(in)exactness of complex numbers

[Mikael Olofsson]

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). 

 >  (_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.

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:

  "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."

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

/Mikael

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E-Mail:  Mikael Olofsson <mikael at isy.liu.se>
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Date:    03-Aug-2001
Time:    09:07:34

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