Re: Computing infinity modulus and zero modulus

"Mariano_Suárez-Alvarez" <mariano.suarezalvarez@xxxxxxxxx> wrote:
On Feb 8, 3:29 pm, ju...@xxxxxxxxxxxxx wrote:
Please bear with me as I am a professional programmer but nothing more
than passionate about Math, so I'll simply go straight to the point in
my own "language".

I'm writing code to support Complex numbers and operations.

Based on what you say later, you will obviously need to use some
_extension_ of the complex numbers. The simplest is the one-point
extension, C*, obtained by adjoining a single undirected infinity, oo, to
the set of complex numbers. (But unfortunately, I doubt that C* will
actually be adequate for your needs.)

My goal is no operation of any kind should ever return NaN, that is no
operation should be undefined.

To be picky: I think that Kahan would tell you that NaN is not the same as
"undefined".

But I think I understand your point:
You want operations to return _numbers_ always.

To make the question simple, I'll focus on the
modulus operator on Reals to start from.

I have my doubts that that's the best way to start, and so I'm not going to
comment on the next paragraph.

I have found something very promising at
http://www.gwiep.net/books/parad09.htm, and - as far as I get it -
with some geometrical reasoning it defines "zero modulus" as x%0 = 0,
and infinity modulus as x%oo = x. The problem for me is it doesn't
cover all other "boundary" cases like oo %y or oo%oo.

My question is, is there any consistent and well founded way to
achieve that goal?

For your general goal, I think most people would answer NO.

And, broadening a bit, what in such constructed
algebra would be 0/0 and 0^0?

Suppose that we're using C*. Then I suggest the following:

1. Adopt the rule that 0*x = 0 for _all_ x.
2. Letting /x denote the reciprocal of x, define /0 = oo and /oo = 0.
3. Define division by x/y = x*(/y).
4. Define log(0) = oo.
5. Define exponentiation by x^y = exp(y*log(x)).

It would then follow that

0/0 = 0*(/0) = 0*oo = 0

and

0^0 = exp(0*log(0)) = exp(0*oo) = exp(0) = 1.

Please note that, as far as I am concerned (and as far as it makes
sense), here both infinity and zero should be taken as
"constants" (yep, it is an actual program what I'm trying to build).

Thanks a lot in advance for any insights,

The answer depends on what properties you want to preserve.

Indeed!

For example, it is trivial to set up a system in which
x / 0, for non zero x, is defined to be infinity, but in
that system it will no longer be true that

(x / y) * y = x for all x and y

You say "no longer". But if we were already in the real (or complex) number
system, we had _already_ lost that property.

Of course,

(x / y) * y = x for all x and y

is true, for example, in the system of _positive_ reals. But as soon as 0
is introduced...

David W. Cantrell
.

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