Download KAY174 MATHEMATICS

KAY174
MATHEMATICS
I
Prof. Dr. Doğan Nadi Leblebici
ALGEBRA REFRESHMENT AND EQUATIONS
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=.75
Nonterminating 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 × x
We 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 NUMBERS
PROPERTY
a – b = a + (-b)
a – (-b) = a + b
-a = (-1)(a)
a(b + c) = ab + ac
a(b - c) = ab - ac
-(a + b) = -a - b
-(a - b) = -a + b
-(-a) = a
a(0) = (-a)(0) = 0
EXAMPLE
2 – 7 = 2 + (-7) = -5
2 – (-7) = 2 + 7 = 9
-7 = (-1)(7)
6(7 + 2) = 6.7 + 6.2 = 54
6(7 - 2) = 6.7 - 6.2 = 30
-(7 + 2) = -7 – 2 = -9
-(7 - 2) = -7 + 2 = -5
-(-2) = 2
2(0) = (-2)(0) = 0
OPERATIONS WITH SIGNED NUMBERS
PROPERTY
(-a)(b) = -(ab) = a(-b)
(-a)(-b) = ab
a/1 = a
a/b = a(1/b)
1
1/-a = -1/a =  a
a

a/-b = -a/b = b
-a/-b = a/b
0/a = 0
a/a = 1
EXAMPLE
(-2)(7) = -(2.7) = 2(-7) = -14
(-2)(-7) = 2.7 = 14
7/1 = 7 or -2/1 = -1
2/7 = 2(1/7)
1
1/-4 = -1/4 =  4
2

2/-7 = -2/7 = 7
-2/-7 = 2/7
0/7 = 0
2/2 = 1
OPERATIONS WITH SIGNED NUMBERS
PROPERTY
a(b/a) = b
a(1/a) = 1
1 1
1
. 
a b a.b
EXAMPLE
2(7/2) = 7
2(1/2) = 1
1 1
1
1
. 

2 7 2.7 14
ab  a 
b
  b  a 
c c
c
2.7  2 
7
  7  2 
3 3
3
a  a  1   1  a 
       
bc  b  c   b  c 
a  a  c  ac
    
b  b  c  bc
a
a
a


b(c) (b)(c) bc
2  2  1   1  2 
       
3.7  3  7   3  7 
2  2  5  2.5
    
7  7  5  7.5
a b ab
 
c c
c
a b a b
 
c c
c
2
2
2


3(5) (3)(5) 3.5
2 3 23 5
 

9 9
9
9
2 3 2  3 1
 

9 9
9
9
OPERATIONS WITH SIGNED NUMBERS
PROPERTY
EXAMPLE
a c ad  bc
 
b d
bd
a c ad  bc
 
b d
bd
a c ac
. 
b d bd
a
ac

b
b
c
a
b a
c
bc
4 2 4.3  5.2 22
 

5 3
5.3
15
4 2 4.3  5.2 2
 

5 3
5.3
15
2 4 2.4 8
. 

3 5 3.5 15
2 2.5 10


3
3
3
5
2
3 2  2
5 3.5 15
a
2
b  a . d  ad
c
b c bc
d
7
3  2 . 5  2.5  10
3 7 3.7 21
5
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
x m .x n  x m  n
23.25  28  256
x 0  1 İf x≠0
20  1
x
n
1
 n
x
1
n

x
x n
1 1
2  3 
2
8
1
3

2
8
3
2
3
EXPONENTS AND RADICALS
LAW
EXAMPLE
xm
1
mn
 x  nm
n
x
x
m
x
1
m
x
İf x≠0
( x m ) n  x mn
( xy) n  x n y n
n
x
x
   n
y
 y
n
212
1
4
4

2

16


2
 16
8
4
2
2
24
1
4
2
(23 )5  215
(2.4)3  2343  8.64
3
23
8
2
   3 
3
27
3
EXPONENTS AND RADICALS
LAW
EXAMPLE
 x
 
 y
 y
 
x
n
 x
1
1
x
x
1
n
n
n
n

x
1
n
1
n
x
x n y  n xy
n
x
x
n
y
y
2
3
 4  16
    
9
4
3
3
n
n
2
1
4
5
1
5 3
2

1
4
3
3
1
2
1
1


4 2
9 3 2  3 18
90 3 90 3

 9
3
10
10
EXPONENTS AND RADICALS
LAW
m n
x
m
EXAMPLE
x  mn x
n
 x 
n
 x
m
m
m
x
3 4
 x
n
m
8
2
3
2  12 2
 8 
3
2
 7  7
8
8
 8
3
2
 22  4
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:
3x  5 x  2
10  x
3
3
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.
5ax  2bx  3
3
Algebraic 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
cn x  cn 1 x
n
n 1
 .....  c1 x  c0
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.
x y  z   xy  xz
2
x  a x  b  x  a  bx  ab
2
ax  cbx  d   abx  ad  cbx  cd
2
2
2
x  a   x  2ax  a
x  a 
2
 x  2ax  a
2
2
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.
 y  a x  a  x  a
3
3
2
2
3
x  a   x  3ax  3a x  a
3
3
2
2
3
x  a   x  3ax  3a x  a
2
2
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.
xy  xz  x( y  z )
x  a  bx  ab  ( x  a)( x  b)
2
FACTORING
Listed below are factorization rules.
abx  (ad  cb) x  cd  (ax  c)(bx  d )
2
2
2
x  2ax  a  ( x  a)
2
2
2
x  2ax  a  ( x  a)
2
2
x  a  ( x  a)( x  a)
3
3
2
2
x  a  ( x  a)( x  ax  a )
3
3
2
2
x  a  ( x  a)( x  ax  a )
2
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:
x23
2
x  3x  2  0
y
7
y 5
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
 b  b  4ac
x
2a
2