© 2005 Baylor UniversitySlide 1

Fundamentals of Engineering AnalysisEGR 1302 - Introduction to Complex Numbers, Standard Form

Approximate Running Time - 20 minutesDistance Learning / Online Instructional Presentation

Presented byDepartment of Mechanical Engineering

Baylor University

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© 2005 Baylor UniversitySlide 2

0 5 1010

0

108.847

6.88

g x( )

100 x

Tout

T

Sinusoidal Response

Complex Numbers

Definitions and Formats Complex Numbers mathematically represent actual physical systems

Tin

SYSTEMTout

Feedback

Exponential Decay

© 2005 Baylor UniversitySlide 3

The General Quadratic Equation

02 cbxax

2

22

4

4)

2(

a

acb

a

bx

a

acb

a

bx

2

4

2

2

Take the Square Root

2

2

2

22

44 a

b

a

c

a

bx

a

bx

Complete the Square

a

cx

a

bx

2

a

acbbx

2

42

The Solution to theGeneral Quadratic Equation

© 2005 Baylor UniversitySlide 4

Solutions of the Quadratic Equation

By solution, we mean“roots”, or where x=0

5 0 510

0

1010

10

f x( )

55 x

2nd Order

5 0 510

0

1010

10

f x( )

55 x

3rd Order

a

bacbx

2

41 2

acb 42 If there is one real roota

acbbx

2

42

5 0 510

0

1010

10

f x( )

55 x

acb 42 If there are no real roots, as shown

acb 42 If there are two real roots, as above

© 2005 Baylor UniversitySlide 5

The Imaginary Number

0222 xxConsider:

12122 xxComplete the square:

11

1)1( 2

x

xTake the square root:

11 xThe solution:

1Because does not exist,

we call this an “imaginary” number,and we give it the symbol “ ”or “ ”.ji

becomes ix 1

© 2005 Baylor UniversitySlide 6

Complex Numbers

Substitute into 0222 xxix 1

02)1(2)1)(1( iii

022221 2 iii

11*1* 2 iii

0)1(1 Checks!

a

ibacbx

2

4 2A general solution for is24 bac

© 2005 Baylor UniversitySlide 7

z=x+iy

Complex Numbers

a

baci

a

bx

2

4

2

2

Definitions

The “Standard Form” Im(z)=yRe(z)=x

z=x+iy and if y=0, then z=x, a real number

i3=-i i4=1i2=-1 i5=i i6=-1

z1=x1+iy1Given z2=x2+iy2and Then ifz1=z2

y1 = y2

x1 = x2

© 2005 Baylor UniversitySlide 8

Algebra of Complex Numbers

z1=x1+iy1Definitions: Given z2=x2+iy2

z1+z2=z3z3= (x1 + x2) + i(y1 + y2)

z1-z2=z3z3= (x1 - x2) + i(y1 - y2)

z1 * z2=z3 z3= (x1 + iy1)(x2 + iy2) = x1x2+ix1y2+ix2y1+i2y1y2

z3= x1x2+i2y1y2 +i(x1y2+x2y1)

Im(z3)= x2y1-x1y2

Re(z3)= x1x2-y1y2

© 2005 Baylor UniversitySlide 9

Dividing Complex Numbers

32

1 zyix

bia

z

z

To divide, must eliminate the “i” from the denominator

We do this with the “Complex Conjugate” - by CHANGING THE SIGN OF i

22

22

2112212122

222222

2

221212121

22

22

22

11 )()(*

yx

yxyxiyyxx

iyiyxiyxx

iyyiyxixyxx

iyx

iyx

iyx

iyx

22

22

21213

)()Re(

yx

yyxxz

22

22

21213

)()Im(

yx

yxxyz

© 2005 Baylor UniversitySlide 10

Reciprocals of Complex Numbers

jjj

j

j

jj 13

3

13

2

13

32

94

32

32

32*

32

1

32

1

13

2)Re( z

13

3)Im( z

Multiply by the Complex Conjugate to put in Standard Form

jj

j

j

j

jj

2

*11

© 2005 Baylor UniversitySlide 11

This concludes the Lecture