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Properties of Real Numbers & Clock Addition

Learn about the subsets and properties of real numbers and how they apply to clock addition. Explore the closure, associative, identity, and inverse properties.

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Properties of Real Numbers & Clock Addition

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  1. CHAPTER 5 Number Theory and the Real Number System

  2. 5.5 • Real Numbers and Their Properties; Clock Addition

  3. Objectives • Recognize the subsets of the real numbers. • Recognize the properties of real numbers. • Apply properties of real numbers to clock addition.

  4. 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.

  5. Example: Classifying Real Numbers • Consider the following set of numbers: • List the numbers in the set that are • natural numbers b. whole numbers c. integers • d. rational numbers e. irrational numbers f. real numbers

  6. Example: Classifying Real Numbers (continued) • Solution:

  7. Properties of the Real Numbers

  8. Properties of the Real Numbers

  9. Properties of the Real Numbers

  10. Example: Identifying Properties of Real Numbers • Name the property illustrated: • (4 + 7) + 6 = 4 + (7 + 6) Commutative property of multiplication Associative property of addition Distributive property of multiplication over addition Inverse property of addition

  11. 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.

  12. 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.

  13. 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.

  14. 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. Locate 3 on the left and 4 on the top of the table.

  15. 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.

  16. 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

  17. Example continued e. Verify two cases of the commutative property:

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