Chapters
- Introduction
- Complex Multiplication Laws
- Real and Imaginary Number Multiplication
- Multiplicative Identity and Multiplicative Inverse
- Conjugate Multiplication and the Modulus
- Equations with Complex Solutions
- Visualization of Complex Number Multiplication
- Multiplication is Angle Addition
- Triangle Method
Introduction
We multiply 2 Complex numbers with by treating each of them as binomials and using the normal form of binomial multiplication, the FOIL process (First, Outer, Inner, Last)
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 conjugate of the product of 2 Complex numbers is equal to the product of the conjugates of 2 Complex numbers.
Example
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
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