KAY174MATHEMATICS

Prof. Dr. Doğan Nadi Leblebici

ALGEBRA REFRESHMENT AND EQUATİONS

PURPOSE: TO GIVE A BRIEF REVIEW OF SOME TERMS AND METHODS OF MANIPULATIVE MATHEMATICS.

SETS AND REAL NUMBERS

A SET is a collection of objects. Example: The set A of even numbers between 5 and 11.

The Set A={6, 8, 10}

An OBJECT in a set is called a member or element of that set.

The Object in the Set A is 6 or 8 or 10.

Positive integers are natural numbers. Example = {0, 1, 2, 3, ….}

Rational numbers are numbers such as ½ and 5/3, which can be written as ratio (quotient) of two integers.

SETS AND REAL NUMBERS

A rational number is one that can be written as p/q where p and q are integers and q≠0. Because we can not divide by 0.

All integers are rational.

All rational numbers can be represented by decimal numbers that terminate or by nonterminating repeating decimals.

Terminating Decimal: 3/4=.75Nonterminating Repeating Decimal: 2/3=.6666….Nonterminating Nonrepeating Decimals are called irrational numbers. Example: ∏ (Pi) and √2 are irrational.

SETS AND REAL NUMBERS

Together, rational numbers and irreational numbers form the set of real numbers.

SETS AND REAL NUMBERS

Real Number Venn Diagram

SOME PROPERTIES OF REAL NUMBERS

1. The Transitive Property

If a = b and b = c, then a = c.

2. The Commutative Property

x + y = y + x or x × y = y × xWe can add or multiply two real number in any order.

3. The Associative Property

a + (b + c) = (a + b) + c or a(bc) = (ab)c

SOME PROPERTIES OF REAL NUMBERS

4. The Inverse Property

Additive Inverse: a + (-a) = 0 or Multiplicative Inverse a.a-1 = 1

5. The Distiributive Property

a(b + c) = ab + ac and (b + c)a = ba + ca

OPERATIONS WITH SIGNED NUMBERSPROPERTY EXAMPLEa – b = a + (-b) 2 – 7 = 2 + (-7) = -5a – (-b) = a + b 2 – (-7) = 2 + 7 = 9-a = (-1)(a) -7 = (-1)(7)a(b + c) = ab + ac 6(7 + 2) = 6.7 + 6.2 = 54a(b - c) = ab - ac 6(7 - 2) = 6.7 - 6.2 = 30-(a + b) = -a - b -(7 + 2) = -7 – 2 = -9-(a - b) = -a + b -(7 - 2) = -7 + 2 = -5-(-a) = a -(-2) = 2a(0) = (-a)(0) = 0 2(0) = (-2)(0) = 0

OPERATIONS WITH SIGNED NUMBERSPROPERTY EXAMPLE(-a)(b) = -(ab) = a(-b) (-2)(7) = -(2.7) = 2(-7) = -14(-a)(-b) = ab (-2)(-7) = 2.7 = 14a/1 = a 7/1 = 7 or -2/1 = -1a/b = a(1/b) 2/7 = 2(1/7)1/-a = -1/a = 1/-4 = -1/4 = a/-b = -a/b = 2/-7 = -2/7 =-a/-b = a/b -2/-7 = 2/70/a = 0 0/7 = 0a/a = 1 2/2 = 1

a

1

4

1

b

a

7

2

OPERATIONS WITH SIGNED NUMBERSPROPERTY EXAMPLEa(b/a) = b 2(7/2) = 7a(1/a) = 1 2(1/2) = 1

baba .

11.1

14

1

7.2

1

7

1.2

1

c

bab

c

a

c

ab

3

727

3

2

3

7.2

c

a

bcb

a

bc

a 11

7

2

3

1

7

1

3

2

7.3

2

bc

ac

c

c

b

a

b

a

5.7

5.2

5

5

7

2

7

2

bc

a

cb

a

cb

a

))(()( 5.3

2

)5)(3(

2

)5(3

2

c

ba

c

b

c

a

9

5

9

32

9

3

9

2

c

ba

c

b

c

a 9

1

9

32

9

3

9

2

OPERATIONS WITH SIGNED NUMBERSPROPERTY EXAMPLE

bd

bcad

d

c

b

a

15

22

3.5

2.53.4

3

2

5

4

bd

bcad

d

c

b

a

15

2

3.5

2.53.4

3

2

5

4

bd

ac

d

c

b

a.

15

8

5.3

4.2

5

4.3

2

b

ac

cba

3

10

3

5.2

532

bc

a

cba

15

2

5.3

2

532

bc

ad

c

d

b

a

dcba

.21

10

7.3

5.2

7

5.3

2

5732

EXPONENTS AND RADICALS

The product “x.x.x” is abbreviated “x3”. In general, for n a positive integer, xn is the abbrevation for the product of n x’s. The letter n in xn is called the exponent and x is called the base.

LAW EXAMPLE

İf x≠0

nmnm xxx . 25622.2 853

10 x 120

nn

xx

1

8

1

2

12

33

nnx

x

182

2

1 33

EXPONENTS AND RADICALSLAW EXAMPLE

İf x≠0

mnnm

n

m

xx

x

x

1

1622

1162

2

2 44

48

12

1m

m

x

x1

2

24

4

mnnm xx )( 1553 2)2(

nnn yxxy )( 64.842)4.2( 333

n

nn

y

x

y

x

27

8

3

2

3

23

33

EXPONENTS AND RADICALSLAW EXAMPLE

nn

x

y

y

x

9

16

3

4

4

322

nn xx 1

2

1

4

1

4

14

21

21

nn

n

xxx

111

1

551

33

nnn xyyx 333 1829

nn

n

y

x

y

x 33

3

3

910

90

10

90

EXPONENTS AND RADICALSLAW EXAMPLE

mnm n xx 123 4 22

mnn mnm

xxx 42888 2233 23

2

xxm

m 778

8

OPERATIONS WITH ALGEBRAIC EXPRESSION

If numbers, represented by symbols, are combined by the operations of addition, substraction, multiplication, division, or extraction of roots, then the resulting expression is called an algebraic expression. For example:

3

3

10

253

x

xx

is an algebraic expression in the variable x.

OPERATIONS WITH ALGEBRAIC EXPRESSION

In the fallowing expression, a and b are constants, 5 is numerical coefficient of ax3 and 5a is coefficient of x3.

325 3 bxaxAlgebraic expression with one term is called monominals, two terms is binominals, three terms is trinominals, more than one term is also called multinominal.

OPERATIONS WITH ALGEBRAIC EXPRESSION

A polynominal in x is an algebraic expression of the form

011

1 ..... cxcxcxc nn

nn

Where n is a positive integer and c0, c1, …..cn are real numbers with cn≠0. We call n the degree of polynominal. Thus, 4x3 – 5x2 + x – 2 is a polynominal in x of degree 3. A nonzero constant such as 5 is a polynominal of degree zero.

OPERATIONS WITH ALGEBRAIC EXPRESSION

Below is a list of special products that can be obtained from distributive property and are useful in multiplying algebraic expressions.

xzxyzyx abxbaxbxax 2

cdxcbadabxdbxcax 2

222 2 aaxxax

222 2 aaxxax

OPERATIONS WITH ALGEBRAIC EXPRESSION

Below is a list of special products that can be obtained from distributive property and are useful in multiplying algebraic expressions.

22 axaxay

32233 33 axaaxxax

32233 33 axaaxxax

FACTORING

If two or more expressions are multiplied together, the expressions are called factors of the product. Thus if c=ab, then a and b are both factors of the product c. The process by which an expression is written as a product of its factors is called factoring. Listed below are factorization rules.

)( zyxxzxy

))((2 bxaxabxbax

FACTORING

Listed below are factorization rules.

))(()(2 dbxcaxcdxcbadabx 222 )(2 axaaxx 222 )(2 axaaxx ))((22 axaxax ))(( 2233 aaxxaxax ))(( 2233 aaxxaxax

EQUATIONS – LINEAR EQUATIONS

An equation is a statement that two expressions are equal. The two expressions that make up an equations are equal. The two expressions that make up an equation are called its sides or members. They are seperated by the equality sign “=“. Examples:

32x0232 xx

75

yy

EQUATIONS – LINEAR EQUATIONS

Each equation contains at least one variable. A variable is a symbol that can be replaced by any one of a set of different numbers. When only one variable is involved, a solution is also called a root. The set of all solutions is called the solution set of the equation.

EQUATIONS – QUADRATIC EQUATIONS

A quadratic equation in the variable x is an equation that can be written in the form

ax2 + bx + c = 0

Where a, b, and c are constants and a ≠ 0

EQUATIONS – QUADRATIC EQUATIONS

Quadratic Formula

a

acbbx

2

42