whats the way to grasp the idea of complex numbers and how it differs from real numbers in plotting?

Answer 1

A complex number is just a single number, but it has two parts: a real part and an imaginary part. To graph a real number, you just graph it on the real number line (ex: to graph 2, draw a real number line and put a dot above 2), a single axis, one dimension. All the real numbers can be contained in one dimension.

To graph a complex number, there are two parts, so you need two axes, which means having two dimensions. It takes TWO dimensions to 'hold' all the complex numbers. To graph a complex number, you treat the real part as an x-value and the imaginary part as a y-value. (ex: z= -1+2i, you would draw a point at (-1,2)).

Hope this helps.

Answer 2

When plotting a single variable function, like y = f(x), a 2D graph is sufficient to show the function. For every x along the x axis, one plots the value f(x) in the y axis, and so we have a function curve.

Plotting a complex variable function, w = f(z), where z is a complex number, actually requires a 4 dimensional space to show the function. Since this is not practical, what's usually done is to do a "conformal mapping", which is a way of representing w = f(z), where w and z are complex numbers of the form x + iy. Complex numbers make a 2D array of values. For every complex number z = x + iy, the function w = f(z) generates a new complex number w = u + iv, and this is plotted on the 2D plane array of complex numbers. Of course, if what you did was to plot all the points, you'll end up with a black screen, so that isn't helpful. Instead, what you can do is to plot lines of x in integer values, and iy in integer values, ending up with new lines which 1) are usually curved, and 2) intersect at right angles (hence the term "conformal mapping", which meets the Cauchy-Riemann criteria). This is the way of plotting or visualizing functions of a complex variables, and below is a link giving an example. The grid on the left is mapped onto the circular grid on the right. Also check out the Wolfram site on conformal mapping, with more examples.

Answer 3

The fascinating thing about complex numbers is that you can represent them with the real part along a horizontal axis and the imaginary part along the vertical axis so that they are a vector:

z = a+bi is the vector (a,b) in this plane

and multiplication is defined! And roots!!!

in fact, the nth roots of -1 are evenly spaced around the unit circle.

etc, etc, etc.

I think that they are a lot of fun to work with.