Graphical explanation of multiplying and dividing complex numbers - interactive applets


The following applets demonstrate what is going on when we multiply and divide complex numbers.

See the previous section, Products and Quotients of Complex Numbers for some background.

The multiplication interactive

Things to do

In this first multiplication applet, you can step through the explanations using the "Next" button. You'll see examples of:

  • Multiplying by a scalar (a real number)
  • Multiplying by the imaginary number j = √(−1)
  • Multiplying by both a real and imaginary number

You can also use a slider to examine the effect of multiplying by a real number.


We start with a complex number 1 + 2j.

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The quotient interactive

The next applet demonstrates the quotient (division) of one complex number by another.

We have a fixed number, 5 + 5j, and we divide it by any complex number we choose, using the sliders.

Things to do

First, read through the explanation given for the initial case, where we are dividing by 1 − 5j.

Then, use the sliders to choose any complex number with real values between − 5 and 5, and imaginary values between − 5j and 5j.

The explanation updates as you change the sliders.


We start with a complex number 5 + 5j.

We divide it by the complex number .

In polar form, the two numbers are:

5 + 5j = 7.07 (cos 45o + j sin 45o)

The quotient of the two magnitudes is:

7.07 ÷ =

The difference between the two angles is:

45o =

So the quotient (shown in magenta) of the two complex numbers is:

(5 + 5j) ÷ ()

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Math used in these applets

Here is some of the math used to create the above applets.

  1. Adding, multiplying, subtracting and dividing complex numbers
  2. Converting complex numbers to polar form, and vice-versa
  3. Converting angles in radians (which javascript requires) to degrees (which is easier for humans)
  4. Absolute value (for formatting negative numbers)
  5. Arrays (complex numbers can be thought of as 2-element arrays, and that's how much ofthe programming is done in these examples
  6. Inequalities (many "if" clauses and animations involve inequalities)