Re: complex numbers


"uucp" <bad_shoes@xxxxxxxxxxxxx> wrote in message news:1118676242.658682.95800@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>
>
> Dirk Van de moortel wrote:
> > "The TimeLord" <mathnphysics-not@xxxxxxxxxxxxx> wrote in message news:UqWdnes_svgUJzHfRVn-gQ@xxxxxxxxxxxxxx
> > > Don Giovanni <laterel0328@xxxxxxxxx> wrote in
> > > <1118580874.108666.125200@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> on Sunday 12 June
> > > 2005 07:54 posted to sci.physics.relativity:
> > >
> > > > i never understod whay complex
> > > > number only have two dimenssion,
> > > > the real and imaginary part
> > > >
> > > >
> > > > whay not more?
> > >
> > > Actually they don't have two dimensions either. A dimension is defined as
> > > the number of linearly independent basis vectors that span a space. I know
> > > that sounds trite, but my point is that in understanding things like
> > > complex numbers, you need to understand how mathematicians define things.
> > >
> > > So, a complex number is a number that when multiplied by itself equals a
> > > real number.
> >
> > That is wrong.
> > A complex number multiplied by itself does not give a real number.
> > It gives a compex number:
> > Having x and y real numbers,
> > ( x + y i )^2 = x^2 - y^2 + 2 x y i
> > Only if x = 0 will the result be a real number, i.o.w. a strictly
> > imaginary number multiplied by itself gives a real number.
>
> wrong, only if the imaginary part or the real part are zero

Indeed. The word "only" should not be there.
A strictly imaginary number multiplied by itself gives a real number.
and of course a strictly real number multiplied by itself gives a
real number as well.

>
> >
> > > As long as the real number is positive, another real will do
> > > as in Sqrt[4]=+-2; both real. If the real is negative then you get the
> > > imaginary part as in Sqrt[-4]=+-2i.
> >
> > That is wrong as well.
> > sqrt(4) = 2
> > - sqrt(4) = - 2
> > sqrt(-4) is nonsense
>
> wrong, sqrt(-4) = sqrt(-1*4) = sqrt(-1)*sqrt(4) = i4,
> which is complex pure imaginary

Other pedagogically unsound authors would write
sqrt(-4) = 2 i
but nevertheless they would be silly.


>
> >
> > Sqrt is a function defined for positive real numbers only.
> > The result is a positive real number (and positive includes zero).
>
> wrong fool

That is the usual definition.

>
> >
> > What you *can* write however, is this:
> > sqrt( x^2 ) = +- x
> > which is an abbreviation for the statement:
> > | sqrt( x^2 ) = x (namely for all real x >= 0)
> > | or
> > | sqrt( x^2 ) = -x (namely for all real x <= 0)
> > You can write this because in both cases the argument and the
> > result of the function are positive values
> > Which one of both equations is valid, depends on the sign
> > of x. That is why you *cannot* write
> > sqrt( 4 ) = +- 2
> > since the case with -2 can never occur.
>
> bullshitisimo

Nice to know that apparetly you are just as stupid as Androcles:
http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/STILL.html
http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/CanSpecify.html
http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/Nearly.html
http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/Quadratic.html
http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/GrowUp.html
http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/Tautology.html
http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/Material.html
http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/GIVEN.html
http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/PythagoRescue.html
http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/SqrtRev.html
http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/NegSqrt.html
http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/SqrtAnswers.html


>
> >
> > >
> > > So then, the answer to your question is that real numbers can be positive or
> > > negative, so you need both real and imaginary parts of complex numbers to
> > > describe them. The real part describes the square root of the positive and
> > > the imaginary part describes the square root of the negative.
> >
> > The last sentence sounds like nonsense.
> >
> > >
> > > To really get a feel for this, consider...
> > > i = Sqrt[-1]
> > > So i^2 = -1
> > > So i^4 = 1
> > > But
> > > Sqrt[1] = +-1
> > > So Sqrt[i^4] = +-1
> > > So i^2 = +-1 since Sqrt[i^4]=(i^4)^(1/2)
> > >
> > > You can see that unless you keep straight just what the square root is
> > > defined to be, you can go very far afield by continuing the square roots to
> > > result in i=1, which is nonsense.
> >
> > Indeed, that is why one should never
> > - put anything but positive numbers under the square root sign.
> > - write something like sqrt(4) = +- 2
>
> bull again fool, this is not about you, but
> about the process under observation
>
> >
> > Only in very limited contexts (when writing about complex
> > numbers and complex functions), one can work with so called
> > "multi-valued" functions and "principal values", i.o.w. with
> > functions from C to the power set of C. In that special case
> > one can write something like
> > (-1)^(1/2) = { i, -i }
>
> wrong moron, (-1)^(1/2) only gives i,
> what a moron ...ahaha AHAHA ahaha...

Not in the context I sketched.
So what do we have here... you are an ignoramus in physics,
you are an ignoramus in manners, now you turn out to be an
ignoramus in mathematics as well :-)

Dirk Vdm


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