Meaning of the Phrase: Divide by Zero.

Date: 01/02/98 at 23:41:18
From: Alfred Coppel
Subject: Meaning of the phrase: Divide by zero.

I am a novelist--28 published books, a dozen of them science-fiction. 
I want to use the phrase 'divide by zero' to indicate a physically
impossible task. How far off the mark am I? In fact, what does the
phrase actually mean? Thanks.

Date: 01/03/98 at 05:03:03
From: Doctor Pete
Subject: Re: Meaning of the phrase: Divide by zero.

Hi,

Well, division by zero is not so much "physically impossible" as it is 
"in violation of mathematical axioms."  You see, the phrase 
"physically impossible" implies a task that cannot be done, no matter 
the amount of exertion of effort, whereas the phrase "in violation of 
mathematical axioms" means that the operation contradicts certain 
basic assumptions regarding the system in question.

Numbers have certain properties and rules; for instance, we say that 
adding, subtracting, multiplying, and dividing two numbers will give 
another number. Subtraction is the opposite of addition, as division 
is the opposite of multiplication.  Any number multiplied by zero 
gives zero. 

There are several of these basic rules, called axioms, and in 
particular, the kinds of numbers we are familiar with, and do basic 
arithmetic with, form what mathematicians call a "field." In this 
field, these rules I have described are called "field axioms." (There 
are others as well.)  

In essence, the field axioms lay down a set of rules, i.e., basic 
assumptions, about how to put numbers together to get other numbers.  
And so, division by zero can be shown to contradict these rules (this 
proof is usually taught in beginning algebra classes.)

Technically speaking, division by 0 is not impossible; rather, it is 
contradictory to assumption. As such, we disallow it as a valid 
operation on numbers. "Physically impossible" is a more fitting 
description of a phenomenon, such as the creation of a perpetual 
motion machine, or the decrease in entropy of a closed system.  
Division by 0 is not so much a phenomenon as it is a supposed 
construction, which is provably contradictory to a given set of rules 
and therefore not permitted within the system upon which the rules are 
imposed.

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