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

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