|r|=-1

[Shadow]

Hey all,

 

I was wondering, if one were to define a number r, for which [math]|r|=-1[/math], would it be possible to logically deduce it's behavior in mathematical operations? I know that the very concept is unimaginable and I also know it would be useless. But if we can have square roots of negative numbers, why not numbers denoting negative length? I'd be interested to know what [math]r^r[/math] would be, for example.

 

Cheers,

 

Gabe

[the tree]

I'm going to say no. |.| is usually defined as mapping from something to [imath]\mathbb{R}^{+}[/imath].

 

Norms and magnitudes are always positive.

 

The thing with the existence of the the complex field is that it can be easily constructed (see Kronecker) in a way that instantly tells us some of it's basic properties.

[Bignose]

The reason that it won't exist is because of the definition of the norm. There is a symmetry present that will always result in a positive number.

 

I.e. [math]| \mathbf{r} | = (\mathbf{r},\mathbf{r}) = \int_\Omega\omega(\mathbf{x}) \mathbf{r}(\mathbf{x}) \mathbf{r}(\mathbf{x}) d \mathbf{x}[/math] where the (,) notation stands for inner product and [math]\omega(\mathbf{x})[/math] is the weight function of the metric space, which is defined to be positive for all x, too. (Depending on the details, often the square root of the right hand side is taken.)

 

But, all the terms on the RHS are positive, because the weight function is required to be positive and when you take the inner product of a term (a number, vector, function) of itself, it will always be positive. That's just the way all these things are defined to be. Now, you can change the definitions, I guess, but that won't really answer the question.

 

So, really, no -- given the current commonly accepted definition there is no such thing.

[Shadow]

Whoa, that's a little too sophisticated an answer for me to absorb, but I understand enough to get the gist. Thanks guys )

[Norman Albers]

Very cool question as I am just at this looking into General Relativity. What do we do with the concept of the vector "norm" when [math]ds^2[/math] is negative? I'm trying to make sense of |ds|= i . Somebody brilliantly defined the square root of minus one. Can we define some sort of fantasy numbers? They'd need to be orthogonal to the complex plane...

[the tree]

The quarterions are orthogonal to the complex plane, they still have positive magnitude. The issue isn't a case of inventing the number, it's a case of being able to say anything about that number apart from the property you've defined it as.

[timo]

Very cool question as I am just at this looking into General Relativity. What do we do with the concept of the vector "norm" when [math]ds^2[/math] is negative?

We [physicists] call the stuff pseudo-norm and pseudo-metric when/if talking to mathematicians.

[Norman Albers]

I figure you mathheads can characterize such things well in terms of symmetries and such. Given i, successive multiplications give us a rotation or toggling between <1, i, -1, -i>, and this is hugely useful as complex analysis. 'twould be cool if this could be extended to the fantasy numbers. Good one, Tree, this makes me want to learn what quaternions are. . . . .Having started reading Wiki on them, this is quite exciting, thank you!!! Is this Hamilton the Hamilton of mechanics and such? I see there was a Quaternion Society, I shall look it up, along with Phlogiston. Atheist, no problem, I've been called worse. Seriousness aside, I don't buy it when people say, here timelike intervals are spacelike and vice versa. This seems like a copout to me and so I have suggested we acknowledge that, say inside a black hole, radial propagation has a positive [math] [\frac {dr}{cdt}]^2 [/math], but not tangential, like [math] [\frac {rd\theta}{cdt}]^2. [/math] I am getting into complexifying GR here: http://www.scienceforums.net/forum/showthread.php?p=484892#post484892 .

Edited April 13, 2009 by Norman Albers

[Norman Albers]

Tree, the question I am looking at is the relevance of quaternions to angular momentum formulation in QM.

[Norman Albers]

The only previous encounter I've had with quaternions was reading Doug Sweetser on another distant forum in a faraway galaxy. If I recall correctly he rejected their usefulness, but my exchanges there were truncated.