So today I was bored in class so I started to think about polynomials, specifically parabolas. I know that it is possible to find imaginary roots of a parabola. My question is when you graph this parabola that has two imaginary roots the curve never hits the x-axis then were are the imaginary roots at? When I asked my instructor he said he had no idea. My only guess was that there was a z-axis there the appeared somewhere but I really have no ideas. So where are the imaginary roots on the graph?

You can't find them graphically as the axes are real.

So if the axes where complex numbers then they would be there correct.

Except that axes are 1 dimensional entities and if you made them complex numbers then they would have to be 2 dimensional!

 

So you would have to plot a 4 dimensional graph, because your axes would become 2 dimensional planes. This is why it's often so much more difficult to do complex analysis, it's hard to conceptualise the extra degrees of freedom.

Since you're only interested in what values x must take for f(x) to be zero: you could do it with three axis, Re(x) Im(x) and |f(x)|.

 

edit see attachment of a plot of |f(x+iy)|, where f(x):=x²+2x+5 and has complex roots.

Edited November 9, 200817 yr by the tree
i dun a pichur