Multiplying Complex Numbers

with

which makes

is the Real part of the product

or

is the Imaginary part of the product

or

multiplied by

Example

and

with

or straight into

form

The

form already takes into account that there will be an

term, which will make

the opposite sign of the original product.

Example

and

with

Complex Multiplication Laws

Commutative Law

Example

and

then

Associative Law

Example

and

then

and

Multiplying 3 or more Complex numbers is the same process as 2 number multiplication, just with extra steps.

Distributive Law

Example

and

then

and

Real and Imaginary Number Multiplication

Real Multiplication

Multiplying a Complex number by a Real number

just magnifies or shrinks the components of the number by the magnitude of the Real number.

Example

and

then

Multiplying a Complex number by

doubles the

and

components and changes their sign.

Example

and

then

Multiplying 2 Real numbers together gives back a Real number as a product.

Imaginary Multiplication

Justification of

Example

and

then

Example

Ex.

and

then

Multiplying a Complex number by an Imaginary number

magnifies or shrinks the components by the magnitude of the Imaginary number, switches the magnitudes of the components and changes the sign of the y component.

Multiplicative Identity and Multiplicative Inverse

Multiplicative Identity

is the Multiplicative Identity of the Complex Numbers.

Example

and

then

Multiplying a Complex number by

gives back the Complex number as the product.

Multiplicative Inverse

The Multiplicative Inverse

of a Complex number

is

Example

and

then

This shows that

gives back the Multiplicative Identity as the product.

Conjugate Multiplication and the Modulus

Conjugate Multiplication gives the Square of the Modulus

When we multiply a Complex number

by its conjugate

, we obtain the square of the modulus

of the Complex number

. If

then

and

Example

and

The modulus of its conjugate

is also equal to

and

then

The Conjugate of a Product

and

then

z_1z_2=(1+3i)(4-2i)=(4+6)+(-2+12)i=10+10i

and

and

then

The Modulus of a Product

We want to show that the identity

holds true

Example

and

then

and then by taking the square root of both sides of the equation

we can deduce that

which is an important and useful identity that shows that the modulus of a product is equal to the product of each modulus.

Equations with Complex Solutions

We may encounter equations that involve the square of an unknown Complex number

set equal to another known Complex number

where we have to solve for the Complex number's Real and Imaginary parts

Example

with

then

we set the Real part of

equal to

and the Imaginary part of

equal to

and have two equations

and

or

we can solve for

and

by finding the modulus of

and by using the identity

we can say

and use this to find

and

and

and

and

and

then the first solution

and

and

then the second solution is

Checking the solutions

Visualization of Complex Number Multiplication

Introduction

Multiplication by

is a 90 degree rotation in the

. Multiplying a positive Real number by

switches the number from the positive

to the positive

Example

and

with

or just

and multiplying a negative Real number by

switches the number from the negative

to the negative

Multiplying a positive Imaginary number by

switches the number from the positive

to the negative

Example

and

and multiplying a negative Imaginary number by

switches the number from the negative

to the positive

Multiplying a Complex number by

switches the

and

components and changes the sign of the switching

component

Example

switches to

and

switches to

Example

The product of the multiplication of 2 Complex numbers is another Complex number

Example

and

then

Multiplication is Angle Addition

Triangle Method