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Why Not Force Dividing by Zero?

Date: 10/22/2002 at 04:09:07
From: Rindert Bolt
Subject: Divide by zero

Hello,

I wonder if there has ever been an attempt to force dividing by zero 
just as it has been done so successfully by forcing the square root of 
-1 to be i. 

Of course it has been attempted, but why is it not used?

Best regards, 
Rindert


Date: 10/22/2002 at 09:46:18
From: Doctor Mitteldorf
Subject: Re: Divide by zero

Dear Rindert,

It's a good question, and there's a good answer. When we define the
square root of -1 as i, we expand the real number system to a complex
number system. Every number has two parts - the real part and the
part that is a multiple of i. We get an interesting and useful system.  
Numbers like 3+2i have meaning, and can be added, subtracted, 
multiplied, and divided with self-consistent results. Problems such as
sqrt(i) and sin(1+i) turn out to have meaning, and the answers are
consistent and obey the same algebraic laws as square roots, and the
same trig identities that sines and cosines of real numbers obey.  

What's more, we don't have to expand the system any more or define
any new levels of imaginary numbers, because all algebraic equations
now have solutions, even if the coefficients in those equations are
complex numbers.

But suppose we tried the same trick for 1/0. We define 1/0 to be z, 
and we say from now on, all numbers will be expressed as two parts, 
as, for example, 2+3z. When we try to calculate in this system, we
run into strange results, leading to contradictions. For example, you
can prove that z+1=z.  Just divide both sides by z.  

    1 + 1/z = 1

Since 1/z=0, this must be true. But if it is true that z+1=z, then we
can subtract z from both sides and we get 1=0. But if 1=0, then it is
certainly true that 2=1, or any real number is equal to any other real
number. So all real numbers are the same, and we shouldn't confuse
things by calling 7 a different name from 100.

The moral of this story is that we take it for granted that if we 
solve a problem in two different ways, we will get the same answer
both ways. Mathematics is consistent. But this is not a fact about 
"the world"; rather this is a design feature of the rules of
mathematics. If we design our mathematics in the "wrong" way, we will
get inconsistencies.

(If you follow this kind of thinking deeper into the philosophy of
mathematics, you will come across the program of Bertrand Russell in
the early part of the 20th century. He tried to prove that the
mathematical system that we all know so well had the simple, desirable
properties that 

   1. Every statement is either true or false.
   2. Every true statement has a proof, and every false statement has
      a disproof.

It seems that these things are almost obvious. We are assuming
something like this every time we set out to solve a math problem. But 
Russell encountered frustration. He couldn't seem to convince himself 
that these simple critera were met. A few years later, along came Kurt 
Godel, and at last we knew why. This is a fascinating story for 
another day.)  

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Logic
Middle School Logic
Middle School Number Sense/About Numbers

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