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Transcript
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 1 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} 2 A = {1, 2, 3, . . .} is the set of natural numbers. 4A –99 A “is an element of ” “is not an element of ” C = { } or C = both denote the empty set. 3 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. 4 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 AC={} A C = {1, 2, 3, 4, 5, 7, 8, 9, 10, 11} A and C are disjoint sets. 5 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. 6 Ex: 3 0.75 4 Rational (Terminates) 5 0.45454545 11 Rational (Repeats) 7 Numbers in which the decimal is neither terminating nor repeating are called irrational numbers. Ex: 3.14.... 2 1.41421356 e 2.71828183... 8 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 9 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: < 10 Ex: For each number, check all that apply. N I Q R -57 3.3719 7.42917 0 1.191191119… 101 11 Negative numbers are used, for example, to measure the water depth under sea level 30 20 10 0 -10 -20 -30 -40 -50 12 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 13 The absolute value of a real number a, denoted by the symbol a , is defined by the rules 14 Ex: Evaluate 3 - 8 - 5 -(-5) 5 5 -2 +ve Notice the opposite sign 5 - 3 5 - 2 - (- 5 -3 ) 2 5 -5 -ve m - 3 2 - 4m ,where m 3 +ve -ve m - 3 - 2 - 4m 5m - 5 15 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 ab a b -3 8 (-5) 3 8 -5 13 16 Ex: Write the following expression without absolute value symbols x7 x x -1 17 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 18 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 19 Ordering of Numbers: a a<b a=b b b a>b b a a 20 Interval Notation: )(الفترات Inequalities, graphs, and notations: Inequality Graph 3 x 7 ( ] 3 7 x 5 Interval ( 3,7 5, 5 1 x3 ] - 1 3 ) or ( means not included in the solution ] or [ means included in the solution 1 -, - 3 21 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 [ ] 22 Basic Definitions A compound inequality is formed when two or more simple inequalities are joined by one of the two words: AND OR 23 “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. 24 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}. 25 “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. 26 Ex: Write each conjunction below as a compound inequality: x > -3 and x < -6 x < 3 and x > -7 x > -5 and x < 0 27 “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. 28 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, ) 3 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, ). 29 “OR” cont’d Disjunctions MAY NOT be written as continued inequalities. The word “OR” will ALWAYS be present. 30 Properties of Real Numbers Commutative Properties( )خاصية االبدال abba ab ba Associative Properties ( )خاصية التجميع a b c a b c a b c a b c a b c a b c 31 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 32 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 33 Multiplicative Inverse ( )خاصية النظير الضربي 1 1 a a 1 if a 0 a a 1 a is called the reciprocal ( )المقلوب a of 34 Multiplication by Zero a0 0 Division Properties 0 0 a a 1 if a 0 a 35 Rules of Signs a - b - ab - a b - ab - - a a a -a a -b b b - a - b ab -a a -b b 36 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 37 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 38 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 39 Factor Numerical coefficient 3 3x 2 Literal coefficient x 2 40 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 41 You’re shining! 42