Associated Topics || Dr. Math Home || Search Dr. Math

### 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?

```

```

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.

Numbers and Infinity:

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

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/
```
Associated Topics:
High School Imaginary/Complex Numbers

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search