COLLEGE ALGEBRA. Properties of Real Numbers with Clock Arithmetic

Transcription

1 COLLEGE ALGEBRA By: Sister Mary Rebekah Cornell-Style Fill in the Blank Notes and Teacher s Key Properties of Real Numbers with Clock Arithmetic 1

2 Topic: Clock Arithmetic Addition Why use Clock Arithmetic? Clock ADDITION Consider the following problem: Tasha left home at 10 o clock. She came back 5 hours later. At what time did she come back? Write an equation whose solution solves the problem. To avoid confusion with ordinary arithmetic we will use the symbol to denote addition in clock arithmetic. What is the set of clock numbers? What would 10 3 equal? 7 11= SKILLS PRACTICE Directions: Perform the following clock addition

3 Topic: Clock Arithmetic Addition Clock Addition Properties of Clock ADDITION Clock arithmetic under the operation can be denoted by the following table. Please complete the table with the correct missing numbers Binary: Closure: Commutative: Associative: Identity: Inverse: 3

4 Topic: Properties Clock Addition Binary Definition. Is the set S binary under? Justify your response. Closure Definition. An operation is said to be closed if the result of performing the operation with every two elements of the set is an element of the set. Is the set S closed under? Justify your response. Commutative Definition. An operation is commutative if the order in which we combine any two elements of the set does not affect the result. To check that an operation is commutative we need to verify that the result of combining every two elements of the set does not depend on the order in which they are combined. In this case, we need to verify that x y = y x for every pair of number and x and y in the set S= {1, 2, 3,, 12}. Is the set S commutative under? Justify your response. Associative Now let s examine the result of combining three elements of the set. How do we find the result of ? We notice that there are two possibilities. First, we can perform 6 8 (2), and then perform 2 11 (1). Second, we can perform 8 11 (7), and then 6 7 (1). In this case we see that (6 8) 11 = 1 and 6 (8 11) = 1. In other words, (6 8) 11 = 6 (8 11). Thus, Is the set S associative under? Justify your response. 4

5 Topic: Properties Clock Addition Identity Direction: Using clock addition, solve for e for each of the following equations below. 1 e = 1 5 e = 5 9 e = 9 2 e = 2 6 e = 6 10 e = 10 3 e = 3 7 e = 7 11 e = 11 4 e = 4 8 e = 8 12 e = 12 Definition. In general, we say that an operation o has an identity in the Set S if there is a unique element in the set S, e, such that x o e = e o x = x, for every element in S 1. Does multiplication of real number have an identity? 2. Does addition of real numbers have an identity? 3. Does division of real numbers have an identity? Inverse Direction: Using clock addition, solve for e for each of the following equations below. 1 x = x 1 = 5 x = x 5 = 9 x = x 9 = 2 x = x 2 = 6 x = x 6 = 10 x = x 10 = 3 x = x 3 = 7 x = x 7 = 11 x = x 11 = 4 x = x 4 = 8 x = x 8 = 12 x = x 12 = Definition. In general, let o be a binary operation defined for a set S. Let x be an element of S. The inverse of x with respect to o, is the unique element x, such that x o x = x o x = e. 1. What is the inverse of 4 under the standard addition of integers? Justify your response. 2. What is the inverse of 4 under the standard multiplication of integers? Justify your response. 3. What is the inverse of 4 under the standard multiplication of rational numbers? Justify your response. 4. Does every number have an inverse under standard multiplication of real numbers? Justify your response. 5

6 Topic: Properties Clock Subtraction Clock Subtraction FOUR APPROACHES Taking away approach: Missing addend approach: Adding the opposite approach: Subtracting the normal way: DEFINING Clock SUBTRACTION In general, we can define the operation as follows: a)let x and y be two elements of the set S. x y = z if an only of x = z y. b)let x and y be two elements of the set S. x y = x y (y is the inverse of y.) [Note: Either approach can define subtraction, but once we take an approach as the definition, the other becomes a theorem.] Write the set of clock numbers in two different ways: SKILLS PRACTICE Directions: Perform the following clock subtraction

7 Topic: Properties Clock Subtraction Binary Is the set S binary under? Justify your response. Closure Is the set S closed under? Justify your response. Commutative Is the set S commutative under? Justify your response. Associative Is the set S associative under? Justify your response. Identity Does the set S have an identity under? If so, what is it? Justify your response. Inverse Does the set S have an inverse under? If so, what is it? Justify your response. 7

8 Topic: Properties Clock Multiplication Modular Arithmetic Definition. Directions: Please evaluate the following expression Mod (4 X 5) Mod Mod 12 Clock Multiplication Our next objective is to define multiplication and division in clock arithmetic. First, what does 4 5 mean? (You have been asked so many times this question that you should know it without hesitation by now.) 4 5 = In the same way, we can define 4 5= = = 3 5= 8 or you can think of multiplication in terms of regular multiplication, mod

9 Topic: Properties Clock Multiplication Binary Is a binary operation? Justify your response. Closure Is S closed under? Justify your response. Commutative Is a commutative operation? Justify your response. Associative Is an associative operation? Justify your response. Identity Does S have an identity for? If so, what is it? Justify your response. Inverse Which elements in S have an inverse under? What type of numbers are those? 9

10 Topic: Properties Clock Division Defining Clock DIVISION Division in 12-clock arithmetic can be defined using two different approaches: a) The missing factor approach: = 2 because 2 11 = 10. b) Multiplying by the multiplicative inverse (reciprocal) of the divisor: = = 2 Examples MISSING FACTOR APPROACH MULTIIPLYING BY THE MULTIPLICATIVE INVERSE APPROACH Skills Practice Directions: Evaluate the following expressions using Clock Division Undefined Expressions Notice, however, that, as in division of whole numbers, some divisions are not defined: 1. Compute, if possible, Compute, if possible, 6 2. As the multiplication table shows, there is not an element x in S such that x 3 = 8. Therefore, 8 3 is undefined. Also, 8 3 = 8 3, but 3 does not exist in S. Again, 8 3 is undefined because the multiplicative inverse of 3 does not exist in S. As the multiplication table shows, 6 = 3 2 and 6 = 9 2. Since we have more than one element in S that multiplied by 2 yields 6, we say that 6 2 is undefined. Also, 6 2 = 6 2, but 2 does not exist in S. Again, 6 2 is undefined because the multiplicative inverse of 2 does not exist in S. 10

11 Topic: Properties Clock Division Binary Is a binary operation? Justify your response. Closure Is S closed under? Justify your response. Commutative Is a commutative operation? Justify your response. Associative Is an associative operation? Justify your response. Identity Does S have an identity for? If so, what is it? Justify your response. Inverse Which elements in S have an inverse under? What type of numbers are those? 11

12 Topic: Distributive Property Distributive PROPERTY Question: If possible, perform the following operations in two different ways: a) 6 (5 11) We see in this example that the distributive property of multiplication over addition hold in 12-clock arithmetic. b) 7 (4 8) We show in this example that addition is not distributive over multiplication in 12-clock arithmetic. Provide two more examples to illustrate that, in 12-clock arithmetic, multiplication is distributive over addition. 12

13 Topic: Properties Using Abstract System Define another Mathematical SYSTEM New Rules: Let s consider the set SS = aa, bb, cc and the operation Δ, triangulate, defined by the following table Δ a b c a c a b b a b c c b c a To triangulate two elements of S we take the first of the elements in the first column and the second in the first row. The result is the intersection of the row of the first element and the column of the second element. Thus, aa aa = cc, bb cc = cc, etc. Binary Is Δ a binary operation? Closure Is the set S closed under Δ? Commutative Is Δ a commutative operation? Associative Is Δ and associative operation? Identity Is there an identity element for Δ? Inverse Does each element of S have and inverse with respect to Δ? 13

14 Topic: Properties Using Abstract System Define another Mathematical SYSTEM New Rules: Define the operation a b = c, if and only if a = b Δ c. a b c a b c a) Find a b, and find b c. b) Investigate the properties of : (binary, commutative, associative, identity, inverse) c) Is distributive over Δ? Why or why not? d) Is Δ distributive over? Explain. 14

15 Topic: Properties Real Numbers Closure ADDITION OF REAL NUMBERS SUBTRACTION OF REAL NUMBERS Commutative Associative Identity Inverse Distributive 15

16 Topic: Properties Real Numbers Closure MULTIPLICATION OF REAL NUMBERS DIVISION OF REAL NUMBERS Commutative Associative Identity Inverse Distributive 16