KAY174 MATHEMATICS Prof. Dr. Doğan Nadi Leblebici.
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Transcript of KAY174 MATHEMATICS Prof. Dr. Doğan Nadi Leblebici.
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