Re: complex numbers


"The Ghost In The Machine" <ewill@xxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote in message news:rqh0o2-c3o.ln1@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> In sci.physics.relativity, Dirk Van de moortel
> <dirkvandemoortel@xxxxxxxxxxxxxxxxxxxxxxxxxxx>
> wrote
> on Mon, 13 Jun 2005 13:36:53 GMT
> <VZfre.119624$ia5.6770755@xxxxxxxxxxxxxxxxxxxxx>:
> >
> > "N:dlzc D:aol T:com (dlzc)" <N: dlzc1 D:cox T:net@xxxxxxxxxx> wrote in message news:uLfre.7177$7s.91@xxxxxxxxxxxxx
> >> Dear jem:
> >>
> >> "jem" <xxx@xxxxxxx> wrote in message
> >> news:FVere.91014$sy6.84187@xxxxxxxxxxxxx
> >> > The TimeLord wrote:
> >> ...
> >> >> You can see that unless you keep straight just what
> >> >> the square root is defined to be,
> >> >
> >> > Sqrt() is defined to be a function so e.g. Sqrt(1) = 1,
> >> > not +-1, and of course i^2 = -1, not +-1.
> >>
> >> sqrt() may be defined by programming languages to be "the
> >> positive square root", allowing the programmer to assign whatever
> >> sign(s) the programmer chooses. But the result of the square
> >> root is bivalued
> >
> > Well, point me to *one* single technical or engineering
> > publication or text in the world where they have an equation
> > where they mean anything other than 3 when they write sqrt(9),
> > and you are in business.
> > sqrt is the positive root of a positive number.
> >
> >> (except for sqrt(0) ). sqrt(-1) is the
> >> conundrum, i^2 is the solution.
> >
> > There is no such thing as sqrt(-1).
> > sqrt(-1) is for bad enceclopedias.
>
> It may depend on the language and the environment.

Sure, whatever, but I'm not talking about softwares or languages.
I'm talking about mathematics :-)
I think complex numbers should be properly introduced as for
instance a set isomorphic with RxR and with appropriate definitions
for addition and multiplication. Then i is defined as the thing that
corresponds to (0,1) and you can prove that (0,1)^2 = (-1,0)
hence i^2 = -1. And *never* write a thing like sqrt(-1).

Dirk Vdm


.

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