From: shep...@tcg.anl.gov (Ron Shepard)
Subject: Re: the sqrt(-i)
Date: 1998/09/24
Message-ID: <shepard-2409981718110001@macrls.tcg.anl.gov>#1/1
X-Deja-AN: 394532992
Distribution: inet
References: <360A986D.CD8EBCC1@oneimage.com>
Organization: Argonne National Laboratory, Chicago IL
Newsgroups: sci.math.num-analysis

In article <360A986D.CD8EB...@oneimage.com>, "Robert Simmons Jr."
<gnu...@oneimage.com> wrote:

[...]
>My question is this. If we try to take the sqrt(-i) what is the result ?

Unlike the situation with real numbers, it is not necessary to extend the
idea of complex numbers in order to find sqrt(-i).  There is a complex
number that satisfies that role.  In fact, there are two of them, just as
there are two distinct numbers that satisfy any square root expression.

The easiest way to think about these square roots is to think in terms of
a polar representation (an angle, and a distance from the origin).  The
complex number -i is one unit down on the y axis, so this is either -90
degrees or 270 degrees.  The two square roots of -i are just half of these
angles, -45 degrees or 135 degrees, with unit distance from the origin.

>Is it yet another dimension in our number line? 

In the case of real numbers, the system *had* to be extended to find
solutions to such equations, the solution simply does not exist within the
real numbers.  In the case of complex numbers, this extension is not
necessary.  That is not to say however, that some new extension could not
be dreamed up that would give new distinct solutions to the question.

>I would LOVE to hear coments on this idea. If it is nieve or otherwise,
>the comments will be appreciated. [...]

No, this gets to a deep and interesting set of questions.  Is mathematics
"discovered" or is it "invented"?  Is it an abstract endeavor that happens
to have practical applications, or is it fundamentally and inescapably a
practical endeavor?

$.02 -Ron Shepard