chapter 2 - number system ii

8/13/2019 Chapter 2 - Number System II

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Chapter 2 :Number System (cont.)

SM0013 Mathematics I

Khadizah GhazaliLecture 4 15/06/2011


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Usingi

Now we can handle quantities thatoccasionally show up in mathematicalsolutions

What about

1a a i a = =

49 18


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???


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Warning

Consider It is tempting to combine them

The multiplicative property of radicals only worksfor positive values under the radical sign

Instead; use imaginary numbers

16 49

16 49 16 49 4 7 28 = + = =

216 49 4 7 4 7 28i i i = = =


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Example 2.12 :

Given that the following complex numbers as,

Identify the real part and the imaginary part of

these complex number.


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Solution :


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Example 2.13 :


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Operations on Complex

Numbers

Complex numbers can be combined withAddition

Subtraction

Multiplication

division

( ) ( )3 8 2i i + +

( ) ( )1 4 2i i +

( ) ( )2 4 4 3i i + 6 7i

11 i

6 2i


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Division technique

Multiply numerator and denominator bythe conjugate of the denominator

3

5 2

i

i2

2

3 5 2

5 2 5 2

15 625 4

6 15 6 15

29 29 29

i i

i i

i ii

i

i

+=

+

+=

+

= = +


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Square roots

Ifz2

= a + bi andz =x + yi, therefore

Using the properties of complex numberequality we have

( )

( ) ( )

2

2 2 2

x yi a bi

x y xy i a bi

+ = +

+ = +

2 2y a = 2xy b=and


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Example 2.14 :Example 2.14 :


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Example 2.14 :Example 2.14 :


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Argand Diagram

Any complex number z = a + bican berepresented by any ordered pair (a, b)

and hence plotted on xy-axes with the realpart measured along x-axis and the

imaginary part along the y-axis. Thegraphical representation of the complex

number field is called an Argand diagram.


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Example 2.15:

Represent the following complex numberson an Argand diagram.

(a) z = 3 + 2i

(b) z = 4 5i


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SolutionIm (y)

Re (x)1 2 3 4 5 6-3 -2 -1

-2

-3

-4

-5

2

1

(3,2)

(4,-5)


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Modulus and argument of

complex number

modulus of a complex number

argument of complex number


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Properties of modulus


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Properties of argument


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Polar CoordinatesPolar Coordinates

With siny r =cos ,x r =

z takes the polar form:

)sin(cos

jrz +=


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Using Polar Representation

Recall that a complex number can berepresented as

Then it follows that

What about z3 ?

( )cos sinz r i = +

( ) ( )

( )

2

2

cos sin cos sin

cos 2 sin 2

z r i r i

r i

= + +

= +


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DeMoivre's Theorem

In general (a + bi)n

is

Apply to

Try

( ) ( )( )cos sinn nz r n i n = +

)( )4

3 cos330 sin 330i+

12

2 2

2 2

i

+


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Using DeMoivre to Find Roots

Again, starting with

also

works when n is a fraction Thus we can take a root of a complex number

( )cos sina bi z r i + = = + ( ) ( )( )cos sinn nz r n i n = +

1/ 1/ 360 360cos sinn n k k

z r in n

+ + = +


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Using DeMoivre to Find Roots

Note that there will be n such roots

One each for k = 0, k = 1, k = n 1

Find the two square roots of

Represent as z = r cis

What is r?

What is ?

1/ 1/ 360 360cos sinn n k kz r in n

+ + = +

1 3i +


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Graphical Interpretation of Roots

Solutions are:

( )( )1/ 2

1 3

2 cos120 sin120

120 1202 cos sin

2 2

120 360 120 3602 cos sin2 2

i

i

i

and i

+

= +

= +

+ + +

2 6

2 2i+

2 6

2 2i

Roots will be equally spaced

around a circle with radius r1/2

Roots will be equally spacedaround a circle with radius r1/2

2


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Graphical Interpretation of Roots

Consider cube root of 27

Using DeMoivre's Theorem27 27 0cis=

1/ 33 0 36027 27 cis3

3 cis 0, 3 cis120, 3 cis 240

k

and

+ =

=

3 0i+

3 3 32 2

+

3 3 3

2 2





S i 2 4


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Section 2.4

Inequalities of Real Numbers

is a statement that shows the relationshipbetween two (or more) expressions with

one of the following five signs:


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2.16 :


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b)


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Section 2.7

The Absolute Value in

Real Numbers

Section 2 5


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Section 2.5The Absolute Value in Real Numbers

is the distance from thatnumber to the origin (zero) on

the number line. that distance is always given asa non-negative number.


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S ti f b l t l


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Some properties of absolute value:

S ti f b l t l ( t )


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Some properties of absolute value (cont.):


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2.17 :


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The Triangle Inequality


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2.18 :

chapter 2 - number system ii

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