Re: complex numbers

The TimeLord wrote: 
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? 

Look up Quaternions.



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.


Dimension in mathematics is defined in a several different ways.


So, a complex number is a number that when multiplied by itself equals a
real number.

If both the Real and Imaginary parts of a complex number are non-zero the square of that number isn't Real.

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.

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.

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,

Sqrt() is defined to be a function so e.g. Sqrt(1) = 1, not +-1, and of course i^2 = -1, not +-1.

you can go very far afield by continuing the square roots to 
result in i=1, which is nonsense. Same way with the nature of a complex
number.

Your post does contain some nonsense, but it's not due to the nature of Complex numbers.


This probably doesn't help, but I thought I'd give it a shot anyway.

.

Relevant Pages

  • Re: complex numbers
    ... Dirk Van de moortel wrote: ... >>> i never understod whay complex ... >>> number only have two dimenssion, ... >> the imaginary part describes the square root of the negative. ...
    (sci.physics.relativity)
  • Re: complex numbers
    ... >> i never understod whay complex ... >> number only have two dimenssion, ... > the imaginary part describes the square root of the negative. ... The last sentence sounds like nonsense. ...
    (sci.physics.relativity)
  • Re: complex numbers
    ... > i never understod whay complex ... > number only have two dimenssion, ... a complex number is a number that when multiplied by itself equals a ... the imaginary part describes the square root of the negative. ...
    (sci.physics.relativity)