Download Lesson 1: Classifying Real Numbers

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.
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2) 9
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Write each decimal as a fraction in simplest form.
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4) 0.6
2
3
3) 4 8
5) 5.75
New stuff: sets

A set is a collection of things.
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In math when we talk about sets, we are talking about collections of
numbers.
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Our number system is made up of these sets.
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We name sets using a capital letter.
New stuff: sets

The things contained in sets are called elements
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We use braces {} to denote sets.
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There are three kinds of sets: finite sets, infinite sets, and the empty set.
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Finite sets contain a finite number of elements: (e.g. A= {1, 2, 3, 6}
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Infinite sets contain an infinite number of elements: (e.g. N={1, 2, 3, 4, 5, …}
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The empty set contains no elements: { } or ∅
Subsets
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For now, we are going to be working with a subset of the complex
numbers called the Real Numbers.
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A subset is a set of numbers that is part of a larger set.
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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.
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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
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A. 2
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B. 5
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C. 3 2
1
Example 2: Identifying sets for RealWorld Situations

Identify the set of numbers that best describes each situation. Explain your
choice.
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A. the value of the bills in a person’s wallet
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B. the balance of a checking account
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C. the circumference of a circular table when the diameter is a rational
number
Intersections
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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
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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
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Find 𝐴 ∩ 𝐵 and 𝐴 ∪ 𝐵.
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a) 𝐴 = {2, 4, 6, 8, 10, 12}; 𝐵 = {3, 6, 9, 12}
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b) 𝐴 = 11, 13, 15, 17 ; 𝐵 = {12, 14, 16, 18}
Closure
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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.
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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.
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a) The set of whole numbers is closed under addition.
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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.
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Find 𝐶 ∩ 𝐷 and 𝐶 ∪ 𝐷.
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g) 𝐶 = 4, 8, 12, 16, 20 ; D = {5, 10, 15, 20}
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h) 𝐶 = 6, 12, 18, 24 ; 𝐷 = {7, 14, 21, 28}
Lesson Practice
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Determine whether each statement is true or false. Provide a
counterexample for false statements.
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i) The set of whole numbers is closed under multiplication.
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j) the set of natural numbers is closed under division.
Homework
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Start with the problems you think will be hardest. If you need help, put your
help card in the corner of the desk.
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Pg. 5-6, #1-30.
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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!