Complex Numbers Basic Operations - Online Math Tutor Juan Castaneda, MS

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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 XY-plane, where x, the x-coordinate of a point (x,y), is the real part of the complex number z = x + iy; while y, its y-coordinate, 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:

  • their Sum is:
S = Z + W= (x + iy) + (u + iv)= (x + u) + i(y + v)
  • their Difference is:
D = Z - W= (x + iy) - (u + iv)= (x - u) + i(y - v)
  • their Product is:
P = Z × W= (x + iy) × (u + iv)= (xu - yv) + i(xv + yu)
  • their Quotient is:
Q = Z / W= (x + iy) / (u + iv)= (xu + yv)/(u2 + v2) + i(uy - xv)/(u2 + v2)

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 X-axis, with identical x-coordinates but opposite y-coordinates), then both S and P will become real (they will place themselves on the X-axis); while D will become purely imaginary (it will be located somewhere on the Y-axis).


If the coordinate grid goes away you can reload the page, or reset the applet (click on its upper right corner), or right-click inside the applet & select the grid in the pop-up menu.


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Last review: April 7, 2017