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Originally Posted by The_Big_K
This is something that happens when the title of the thread does not give any information about what's being discussed in the thread.
Let's first choose topic - Complex Integrals, Electro Magnetics or 4D.
-The Big K-
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the big K, thanks for your great concern.
now let me explain a bit about 4D imagination.
4d geometry is a good thinking exercise to try and visualise. Right, imagine a function that takes and returns a real number (ie it exists on the 'normal' number line). You can represent this as a line on a 2d graph, with aeis, say, x and y being at a 90^\circ angle to each other. In this 2 dimensional space, the axes x and y are said to be orthogonal - finding out the x position of a point gives you no information on the y position. You then extend this to three dimensions, we get the x, y and z axes. Again, knowledge of a point in 3D's position on one axis contains no information about the point's position on the other two axes - the three axes are again orthogonal.
Now think of this - you can project a 3D graph onto a 2D graph by taking a slice through at, say, a given Z value. For example, a sphere in 3d with describe either nothing, a circle or a point when projected for a given 'slice' into 2D.
This all works nice and easy because we live in 3D. The conceptual leap comes when you consider 4 orthogonal axes. In a function taking and returning complex values we require 4 values to describe the function's behaviour :
f(z_{1}) = f(a+bi)= z_{2} = c+di
Where z_{1,2} are complex and a, b, c, d are real. These four numbers can represent four orthogonal axis, just as for f(x)=y, x and y represent values on two orthogonal axes.
Try to imagine slices through this 4D space. They will be able to be represented as 3D graphs that you can visualize.