"the" maths question

[Guest uut]

f(x)=(-2)^x

yep, that's the thing that have been bugging me for some weeks now. maybe you guys have an answer to this. Here goes:

consider f(x) is an harmless function (it's not exactly a function really, keep readin) with a domain of positive natural numbers. If you graphed it it would be something like:

the distance between the dots keep increasing as you see.

(-2)^1=-2

(-2)^2=+4

(-2)^3=-8

now, what would happen if the function was not defined in positive natural numbers but all the real numbers? The range would be complex. (-2)^(1/2) for example.

Let's say the imaginary axis is the z-axis. As the domain is still real numbers its in one dimention the graph would be 3-dimentional (not 4). So what would it look like ladies and gens? My crazy thoughs are it would be a spiral or.. err.. put infinite spheres on top of each other. The tangent points being natural numbers. I favor the second one more. Hope I could explain. Any ideas?

[Frei]

I compiled a table of xyz values to simplify the process of

visualising the data.

The list displays the real part of f(x)=(-2)^x as y and the imaginary

part as z, in reverse order.

x increment is 0.1, and the list goes from x=0 to x=4

Here's a 3d graph made by some old software I found.

I know it's not an answer to your question, it's just a stepping

stone, but it could help. At least it could help start a discussion.

Good luck!

Edited January 3, 2008 by Frei

[Guest uut]

Hey thanks for the answer! The graph is a bit unexpected by anyway.

Yet I think there is another problem. The table above shows only one root for each value of x. You see, f(x) is not really a function. (-2)^(1/2) has two roots for example. (so 1.1 would have 10 roots) As the denominator of the power of -2 increases, the frequency of the points would get closer to being a circle. And as there I can divide 1 by many many many times, there could be infinitely number of points at any place. At the same time they would merge into 10 roots at 1.1. When x is natural number, they would all merge into one real point.

(by the way I'm still sure that the graph would still be 3D. Real numbers go to complex numbers, it would only be 4 dimentional if it was C to C. I guess)