Graphical representation of complex roots to equations

[RK1992]

I've never properly studied complex numbers but I will soon (in September). Basically:

We get taught from a young age that:
the real root of f(x)=x²-4 is where the graph of y=f(x) cuts the x axis

But is there a graphical representation of a complex root?

What's so special about the value x= +/- 2i if g(x)=x²+4 ? Is there a 3D graphical representation of this root?

Thanks in advance.

 

[Hurkyl]

Well, remember that "real root of f(x)" really means a number a such that f(a)=0. It doesn't need any geometric interpretation to make sense.

Some people like to think in terms of geometry rather than algebra; so the particular correspondence between them you invoked says that the roots of f(x) correspond to the intersection of the parabola defined by y=f(x) and the line y=0, just like you described.

The similar geometric interpretation for complex-valued functions of complex numbers requires 4 dimensions to draw. We can often get away with projecting onto a three-dimensional image, though. (Of course, we have to project that onto a two-dimensional image so that we can draw it, and things get messy)

Another common way is to instead draw a before and after picture of the complex plane. e.g. put a grid on the "before" picture, and in the "after" picture we see a picture of how f transformed the grid.


For this function, a good "before" picture is to make the grid out of rays emanating from the origin and circles whose center is the origin.

The after picture consists of lines passing through the point -4 + 0i, and circles centered on that point. The spacing between the circles is unchanged. However, the spacing between the lines has doubled, and the grid overlaps itself -- e.g. the two rays emanating from the origin at angles x° and (x+180)° both map to the same ray emanating from -4 + 0i at an angle of (2x)°.

In this picture, we can estimate the roots of f by looking at 0 + 0i in the "after" picture, identifying the grid points that lie there, and then finding where they came from in the "before" picture.

 

[jackmell]

I've never properly studied complex numbers but I will soon (in September).

But is there a graphical representation of a complex root?

Try and understand this Wolfram Demonstration:

http://demonstrations.wolfram.com/LocationOfComplexRootsOfARealQuadratic/

 

[HallsofIvy]

A simpler (or more simple minded) way of looking at it:

The equation $x^2- 2ax+ a^2+ b^2= (x-a)^2+ b^2= 0$ has roots \(\displaystyle x= a\pm bi\). It has complex roots, of course, because its graph does not cross the x-axis.

The vertex of the graph is where x= a so that itex]y= (x-a)^2+ b^2= b^2[/itex]. That is, the graph goes down to $(a, b^2)$ and then back up again. In general, if the graph of $y= x^2- ax+ b$ lies entirely above the x-axis, and its vertex is at $(x_0, y_0)$, then its roots are $x_0\pm i\sqrt{y_0}$.

 

[CellCoree]

I find the concept of complex roots fascinating and essential in understanding many mathematical and scientific phenomena. While we are often introduced to real roots and their graphical representations at a young age, complex roots and their graphical representations are typically explored in more advanced mathematics courses.

To answer your question, yes, there is a graphical representation of complex roots. Complex roots correspond to points on the complex plane, which is a two-dimensional plane where the horizontal axis represents the real numbers and the vertical axis represents the imaginary numbers. The value of x=+/-2i in the equation g(x)=x²+4 can be represented as points on the y-axis at 2i and -2i. These points are known as the imaginary roots of the equation.

In terms of a 3D graphical representation, it is possible to plot complex roots in three dimensions, but it requires a different type of graph known as a Riemann surface. This type of graph is commonly used in complex analysis to visualize complex functions and their roots.

Overall, understanding complex roots and their graphical representations is crucial in many areas of science, from quantum mechanics to electrical engineering. I encourage you to continue your studies in this area, and I am confident that you will find it both challenging and rewarding. Best of luck in your studies!