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# Graphical explanation of multiplying and dividing complex numbers - interactive applets

## Introduction

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.

scale

## 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.

real
j

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) ÷ ()