Transcript A and B

P.1 THE REAL NUMBERS SYSTEM
)‫(االعداد الحقيقية‬
Objectives:
Sets
Union and Intersection of sets
Absolute Value and Distance
Interval Notation
Order of Operations
Simplifying Variable Expressions
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Def: A set (‫ )المجموعة‬is a collection of distinct objects. The
objects in the set are called elements.
Typically, sets are represented by set builder notation:
{ x | x has some property }
The set of
such that
x has the given
property
all elements x
Ex:
{ x | 0 < x < 5, x is an integer}
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A = {1, 2, 3, . . .} is the set of natural numbers.
4A
–99  A
“is an element of ”
“is not an element of ”
C = { } or C =  both denote the empty set.
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The union of two sets, A  B, is the set of all elements that
belong to either A or B or both.
The intersection of two sets, A  B, is the set of all elements that
are common to both A and B.
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Ex: A = {1, 2, 3, 4, 5} B = {4, 5, 6, 7, 8}
C = {7, 8, 9, 10, 11}
1
2
3
A  B = {4, 5}
A
4
6
5
7
8
B
9
10
11
C
AC={}
A  C = {1, 2, 3, 4, 5, 7, 8, 9, 10, 11}
A and C are disjoint sets.
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Sets of Numbers:
The natural numbers (‫ )االعداد الطبيعية‬N are the numbers 1, 2, 3, ...
The whole numbers (‫ )االعداد الكلية‬W are 0, 1, 2, 3, …
The integers (‫ )االعداد الصحيحة‬I are …, -3, -2, -1, 0, 1, 2, 3, …
A rational number (‫النسبية‬
a ‫ )االعداد‬Q is a number that can be
expressed as a quotient . The integer a is called the
b
numerator, and the
integer b, which cannot be 0, is called the denominator.
Rational numbers have decimals that either terminate or repeat.
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Ex:
3
 0.75
4
Rational
(Terminates)
5
 0.45454545
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Rational
(Repeats)
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Numbers in which the decimal is neither terminating nor
repeating are called irrational numbers.
Ex:   3.14....
2  1.41421356
e  2.71828183...
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
Even Numbers : Any number that can be divided by 2

Odd Numbers : Any number that cannot be divided by 2
Prime
Number : A positive integer other than one whose only
factors are 1 and itself.
2,3,5,7,11,13,17,19

Composite Number : An integer that can be divided by at least
one other number (a factor) other than itself.( not prime)
56 = 7 x 8
Factors
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The set of all rational and irrational numbers form the set
of real numbers R.
-1.87
2
-
-6 -5 -4 -3 -2 -1 0
1
4.55
+
2
3
4
5
6
Greater Than: >
Less Than: <
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Ex: For each number, check all that apply.
N
I
Q
R
-57
3.3719
7.42917
0
1.191191119…
101
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Negative numbers are used, for example, to measure
the water depth under sea level
30
20
10
0
-10
-20
-30
-40
-50
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Absolute Value and Distance )‫(القيمة المطلقة‬
For a real number x , | x | is the distance between x and 0.
-8 -7 -6 -5 -4 -3 -2 -1
x
-8  8
0
1
2
3
4
5
6
7
8
8 8
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The absolute value of a real number a,
denoted by the symbol a , is defined by
the rules
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Ex: Evaluate
3 - 8  - 5  -(-5)  5
Notice the
opposite sign
5 -2 -
5 - 3  5 - 2 - (-
+ve
-ve


5 -3 )  2 5 -5
m - 3  2 - 4m ,where m  3
+ve
-ve
 m - 3   -  2 - 4m    5m - 5
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Absolute Value Properties:
If a and b are any real numbers, then
Example
-a  a
-4  -  -4  4  4
ab  a b
 2  -3  -6  2
a
a

b
b
-2
-2
2
 
3
3
3
 b  0
a b  a  b
-3
8  (-5)  3  8  -5  13
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Ex: Write the following expression without absolute
value symbols
x7
x  x -1
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If P and Q are two points on a real number line with
coordinates a and b, respectively, the distance between P
and Q, denoted by d (P, Q), is
d  P , Q  b - a
Def: Let P and Q be points on a real number line with
coordinates -3 and 8, respectively. Find the distance
between P and Q.
d  P, Q   8 - (-3)
 11  11
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Ex: Use the absolute value symbol to describe the
following statement
x is morethan 2 units from 4 but less than 7 units from 4
Thedistancebetween y and - 3is greater than 6
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Ordering of Numbers:
a
a<b
a=b
b
b
a>b
b
a
a
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Interval Notation: )‫(الفترات‬
Inequalities, graphs, and notations:
Inequality
Graph
3 x 7
(
]
3
7
x5
Interval
(
3,7
5,
5
1
x3
]
-
1
3
) or ( means not included in the solution
] or [ means included in the solution
1

 -, - 
3

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Intervals:
Interval
Graph
Example
(a, b)
a
b
[a, b]
a
[
b
]
(a, b]
a
(
b
]
[a, b)
a
b
[
)
(a,  )
a
(
(- , b]
(3, 5)
[4, 7]
(-1, 3]
[-2, 0)
(1,  )
(
b
(- , b)
[a,  )
)
)
a
[
b
]
3
5
(
)
4
7
[
]
-1
3
(
]
-2
0
[
)
1
(
2
(- , 2)
)
[0,  )
0
(- , -3]
-3
[
]
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Basic Definitions

A compound inequality is formed when two or more
simple inequalities are joined by one of the two
words: AND
OR
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“AND” as a Connector

IF Two simple inequalities joined by the word “and”,
then the solution of the compound inequality is the
intersection )‫(تقاطع‬of the solutions of each simple
inequality, in other words, all numbers that satisfy
BOTH inequalities.
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Ex:
Suppose one inequality has as its solution the set {x | - 2 < x }
(
-2
Interval notation: (-2,  )
)
And suppose the other inequality has as its solution {x | x < 1}
1
Interval notation: (  , 1)
The symbol for the intersection of these two sets is
{x | x > -2 } { x | x < 1}.
Graphically this would be
(
)
-2
1
and is used as follows:
Interval notation: ( -2 , 1)
The set notation can be simplified to {x | -2 < x and x < 1 }
which can then be simplified to { x | -2 < x < 1}.
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“AND” cont’d




Conjunctions may be written with the word “AND” clearly
visible (as in the previous example).
They may also be written as CONTINUED inequalities:
2 < x < 5.
This statement is read from the middle to the left AND then to
the right. The word “AND” is implied in the notation.
x is greater than 2 AND x is less than 5.
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Ex:




Write each conjunction below as a compound inequality:
x > -3 and x < -6
x < 3 and x > -7
x > -5 and x < 0
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“OR” connector

If two simple inequalities joined by the word “or”, then the
solution of the compound inequality is the union(‫ )االتحاد‬of the
solutions of each simple inequality, in other words, all
numbers that satisfy one (or both) of the inequalities.
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Ex:
Suppose one solution to a disjunction is {x | x < -3}
)
-3
Interval notation ( , -3)
Suppose the other solution is { x | x > 3 }
(
Interval notation (3,  )
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The solution is the union of the sets, symbolized as , and written as
{x | x < -3 } { x | x > 3 }
The graph combines all possible solutions:
)
(
-3
3
Thus producing a final set notation of {x | x < -3 or x > 3} and an interval
notation of (- . -3 ) (3, ).
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“OR” cont’d


Disjunctions MAY NOT be written as continued inequalities.
The word “OR” will ALWAYS be present.
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Properties of Real Numbers
Commutative Properties( ‫)خاصية االبدال‬
abba
ab  ba
Associative Properties ( ‫)خاصية التجميع‬
a   b  c   a  b   c  a  b  c
a   b  c   a  b   c  a  b  c
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Distributive Property ( ‫)خاصية التوزيع‬
a   b  c  a  b  a  c
 a  b  c  a  c  b  c
Ex : 3   x - 5  3  x - 3  5
 3x - 15
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Identity Property ( ‫)خاصية العنصر الحايد‬
0 a  a  0 a
a  1  1 a  a
Additive Inverse Property ( ‫)خاصية النظير الجمعي‬
a   - a  - a  a  0
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Multiplicative Inverse ( ‫)خاصية النظير الضربي‬
1 1
a    a  1 if a  0
a a
1
a
is called the reciprocal ( ‫)المقلوب‬
a of
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Multiplication by Zero
a0  0
Division Properties
0
0
a
a
 1 if a  0
a
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Rules of Signs
a - b  -  ab  - a b  -  ab
-  - a  a
a -a
a

-b b
b
 - a  - b  ab
-a a

-b b
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Basic Terminologies: ( ‫)مصطلحات اساسية‬
For two real numbers a and b:
minus
The difference is
a - b = a + (-b)
The sum is a + b
plus
The product is a . b
subtraction ( ‫)الطرح‬
addition ( ‫)الجمع‬
times
multiplication ( ‫)الضرب‬
Divided by
a
1
 a
The quotient is a / b defined as
b
b
division
( ‫)القسمة‬
if b  0
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Algebraic expression: ( ‫)تعبير جبري‬
Is the mathematical operations to be carried out on
combination of numbers and variables
An equation: ( ‫)معادلة‬
is a statement of equality between two numbers or two
expressions
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Terms: ( ‫)الحدود‬
The components of an algebraic expression that are separated
by addition or subtraction
terms
( ‫)عوامل‬Factors
2x2
–3x
–1
The components of a term separated by multiplication or
division
-2xy = -2 x y
36x2y
60xy2
factors
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Factor
Numerical coefficient
3
3x
2
Literal coefficient
x
2
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Property of Equation
Let a, b and c be real numbers.
Reflexive
a=a
Symmetric
if a = b , then b =a
Transitive
if a =b and b =c , then a =c
Substitution
if a = b, then a may replaced by b in any
expression that involves a
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You’re
shining!
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