CHAPTER 5 Number Theory and the Real Number

5. 5 Real Numbers and Their Properties; Clock Addition ALWAYS LEARNING Copyright © 2019,

Objectives 1. Recognize the subsets of the real numbers. 2. Recognize the properties of real numbers. 3. Apply properties of real numbers to clock addition. ALWAYS LEARNING Copyright © 2019, 2015, 2011 Pearson Education, Inc. Slide 3

The Set Real Numbers The union of the rational numbers and the irrational numbers is the set of real numbers. The sets that make up the real numbers are called subsets of the real numbers. ALWAYS LEARNING Copyright © 2019, 2015, 2011 Pearson Education, Inc. Slide 4

Example: Classifying Real Numbers Consider the following set of numbers: List the numbers in the set that are a. natural numbers b. whole numbers c. integers d. rational numbers e. irrational numbers f. real numbers ALWAYS LEARNING Copyright © 2019, 2015, 2011 Pearson Education, Inc. Slide 5

Example: Classifying Real Numbers (continued) Solution: a. natural numbers Because b. whole numbers , -7 because integers include whole numbers and the negative natural numbers , -7, -¾, 0. 6, & 7. 3 because these numbers can be expressed as a quotient or as a terminating or repeating decimal 0, d. rational numbers 0, e. irrational numbers f. real numbers because whole numbers include 0 and the natural numbers 0, c. integers because neither terminate nor have blocks of repeating digits , π 0, , -7, -¾, 0. 6, ALWAYS LEARNING =9 , 7. 3, & π because real numbers have all the above numbers as subsets Copyright © 2019, 2015, 2011 Pearson Education, Inc. Slide 6

Properties of the Real Numbers ALWAYS LEARNING Copyright © 2019, 2015, 2011 Pearson Education,

Example: Identifying Properties of Real Numbers Name the property illustrated: a. Commutative property of multiplication b. (4 + 7) + 6 = 4 + (7 + 6) Associative property of addition c. Distributive property of multiplication over addition d. Inverse property of addition ALWAYS LEARNING Copyright © 2019, 2015, 2011 Pearson Education, Inc. Slide 10

Rotational Symmetry A symmetry of an object is a motion that moves the object back onto itself. In symmetry, you cannot tell, at the end of the motion, that the object has been moved. If it takes m equal turns to restore an object to its original position and each of these turns is a figure that is identical to the original figure, the object has m-fold rotational symmetry. ALWAYS LEARNING Copyright © 2019, 2015, 2011 Pearson Education, Inc. Slide 11

Clock Arithmetic & Groups Clock addition is defined as follows: Add by moving the hour hand in a clockwise direction. The symbol is used to designate clock addition. ALWAYS LEARNING Copyright © 2019, 2015, 2011 Pearson Education, Inc. Slide 12

Example: Properties of the Real Numbers Applied to the 6 -Hour Clock System The table for clock addition in a 6 -hour clock system, is shown. a. How can you tell that the set {0, 1, 2, 3, 4, 5} is closed under the operation of clock addition? The Closure Property. The set {0, 1, 2, 3, 4, 5} is closed under the operation of clock addition because the entries in the body are all elements of the set. ALWAYS LEARNING Copyright © 2019, 2015, 2011 Pearson Education, Inc. Slide 13

Example: continued b. Verify the associative property: We were asked to verify one case of the associative property. Locate 2 on the left and 3 on the top of the table. ALWAYS LEARNING Locate 3 on the left and 4 on the top of the table. Copyright © 2019, 2015, 2011 Pearson Education, Inc. Slide 14

Example 2 continued c. What is the identity element in the 6 -hour clock system? The Identity Property. Look for the element that does not change anything when used in clock addition. Notice the identity is 0. ALWAYS LEARNING Copyright © 2019, 2015, 2011 Pearson Education, Inc. Slide 15

Example continued d. Find the inverse of each element in the 6 -hour clock system. The Inverse Property. When an element is added to its inverse, the result is the identity element. Because the identity element is 0, we can find the inverse of each element in {0, 1, 2, 3, 4, 5} by answering the question: What must be added to each element to obtain 0? element + ? = 0 ALWAYS LEARNING Copyright © 2019, 2015, 2011 Pearson Education, Inc. Slide 16