Re: complex numbers

jem <xxx@xxxxxxx> wrote in <FVere.91014$sy6.84187@lakeread04> on Monday 13
June 2005 07:20 posted to sci.physics.relativity:

> 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.

Yeah, but that doesn't really answer his question.

>
>>
>>
>> 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.

Those definitions usually boil down to a variant of what I said.

>
>>
>> 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.

After I sent the post out, I realized I should have added the stipulation
about complex times its conjugate. - My bad.

[...]
>> 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.

But there are _always_ two possibilities in taking the square root of a
number. So Sqrt[1] = +-1.

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

[smile]

>
>>
>> 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!
.