Review of Complex - PowerPoint Presentation

Slide1

Review of Complex numbers

1

Exponential Form:

Rectangular Form:

Real

Imag

x

y

f

r

=|z

|

 Slide2

 The real and imaginary parts of a complex number in rectangular form are real numbers:

Real

Imag

x

=

Re(z)

y

=

Im

(z)

Therefore, rectangular form can be equivalently written as:

Real & Imaginary Parts of Rectangular FormSlide3

Real

Imag

x

f

r

=|z

|

The real and imaginary components of exponential form can be found using trigonometry:

Geometry Relating the Forms

y

Real

Imag

f

r

=|z

|

 Slide4

Geometry Relating the Forms: Real & Imaginary Parts

Real

Imag

r

=|z

|

The real and imaginary parts of a complex number can be expressed as follows:Slide5

Geometry Relating the Forms: Quadrants

In exponential form, the positive angle, , is always defined from the positive real axis. If the complex number is not in the first quadrant, then the “triangle” has lengths which are negative numbers. 

Real

Imag

x

y

f

r

=|z

|

Real

Imag

 Slide6

Real

Imag

x

y

r

=|z

|

Use Pythagorean Theorem

t

o find

in terms of

and

:

Geometry Relating the Forms:

in terms of

and

 Slide7

Geometry Relating the Forms: in terms of and  

Real

Imag

x

y

f

r

=|z

|

adj

opp

hyp

Use trigonometry

to find

in terms of

and

 Slide8

Summary of Algebraic Relationships between Forms

Real

Imag

x

y

f

r

=|z

|

 Slide9

Euler’s FormulaSlide10

)

Rectangular Form

:

Exponential Form

:

Consistency argument

If these represent the same thing, then the assumed Euler relationship says:

 Slide11

11

Euler’s Formula

Can be used with functions:Slide12

Addition and subtraction of complex

numbers is easy in rectangular form

12Addition & Subtraction of Complex Numbers

Addition and subtraction are analogous to vector addition and subtraction

Real

Imag

a

b

d

c

x

y

a

b

d

c

 Slide13

Multiplication of Complex Numbers

13

Multiplication

of complex numbers is easy in exponential form

Multiplication by a complex number,

, can be thought of as scaling by

and rotation by

Real

Imag

Magnitude scaled by

Angle rotated counterclockwise by

 Slide14

14

Division of Complex Numbers

Division of complex numbers is easy in exponential form

Division of complex numbers is sometimes easy in rectangular form

Multiply by 1 using the complex conjugate of the denominatorSlide15

Complex Conjugate

Another important idea is the COMPLEX CONJUGATE of a complex number. To form the c.c.:

change i -i

Real

Imag

x

y

f

r

=|z

|

The complex conjugate is a reflection about the real axisSlide16

The product of a complex number and its complex conjugate is REAL

.

Common Operations with the Complex Conjugate

Addition of the complex number and its complex conjugate results in a real number

Real

Imag

x

y

f

r

=|z

|

x