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1.1 Algebra

1.1.1 Laws of Algebraic Operations

Commutative law:a + b = b + a

ab = ba

Associative law:a + (b + c) = (a + b) + c

a(bc) = (ab)c

Distributive law: a(b + c) = ab + ac

1.1.2 Powers and Roots

0if10 = aa

a

a

x

x

=1

)( yxyxaaa

+=

a

aa

x

y

x y=

( )

xxxbaab =)(

xyyxaa =)(

x

xx

b

a

b

a=

x

x

x

b

a

b

a=

xx aa =

1

xxx baab = xyx y aa =

y xy

x

aa =

1.1.3 Proportions

Ifa

b

c

d= then the following are true;

a b

b

c d

d

+=

+

,

a b

b

c d

d

=

,

a b

a b

c d

c d

+=

+

1.1.4 The Binomial Formula

The Binomial Formula for a positive integer n;

nnnnnn yyxnnnyxnnynxxyx +++++=+ ....!3

)2)(1(!2

)1()( 33221 (1-1)

Special cases of the binomial formula:

(x + y)2

= x2

+ 2xy + y2

(x - y)2

= x2

- 2xy + y2

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HANDY FORMULAS REV005 2

(ax - by)2

= a2x2

- 2abxy + b2y2

(x + y)3

= x3

+ 3x2y + 3xy

2+ y

3

(x - y)3

= x3

- 3x2y + 3xy

2- y

3

(ax - by)3

= a3x3

- 3a2bx

2y + 3ab

2xy

2+ b

3y3

(x + y)4

= x4

+ 4x3y + 6x

2y2

+ 4xy3

+ y4

(x - y)

4

= x

4

- 4x

3

y + 6x

2

y

2

- 4xy

3

+ y

4

(x + y)

5= x

5+ 5x

4y + 10x

3y2

+ 10x2y3

+ 5xy4

+ y5

(x - y)5

= x5

- 5x4y + 10x

3y2

- 10x2y3

+ 5xy4

- y5

(x + y)6

= x6

+ 6x5y + 15x

4y2

+ 20x3y3

+ 15x2y4

+ 6xy5

+ y6

(x - y)6

= x6

- 6x5y + 15x

4y2

- 20x3y3

+ 15x2y4

- 6xy5

+ y6

Special Products;

x2

- y2

= (x - y)(x + y)

x3

- y3

= (x - y)(x2

+ xy + y2)

x3

+ y3

= (x + y)(x2

- xy + y2)

x

4

- y

4

= (x - y)(x + y)(x

2

+ y

2

)x5

- y5

= (x - y)(x4

+ x3y + x

2y2

+ xy3

+ y4)

x5

+ y5

= (x + y)(x4

- x3y + x

2y2

- xy3

+ y4)

x6

- y6

= (x - y)(x + y)(x2

+ xy + y2)(x

2- xy + y

2)

x4

+ x2y2

+ y4

= (x2

+ xy + y2)(x

2- xy + y

2)

x4

+ 4y4

= (x2

+ 2xy + 2y2)(x

2- 2xy + 2y

2)

1.1.5 Zero and Infinity Operators

a0 = 0 a = 0 indeterminate

00 =a

=0

a

0

0

indeterminate

=a

a

= 0

indeterminate

10 =a 00 =

a

00

indeterminate

a = - a = 0

indeterminate

a - a = 0 - indeterminate

Ifa > 1 then; =

a 0=

a

Ifa = 1 then;1=

a 1=

a Ifa < 1 then; 0=

a =

a

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1.1.6 Definition of Imaginary and Complex Numbers1

Roots of negative numbers have been used

since the 1750's, when the concept imaginary

was devised. Mathematicians of the 17th

century used a book by Raffaele Bombelli,

written in 1572, containing the theory of

imaginary numbers. The theory was further

advanced by the contributions of Johann

Bernoulli (1667-1748), Leonhard Euler (1707-

1783), and Carl Friedrich Gauss (1777-1783).

The representation of complex numbers in

the plane is attributed to Caspard Wessel

(1745-1818) and Jean Robert Argand (1768-

1822).

General Definition of Imaginary Numbers

A complex number is generally written as

a+ib, where a and b are real numbers, and i,

called the imaginary unit, has the propertyi= -1. The real numbers a and b are called thereal and imaginary parts of the complex

number a+ib, respectively.

Figure 1-1: The Gaussian Plane.

Complex Conjugates

The complex numbers a+ib and a-ib are called complex conjugates of each other.

Graphing a Complex Number

A complex number a+ib can be plotted as a point (a, b) on thex-y plane, as shown in Figure 1-1. The diagram iscalled theArgand diagram or the Gaussian plane. The imaginary number can thus be interpreted as the vector OP.

Polar Form of the Complex Number

The point P in Figure 1-1 can also be represented in a polar form. From the figure we see that the terminal point of

the vector OP is a = rcos and b = rsin . Hence, we may write

( )+=+ sincos iriba

The modulus,r, and the amplitude, , or argumentofa+ib are given by22

bar += and ( )ab1tan=

Representation of Complex Numbers

The following representations are commonly used for the complex number a+ib in Figure 1-1:

1 The VNR Concise Encyclopedia of Mathematics, 2nd Ed., pg 77-80, and Handbook of Mathematical, Scientific, and Engineering Formulas,

Tables, Functions, Graphs, Transforms, 1984. Pg. 288-289.

y

x

r

P

O

y

x

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( ) ( ) ( ) ===+=+= rrreiribaba i ,sincos,

1.1.7 Euler's Theorem of Complex Numbers

Refer to Figure 1-1:

=+i

ei sincos (1-2)

DERIVATION:

Using Taylor expansion we can write each term in Equation (1-2) as follows:

...!5!4!3!2

15432

+

+

+

+

++=e

...!6!4!2

1cos642

+

+

=

...

!7!5!3

sin753

+

+

=

Combining those yields:

=

+

+

+=

+

+

+

+

+

=+

ieii

ii

...!5!4!3!2

1

...!7!5!3

...!6!4!2

1sincos

5432

753642

QED

1.1.8 Arithmetic Operations Using Complex Numbers2The vectors A and C are defined as follows;

==+=

==+=

rseidc

rreiba

i

i

C

A

Where; a, b, c, d, k, n, r = Constants.

, = Angles in radians.

Then, the following arithmetic operations can be derived for complex numbers.

Equality of dbcaidciba ==+=+ and

Addition of ( ) ( ) ( ) ( )dbicaidciba +++=+++=+ CA

2Handbook of Mathematical, Scientific, and Engineering Formulas, Tables, Functions, Graphs, Transforms, 1984. Pg. 288-289.

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(f) For n = 0 we have; = 32.1034)96.3034( 33/131/

C

For n = 1 we have;+=+ 12032.1034)36096.3034( 3

3/131/C

For n = 2 we have;+=+ 24032.1034)360296.3034( 3

3/131/C

The three roots are depicted in Figure 1-2:

Figure 1-2: Graphical solution to part (f).

x

y

120

120

120

10.32