Imaginary Numbers, Division By Zero

Date: 07/03/2000 at 19:22:27
From: Marc
Subject: Imaginary Numbers/Division By Zero

Dr. Math,

Here is my question. If we can create a new set of numbers for 
something that doesn't exist in our real number system for the square 
roots of negative numbers (imaginary numbers), then why can't we do 
the same thing for division by zero? The number i was created to equal 
the square root of -1, so why can't we define another variable, for 
example, x, as equal to 1/0, 2x as equal to 2/0, 3x as equal to 3/0 
and so on? Why is there a set of numbers that uses the square root of 
negative numbers, something that is impossible in our "real" number 
system, but NOT one that uses division by zero, something that is also 
impossible in our real number system? 
 
I have one more question. Are there any real life uses for imaginary 
numbers? And if not, why were they created? How can you use a number 
that has no "real" value for any useful purpose?

Thank you for your time.

Date: 07/03/2000 at 21:08:42
From: Doctor Peterson
Subject: Re: Imaginary Numbers/Division By Zero

Hi, Marc.

In creating the complex numbers, what we are doing is extending the 
set of real numbers by adding one new number i, defined so that i^2 = 
-1, and then applying the properties of the operations on real numbers 
to complete the set; for instance, we can multiply i by any real 
number, giving the imaginary numbers, and then we can add any of these 
to any real number to produce a complex number. If at any point the 
properties of operations led to a contradiction (for instance, if it 
turned out that multiplication as defined were not commutative), we 
would have to give up the idea of complex numbers, or at least 
recognize that they are not an extension of the real numbers as we 
know them.

If we tried to do the same thing by adding an "infinite" number (a 
number, not a variable!), say I = 1/0, we would find lots of 
contradictions. For example, I would be defined more specifically as 
the number for which

     I * 0 = 1,

but since

     (I + 1) * 0 = I * 0 + 1 * 0 = 1 + 0 = 1

so that

     (I + 1) * 0 = 1,

we would have to say that

     I + 1 = I

By subtracting I from both sides, we would find that

     1 = 0

Now, any system of numbers for which this is true is not very useful, 
so we just can't add this "I" to the system and still follow the rules 
of the real numbers.

You can read more about this in our FAQs on Dividing by 0 and Large 
Numbers and Infinity:

   http://mathforum.org/dr.math/faq/faq.divideby0.html   
   http://mathforum.org/dr.math/faq/faq.large.numbers.html   

To answer your other question, imaginary numbers are indeed used in 
many ways, particularly in physics and engineering. The first use was 
in solving cubic equations, where it was found that by passing 
temporarily into the complex realm, problems whose ultimate solutions 
were real could be solved! But many variables in physics turn out to 
be complex; physical quantities can often be thought of as just the 
real part of a complex number. Here are a couple answers we've given 
with more details:

   Applications of Imaginary Numbers
   http://mathforum.org/dr.math/problems/zakrzewski10.14.97.html   

   Imaginary Numbers
   http://mathforum.org/dr.math/problems/matt02.28.99.html   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/