(in)exactness of complex numbers

[David C. Ullrich]

On Fri, 03 Aug 2001 11:31:44 +0200 (MET DST), Mikael Olofsson
<mikael at isy.liu.se> wrote:

>
>On 02-Aug-2001 David C. Ullrich wrote:
> >  On Thu, 02 Aug 2001 09:28:05 +0200 (MET DST), Mikael Olofsson
> >  <mikael at isy.liu.se> wrote:
> > >
> > >On 01-Aug-2001 David C. Ullrich wrote:
> > > >  (Not that it makes any difference here, but
> > > >  since you undertstand the math so much better
> > > >  than I do: Exactly what definition of
> > > >  "complex number" do you have in mind here?
> > > >  The _standard_ definition _is_ "pair of
> > > >  real numbers".)
> > >
> > >I'd say it's not the standard definition, but rather the standard
> > >representation.
> >  
> >  Well, it _is_ the _definition_ in many contexts. Honest.
>
>In the sense that it is the only representation most people know of,
>yes (people here refers only to those who knows anything about complex
>numbers). 

Um. I know of several representations. This one seems like the
standard one to me, simply because it _is_ the standard definition
in complex analysis. Really: Find a book on complex analysis that
defines complex numbers in some other fashion.

> >  (_Not_ that I see what it _matters_ what "the" definition
> >  in mathematics is! The idea that that should be an
> >  important criterion in determining the way Python
> >  complexes should work seems silly.)
>
>Well, Pythons complex type should - at least approximately within some
>limits - behave as the corresponding mathematical concept. If it did 
>not, it would be totally useless.
>
><snip my words about complex numbers from x**2 + 1 and complex numbers
>from other irreducible polynomials>
>
> >  Something like that could be very interesting, but surely the
> >  sort of thing that could not possibly belong in Python proper
> >  instead of some specialized module.
>
>Absolutely. I was merely pointing out that there are more than one
>way to do it in math as well. But! There is only one obvious way,
>as you are pointing out. However, the obvious way may change from 
>time to time, just as it does in Python. :o)
>
> > > I mean, how many acually uses complex numbers
> > >at all in their programs. Most people still regard complex numbers as
> > >a very obscure corner in math. About only a 150 years ago, when complex
> > >numbers were new,
> >  
> >  ??? Not that it matters, but I woulda thought they'd been around a
> >  little longer than that.
>
>After consulting "Mathematical thoughts from ancient to modern times" 
>by Morris Kline, I must admit that I was not checking things correctly. 
>Mathematicians were experimenting with roots of negative numbers 
>in the sixteenth century, when they still had very strange ideas about
>negative numbers. According to the very same book, many mathematicians 
>were still utterly confused about complex numbers in the beginning of
>the nineteenth century.

The reason I didn't just say flat out that you were wrong about the
date is that while the time that people first started using complex
numbers in one way or another is clearly much more than 150 years
ago you might have been referring to the date at which people
first started giving a mathematically coherent definition of what
these complex number thingies actually _are_. For some time people
were "using" them althought they were just "imaginary"; the exact
date at which complex numbers became "real" is not at all clear
to me. (Could very well be some time around 150 years ago, that
being when a lot of modern points of view started to sort of
appear faintly on the horizon.)

>By the way, the mathematicans of the sixteenth century also had strange
>ideas about irrational numbers, and they had been known for about 2000 
>years at the time. According to the referred book, in the fifth century
>BC, Hippasus, one of Pythagoras' disciples, proved that there exist 
>numbers that are not rational. Citing the book:

I think it's much more accurate to say they proved that some 
"quantities", certain lengths in particular, could not be represented
using the notion of "number" that they had. Saying that they proved
that sqrt(2) is irrational implies that they had a notion of number
that included sqrt(2), and they shpwed that number was not
rational. I really don't think that's an accurate picture of what
happened (although it's what you read in a lot of books.) I think
it's more accurate to say they showed that there was no number
whose square is 2.

Eudoxus' theory of incommeasurables(?) came very close to the
modern notion of "real number", but that was much later than
the Pythagoreans.

>  "The Pythagoreans were supposed to have been at sea at the time and 
>  to have thrown Hippasus overboard for having produced an element in 
>  the universe which denied the Pythagorean doctrine that all phenomena 
>  in the universe can be reduced to whole numbers and their ratios."

This for example is not a bad way to put it: Klein is not stating
that they decided that there was a _number_ which is not a ratio
of two integers.

>I'm glad that I don't have to fear that my colleagues execute me if I
>find exceptional results.

Just because it hasn't happened in a while doesn't mean it can't
happen... my advice would be to stick to unexceptional results
just to be on the safe side.

>/Mikael
>
>-----------------------------------------------------------------------
>E-Mail:  Mikael Olofsson <mikael at isy.liu.se>
>WWW:     http://www.dtr.isy.liu.se/dtr/staff/mikael               
>Phone:   +46 - (0)13 - 28 1343
>Telefax: +46 - (0)13 - 28 1339
>Date:    03-Aug-2001
>Time:    09:07:34
>
>         /"\
>         \ /     ASCII Ribbon Campaign
>          X      Against HTML Mail
>         / \
>
>This message was sent by XF-Mail.
>-----------------------------------------------------------------------
>


David C. Ullrich