Defining Distance Mathematically

Date: 10/16/96 at 11:17:55
From: Jean-Pierre Chanet
Subject: Distance Definition

I'm trying to calculate the distance between X and X'.
The real definition of distance is  D = abs ( X - X' ), but what is 
wrong with D' = sqrt(X^2 - X'2)?

Date: 10/16/96 at 13:40:44
From: Doctor Tom
Subject: Re: Distance Definition

Hello Jean-Pierre,

There is no unique mathematical way to define distance.  To give some 
ideas of why different definitions make sense, consider the following 
examples:

To find the distance between two points on a plane, find the length
of the straight line connecting them.  But if you're on the earth, and 
trying to get from Paris to New York City, that straightest line goes 
deep under ground, so a "better" measure would be along a (curved) 
path that's shortest, but on the surface of the earth.

If you are in Manhattan (part of New York City where all the streets 
are north-south or east-west and pretty evenly spaced, and all the 
blocks are filled with very tall buildings), the "shortest distance 
between two points might mean the length of the path along city 
streets, since there are buildings in the way of a straight path.  
(This is called by mathematicians the "Manhattan distance".)

And I think you'll appreciate this final example, called the "French
railway distance." It's based on the fact that (at least in the past) 
most of the railways in France headed straight to Paris. So the 
shortest distance between two points is the usual distance (if they're 
on the same straight line heading to Paris), or it's the sum of the 
distances to Paris otherwise. (In other words, you have to go to 
Paris, change trains, then go to the other town.)

To be a distance, a measure has to satisfy some properties:

(a) D(x,x) = 0, and D(x,y) = 0 means x=y
(b) D(x,y) = D(y,x)
(c) D(x,y) <= D(x,z)+D(z,y)

(a) means that the distance from any point to itself is zero.
(b) means it's equally far from x to y as from y to x.
(c) means it can never be shorter to go to an intermediate
    point than to go directly (but it may be the same distance)

You'll find that your D' definition is not a particularly useful
measure because it measures the distance from 5 to -5 as zero.

I think what you're looking for is this:

D' = sqrt((x - x')2)

It's exactly the same as your D in 1 dimension, but it is the usual
Euclidean (straight-line) distance in any number of dimensions.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/