Re: complex numbers
- From: "Dirk Van de moortel" <dirkvandemoortel@xxxxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: Mon, 13 Jun 2005 13:29:24 GMT
"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.
> 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
Sqrt is a function defined for positive real numbers only.
The result is a positive real number (and positive includes zero).
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.
>
> 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
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 }
Dirk Vdm
> Same way with the nature of a complex
> number.
>
> This probably doesn't help, but I thought I'd give it a shot anyway.
>
> --
> // The TimeLord says:
> // Pogo 2.0 = We have met the aliens and they are us!
.
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