# 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.

We start with a complex number 1 + 2*j*.

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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 + 5*j*, 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 − 5*j*.

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

The explanation updates as you change the sliders.

We start with a complex number 5 + 5*j*.

We divide it by the complex number .

In **polar form**, the two numbers are:

5 + 5*j* = 7.07 (cos 45^{o} + *j* sin 45^{o})

The quotient of the two magnitudes is:

7.07 ÷ =

The difference between the two angles is:

45^{o} − =

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

(5 + 5*j*) ÷ ()

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## Math used in these applets

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

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