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Transcript
Lesson 1: Classifying Real Numbers THE NUMBER SYSTEM Warm up 1) A ______________ (Venn diagram, line plot) shows the relationship between sets. Write each fraction as a decimal. 2) 9 Write each decimal as a fraction in simplest form. 4) 0.6 2 3 3) 4 8 5) 5.75 New stuff: sets A set is a collection of things. In math when we talk about sets, we are talking about collections of numbers. Our number system is made up of these sets. We name sets using a capital letter. New stuff: sets The things contained in sets are called elements We use braces {} to denote sets. There are three kinds of sets: finite sets, infinite sets, and the empty set. Finite sets contain a finite number of elements: (e.g. A= {1, 2, 3, 6} Infinite sets contain an infinite number of elements: (e.g. N={1, 2, 3, 4, 5, …} The empty set contains no elements: { } or ∅ Subsets For now, we are going to be working with a subset of the complex numbers called the Real Numbers. A subset is a set of numbers that is part of a larger set. The next page shows the subsets of the Real Numbers. Subsets of Real Numbers Natural Numbers 𝑁 The numbers we use to count things (these are also called the counting numbers): 𝑁 = {1, 2, 3, 4, … } Whole numbers 𝑊 All the natural numbers and zero: 𝑊 = {0, 1, 2, 3, … } Integers 𝑍 The whole numbers and the opposites of the natural numbers: 𝑍 = {… , −3, −2, −1, 0, 1, 2, 3, … } Rational numbers 𝑄 Numbers that can be written in the form 𝑏, where 𝑎 and 𝑏 are integers and 𝑏 ≠ 0. In decimal form, rational numbers either 1 2 terminate or repeat. Examples: 2 , 0. 3, − 3 , 0.125 Irrational numbers 𝑅 − 𝑄 Numbers that can’t be written as the quotient of two integers. Irrational numbers do NOT terminate or repeat in decimal form. 3 Examples: 5, 2, − 7, 3 3, 𝜋, 3𝜋, 𝑒 Real numbers 𝑅 The set including all rational and irrational numbers. 𝑎 Real Numbers, 𝑹 Rational Numbers, 𝑸 Whole Numbers, 𝑾 Natural Numbers, 𝑵 Irrational Numbers 𝑹−𝑸 Example 1: Identifying sets For each number, identify the subset(s) of real numbers to which it belongs A. 2 B. 5 C. 3 2 1 Example 2: Identifying sets for RealWorld Situations Identify the set of numbers that best describes each situation. Explain your choice. A. the value of the bills in a person’s wallet B. the balance of a checking account C. the circumference of a circular table when the diameter is a rational number Intersections The intersection of sets A and B, denoted 𝐴 ∩ 𝐵, is the set of elements that are contained in both A and B A B B 𝐴∩𝐵 Unions The union of sets A and B, denoted 𝐴 ∪ 𝐵, is the set of all elements that are in either A or B. A B Example 3: Intersections and Unions of Sets Find 𝐴 ∩ 𝐵 and 𝐴 ∪ 𝐵. a) 𝐴 = {2, 4, 6, 8, 10, 12}; 𝐵 = {3, 6, 9, 12} b) 𝐴 = 11, 13, 15, 17 ; 𝐵 = {12, 14, 16, 18} Closure A set of numbers is closed, or has closure, under a given operation if the outcome of the operation on any two members of the set is also a member of the set. For example, the sum of any two natural numbers is also a natural number. Therefore, the set of natural numbers is closed under addition. To prove a statement false, we just need to find one example. This is called a counterexample. Identifying a Closed Set Under a Given Operation Determine whether the statement is true of false. Give a counterexample for false statements. a) The set of whole numbers is closed under addition. b) The set of whole numbers is closed under subtraction. Lesson Practice Let’s work a few more examples together. Be sure to put these in your notebook. Find 𝐶 ∩ 𝐷 and 𝐶 ∪ 𝐷. g) 𝐶 = 4, 8, 12, 16, 20 ; D = {5, 10, 15, 20} h) 𝐶 = 6, 12, 18, 24 ; 𝐷 = {7, 14, 21, 28} Lesson Practice Determine whether each statement is true or false. Provide a counterexample for false statements. i) The set of whole numbers is closed under multiplication. j) the set of natural numbers is closed under division. Homework Start with the problems you think will be hardest. If you need help, put your help card in the corner of the desk. Pg. 5-6, #1-30. Make sure you show any necessary work. You homework needs to be neat. If you can’t fit all your work into the box, put it on a separate sheet of paper and just put your answers in the box. Remember, if I can’t hear the music, it’s too loud. This is an individual activity. If you must communicate, you should be whispering. We will be correcting this tomorrow. Make sure you have your red correcting pens!