Chapter 1 - The Real Number System PDF

CHAPTER 1 THE REAL NUMBER SYSTEM In this chapter, we will discuss The set of real numbers Rational and Irrational numbers All subsets of rational and Irrational numbers The different properties of Real numbers Target Skills Identify and classify the subsets of the set of...

CHAPTER 1 THE REAL NUMBER SYSTEM In this chapter, we will discuss The set of real numbers Rational and Irrational numbers All subsets of rational and Irrational numbers The different properties of Real numbers Target Skills Identify and classify the subsets of the set of real numbers. Discuss the schematic diagram of the real number system. Enumerate and apply the fundamental properties of real numbers in arguments of mathematical statements from the given examples.  TOPIC 1 SETS AND SUBSETS OF REAL NUMBERS SETS AND SUBSETS OF Fraction REAL NUMBERS REAL NUMBERS (ℝ) Union of Rational and Irrational numbers Includes all numbers on the number line Excludes imaginary numbers Expressed as an infinite expansion of DECIMALS SETS AND SUBSETS OF Fraction REAL NUMBERS IRRATIONAL NUMBERS (ℚ ) ′ Numbers that cannot be expressed as a quotient of two integers. (fraction) “Not rational or no ratio” Non-terminating and non-repeating decimals Examples: SETS AND SUBSETS OF Fraction REAL NUMBERS RATIONAL NUMBERS (ℚ) Numbers that can be expressed as a quotient or ration of two integers. a , where a and b are integers , b ≠ 0 b 55582613 “non-repeating, terminating decimals” 0.96587134659786431 = 57546601 “Repeating, non–terminating decimals” 0.3333333333333333… = 1 3 111111 “Repeating, terminating decimals” 0.555555 = 200000 SETS AND SUBSETS OF Fraction REAL NUMBERS FRACTIONS a , where and are integers , b ≠ 0 b Cannot be expressed as integers Examples: 11 18 45 112 SETS AND SUBSETS OF Fraction REAL NUMBERS INTEGERS (ℤ) Non – fractions Includes the negative and positive non- fraction numbers NOTE: When we add, subtract, and multiply integers, the answer is also an integer SETS AND SUBSETS OF Fraction REAL NUMBERS WHOLE NUMBERS Non-negative integers Includes 0 SETS AND SUBSETS OF Fraction REAL NUMBERS NATURAL NUMBERS (ℕ) Counting numbers starting from 1 Excludes 0 RECAP 27 32 −28 Fraction 7 415 0.4777777777777777 0.88888 Practice: T or F 1. All Irrational Numbers are Real numbers. 2. All Real numbers are Natural numbers. 3. All Integers are Rational Numbers. 4. All Decimals are Irrational numbers. 5. −45 is a whole number.  TOPIC 2 PROPERTIES OF REAL NUMBERS PROPERTIES OF REAL NUMBERS A. Closure Property B. Associative Property C. Commutative Property D. Distributive Property E. Identity Property F. Inverse Property A. CLOSURE PROPERTY Let and be any real numbers, then, + , − , , and where b ≠ 0, the result is a real number. 8−4=4 8∈ℝ 4∈ℝ (−6) × 11 =− 66 −6 ∈ ℝ 11 ∈ ℝ −66 ∈ ℝ B. ASSOCIATIVE PROPERTY Grouping doesn’t matter Let , and be any real numbers, then Addition: (a + b) + c = a + (b + c) Multiplication: (ab)c = a(bc) Example: a = 4, b = 8, c = 5 ADDITION: (a + b) + c = a + (b + c) (4 + 8) + (5) = 4 + (8 + (5)) 12 + (5) = 4 + 13 = MULTIPLICATION: (ab)c = a(bc) (4)(8) (5) = (4) (8)(5) (32)(5) = (4)(40) = C. COMMUTATIVE PROPERTY Order doesn’t matter Let and any real numbers, then Addition: a+b=b+a Multiplication ab = ba Example: a = 4, b = 8, c = 5 ADDITION: a+b=b+a 4 + (8) = 8 + 4 = MULTIPLICATION: ab = ba (4)(8) = (8)(4) = D. DISTRIBUTIVE PROPERTY Let , , and be any real numbers, then ( + ) = + Example: a = 12, b = 4, c = 3 a(b + c) = ab + ac 12(4 + 3) = (12)(4) + (12)(3) 12(7) = 48 + 36 = E.IDENTITY PROPERTY Let a be a real number, then ADDITION a+0=0+a=a MULTIPLICATION a∙1=1∙a=a Example: Let a = 12, Show that 1. a + 0 = 0 + a = a 2. a ∙ 1 = 1 ∙ a = a Solution: 1. a + 0 = 0 + a = a 2. a ∙ 1 = 1 ∙ a = a 12 + 0 = 0 + 12 = 12 12(1) = 1(12) = 12 F. INVERSE PROPERTY Let a be a real number, then ADDITION: a + (−a) = (−a) + a = 0 1 1 MULTIPLICATION a∙ = ∙a=1 a a EXAMPLES: Let a = 12, Show that Solution: 1 1 1. a + (−a) = (−a) + a = 0 2. a ∙ = ∙ a = 1 a a 12 + (−a) = (−a) + 12 = 0 1 1 (12) = (12) = 1 12 12 1 1 12 + (−12) = 0 (−12) + 12 = 0 (12) =1 (12) = 1 12 12 0=0 0=0 1=1 1=1 RECALL: 1. Closure Property 2. Associative Property 3. Commutative Property 4. Distributive property 5. Identity Property a. Addition b. Multiplication 6. Inverse Property a. Addition b. Multiplication Exercise 1 Determine the property that is being shown in each number. Write the complete property. 1. 4 + (−4) = 0 9. 3(11 − 4) = 33 − 12 2. 3(−12) =− 36 10. 0 + 5 = 5 3. (11)(13) = (13)(11) 4. −2 + 0 =− 2 5. (12 + 13) − 9 = 12 + (13 − 9) 1 6. (4) =1 4 7. 7 (11)(3) = (7)(11) 3 8. −1 − 1 =− 2