\ne0$ }\\
0 & \quad \text{if $x=0$}
\end{array} \right.\]

 

The quantity ordp(x) is called the order of x.  It is the highest power of p that divides x.  For example, let p=5 and let x=75.  Then ord5(75)=2.  Since 25=52 and 25 is the highest power of 5 that divides 75, then the order of x is 2.  Confused yet?  Here are some examples of p-adic absolute values:

|75|5 = |52 ∙ 3|5 = 5-2

|10|5 = |51 ∙ 2|5 = 5-1

|13|5 = |50 ∙ 13|5 = 1

$|\frac{2}{75}|5 = |5-2 ∙ \frac{2}{3} |5 = 52$

|-375|5 = |53 ∙ -3|5 = 5-3

In these examples, we were looking at 5-adic absolute values.  We broke the number up into the form 5n ∙ a.  The 5-adic absolute value of 5n ∙ a is 5-n.

Conjecture 1:  Given a rational number x, the p-adic absolute value of x is p-n, where pn divides x.

|x|p = |pn ∙ a|p = p-n

In the introduction post (http://math.boisestate.edu/m287/quick-intro-to-p-adics/), it was mentioned that large p-adic numbers are closer to 0 than smaller p-adic numbers.  If one was to think of absolute value as a measure of distance, notice in the above examples that the larger the exponent on p, the resulting absolute value (or distance from 0) gets smaller.

2 thoughts on “Absolute Values on Q (Part I)

  1. Pingback: Absolute values on Q (Part II) | MATH 287

  2. Samuel Coskey

    Very interesting post. We all learn about the (usual) absolute value and its basic three properties, and most students probably just assume it is the only one. It is really surprising, therefore, to see that there are WEIRD p-adic examples of absolute values.

    One remark: your conjecture #1 is really just a restatement of the definition, so we can call it a fact. And it does connect back to your earlier post where you said numbers that are highly divisible by $p$ are closer to $0$. This codifies that into a definition!

Comments are closed.