Re: Geometrical meaning of imaginary roots?

In article <1164124905.991803.107060@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<131208@xxxxxxxxx> wrote:
Consider two graphs y = x^2 + 1 and y = 0.

These two graphs have no intersection points in the real coordinate
system.

But these equations have common solutions ( root(-1), 0 ), ( -root(-1),
0 )
if we extend a range of x to the complex number.

How can we explain these complex points geometrically?
That is, what's the geometrical meaning of these common points?....

We think of the graph of a real function of real variable as a
collection of points on the plane because we think of the real numbers
as the line; so pairs of points correspond to coordinate points on R x
R, the plane.

Geometrically, we think of complex numbers as the plane as well, with
the horizontal axis representing the real part, and the vertical axis
representing the imaginary part. But that means that if you want to
"graph" f(x) = x^2 + 1 as a complex function of complex variable (that
is, x can be complex and f(x) is also complex), then you need points
in C x C, which is a 4-dimensional (real) space. If you graph that
function, and the function y = x on this 4-dimensional real space,
then you get two points of intersection, but it is hard to visualize
what you are doing since you need to imagine 4-d space.

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

.

Relevant Pages

  • Re: Exotic functions (elementary)
    ... A FUNCTION WHOSE GRAPH IS DENSE IN THE PLANE ... nonlinear additive functions can map onto ...
    (sci.math)
  • Re: Need help graphing an equation
    ... Think how you'd go about doing it with a calculator, a pen and a sheet of grid paper. ... That means they won't have intersection points, ... If you had only two different suppliers to compare, I'd suggest graphing four lines of z against x for four values of y, for each of two suppliers, and joining the four points of intersection to show the line. ... But the intersection line will have variable y, making it hard to interpret on a graph of z against x. ...
    (microsoft.public.excel.charting)
  • Re: Mike: One last explanation, I hope!
    ... If I want to make a graph which has an x range that goes from xmin to xmax ... If you want x coordinate zero to be at some position other than the ... Does that make sense for the horizontal plane? ...
    (microsoft.public.vb.general.discussion)
  • Re: chromatic number of the plane: open problem
    ... Is there a graph in the plane with finitely many ... for the edges and vertices of the hexagons. ... it might be known that a coloring ...
    (sci.math)
  • chromatic number of the plane: open problem
    ... Problem of chromatic number of the euclidean plane: ... Ron Graham: ``Euclidean Ramsey Theory", ... So D, being at unit distance from B and C, ... By deforming the Moser graph, one can fit it into a plane. ...
    (sci.math)