# (in)exactness of complex numbers

Mikael Olofsson mikael at isy.liu.se
Mon Aug 6 12:16:24 CEST 2001

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

Thanks for making me do my homework myself. I'm glad that I got this
sorted out.

> >By the way, the mathematicans of the sixteenth century also had strange
> >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.

Agreed.

> >  "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.

Well, they did decide on one thing: To remove the symptom of the problem,
the messenger. The real problem was of course, that they had assumed that
"all phenomena in the universe can be reduced to whole numbers and their
ratios."

I have often said that the difference between theory and practice is
smaller in theory than in practice. However, in this particular case, it
is the other way around. If you want to manufacture a physical square
item, and you need to produce something that fits the diagonal, you do
not for a moment care if that diagonal is a rational or not. It reminds
me of a pupil in a basic carpentry class, who asked his teacher how many
digits he needed for pi to make that circular thing that he was supposed
to make. Was the eight digits given by his pocket calculator enough?
Pi, his teacher replied, is just a little bit more than three.

> >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.

:o)

This thing is getting a little bit too off-topic, don't you think?

/Mikael

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Date:    06-Aug-2001
Time:    11:43:48

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