This page is not a tutorial on complex arithmetic. It is a page that will let students pracitce their skill in basic complex arithemtic (addition, subtraction, multiplication and division) while aimultaneously presenting a graphical representation of the operations. For a description of these operations check another page such as Wikipedia, or Wolfram.

**To use this demo:** Simply enter the complex numbers *x* and *y* in either rectangular or polar form. The page shows the numerical results of addition, subtraction, multiplication and division. To see the results on the graph, simply put the cursor over the corresponding result (i.e., addition, subtraction, multiplication or division). The number x is represented by a red dot on the graph, the number y by a blue dot, and the result of the chosen operations is shown as a magenta (●) dot.

Enter x. , , or , .

Enter y. , , or , .

x = a + *j*b = 1 + 1*j* = M_{x}∠θ_{x} = 1.414∠45°

y = c + *j*d = -1 + 1*j* = M_{y}∠θ_{y} = 1.414∠-45°

Note: For x we get $M_x=\sqrt{a^2+b^2}, \quad \theta_x=\mathrm{atan2}(b,a), \quad a=M_x\cdot cos(\theta_x), \quad b=M_x\cdot sin(\theta_x).$

We get similar expressions for y

Move x by dragging red dot with mouse. Move y with shift+mouse.

p = x + y = (a+c) + *j*(b+d)

p = 0 + 2*j* =
2∠90°

Add the blue vector (representing y) to the red vector (representing x) to get the resulting magenta vector (representing p).

yis shown in faint blue line added to x.

q = x - y = (a-c) + *j*(b-d)

q = 2 + 0*j* =
2∠0°

Subtract the blue vector (representing y) from the red vector (representing x) to get the resulting magenta vector (representing p)

yis shown in faint blue subtracted from x.

r = x·y = (M_{x}·M_{y})∠(θ_{x}+θ_{y})

r = 2 + 0*j* =
2∠0°

Multiply magnitudes, and add phases.

Note: you can also multiply with rectangular form

r = x·y = (a·c-b·d)+j(a·d+b·d)

s = x/y = (M_{x}/M_{y})∠(θ_{x}-θ_{y})

s = 0 + 1*j* =
1∠90°

Divide magnitudes, and subtract phases.

Note: you can also divide with rectangular form

s = x/y = ((a·c+b·d)+j(a·d-b·d)) / (c^{2} + d^{2})

© Copyright 2005 to 2015 Erik Cheever This page may be freely used for educational purposes.

Erik Cheever Department of Engineering Swarthmore College