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

### Complex Numbers

```
Date: 11/23/97 at 19:53:47
From: Stuart Zykorie
Subject: Complex numbers

Do all complex numbers have a multiplicative inverse? In other words,
if you have a region R defined by R=(a+b(square root of -5)) where a
and b are defined by all real integers, does every element of R have a
multiplicative inverse. In my findings, it would seem that this is
true. However, I've been told that some do not and those that do are
called the units of the complex numbers.

Stuart Z.
```

```
Date: 11/24/97 at 16:20:23
From: Doctor Bruce
Subject: Re: Complex numbers

Hello Stuart,

You are combining two very different questions here. You asked if all
complex numbers have a multiplicative inverse. Except for  0,  the
answer is yes. Let x + iy  be a complex number, so x and y are real
numbers and at least one of x,y is not zero. Then the number

x/(x^2 + y^2)  - iy/(x^2 + y^2)

is the multiplicative inverse of  x + iy.

But you also mentioned the set

R  =  {a + b*sqrt(-5) | a,b  are real integers}.

Every nonzero element of R has a multiplicative inverse as a complex
number, by what we just said. But in general that inverse is not
itself an element of R. The only elements of R with this property are
+1 and -1; there are no others. It is pretty easy to prove this: just
show that if

a/(a^2 + b^2)  - ib/(a^2 + b^2)  is an element of  R,

then we must have b = 0 and a = +1 or -1.

In the study of Algebraic Number Theory, we introduce the concept of
the Norm of a complex number. The Norm of a + bi, for example, is
a^2 + b^2. We show that the Norm operates on elements of sets like
your R and gives ordinary integers for values. Furthermore, the Norm
is multiplicative. This means

Norm(a + b*sqrt(-5)) * Norm(c + d*sqrt(-5))

= Norm[(a + b*sqrt(-5)) * (c + d*sqrt(-5))].

Elements of a set of complex numbers which have Norm equal to 1 are
called "units" of the set. So we see that elements of a set which have
a multiplicative inverse in the set are just the units of the set.

-Doctor Bruce,  The Math Forum
Check out our web site!  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