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The Four Basic Arithmetic Operations with Complex Numbers
Complex numbers are represented as points in the complex plane. The complex plane is the XYplane, where x, the xcoordinate of a point (x,y), is the real part of the complex number z = x + iy; while y, its ycoordinate, is the imaginary part of z.
Move around complex numbers Z and W (shown as hollow purple points) to see how the points resulting from their four basic arithmetic operations move accordingly:
Points S, D, P, and Q represent the four complex numbers resulting from Z and W's sum, difference, product, and quotient, so they move when Z or W move.
In the interactive worksheet above, points Z and W represent two arbitrary complex numbers. You can move them around in the screen.
This will cause the other four points S, D, P,and Q to automatically move, according to the results of the four basic arithmetic operations of addition, subtraction, multiplication, and division with complex numbers Z and W.
If we write complex number Z as x + iy, and W as u + iv, then:
 S = Z + W  = (x + iy) + (u + iv)  = (x + u) + i(y + v) 
 D = Z  W  = (x + iy)  (u + iv)  = (x  u) + i(y  v) 
 P = Z × W  = (x + iy) × (u + iv)  = (xu  yv) + i(xv + yu) 
 Q = Z / W  = (x + iy) / (u + iv)  = (xu + yv)/(u^{2} + v^{2}) + i(uy  xv)/(u^{2} + v^{2}) 
If you put W exactly on point (1,0), this makes W = 1, and you will notice then both P and Q get on top of Z, because (Z)(1) = Z/1 = Z.
When you put W on point (1,0), this makes W = 1, and now P and Q will also coincide but they will be equal to Z, not to Z.
If you put W on point (0,1), this makes W = i, the imaginary unit. In this case P and Q will be directly opposite to each other across the origin (0,0) because 1/i, the complex reciprocal of i, equals i, and this makes
P = ZW = Z(i) = Z(i) = Z(1/i) = Z/i = (Z/W) = Q
You will also notice that S, D, and P will be points of the grid (with integer coordinates) whenever both Z and W are points of the grid with integer coordinates. However, the quotient Q = Z / W may not have integer coordinates, even when both Z and W do.
The distance from S to Z will always be the same as the distance from D to Z. Both S and D are equidistant from Z, directly opposite each other across Z. Their common distance to Z equals the norm of W, the distance from W to the origin (0,0).
When the norm of W is 1, that is, when the red circle goes through the unit points (1,0), (1,0), (0,1), and (0,1), then both P and Q will be on the blue circle, because their norm will be exactly equal to the norm of Z.
When you start moving both Z and W away from the origin (0,0), then very soon P will exit the screen, because the norm of the complex product P equals the product of the norm of Z times the norm of W.
When you put point Z on the origin (0,0), this makes Z = 0, and then P and Q will also be zero. They will coincide with Z in the origin (0,0); while S will equal W, and D will equal W.
When you put Z and W conjugate to each other (vertically symmetrical across the Xaxis, with identical xcoordinates but opposite ycoordinates), then both S and P will become real (they will place themselves on the Xaxis); while D will become purely imaginary (it will be located somewhere on the Yaxis).
If the coordinate grid goes away you can reload the page, or reset the applet (click on its upper right corner), or rightclick inside the applet & select the grid in the popup menu.

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