Math 110 FOUNDATIONS OF THE REAL NUMBER SYSTEM FOR ELEMENTARY AND MIDDLE SCHOOL TEACHERS

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2-1Numeration Systems Hindu-Arabic Numeration System Tally Numeration System Egyptian Numeration System Babylonian Numeration System Mayan Numeration System Roman Numeration System Other Number Base

1 2-1Numeration Systems Hindu-Arabic Numeration System Tally Numeration System Egyptian Numeration System Babylonian Numeration System Mayan Numeration System Roman Numeration System Other Number Base Systems Numerals Written symbols to represent cardinal numbers. Numeration system A collection of properties and symbols agreed upon to represent numbers systematically. Hindu-Arabic Numeration System 1. All numerals are constructed from the 10 digits 2. Place value is based on powers of 10 Place value: assigns a value to a digit depending on its placement in a numeral. Write 5678 on expanded from 2.1 Numeration System Page 1

Definition of a n Base-ten blocks 1 long 10 1 = 1 row of 10 units 1 flat 10 2 = 1 row of 10 longs, or 100 units 1 block 10 3 = 1 row of 10 flats, or 100 longs, or 1000 units Example 1 a.

2 Definition of a n Base-ten blocks 1 long 10 1 = 1 row of 10 units 1 flat 10 2 = 1 row of 10 longs, or 100 units 1 block 10 3 = 1 row of 10 flats, or 100 longs, or 1000 units Example 1 a. What is the fewest number of pieces you can receive in a fair exchange for 14 flats, 12 longs, and 22 units? b. Write a Hindu-Arabic number to represent this fewest number of blocks, c. Write this number in expanded form. 2.1 Numeration System Page 2

numeration system used a place value system. Numbers greater than 59 were represented by repeated groupings of 60, much as we use groupings of 10 today.

3 Tally Numeration System Uses single strokes (tally marks) to represent each object that is counted. = 13 Egyptian Numeration System Example 2 Use the Egyptian numeration system to represent 2,345,123 Babylonian Numeration System The Babylonian numeration system used a place value system. Numbers greater than 59 were represented by repeated groupings of 60, much as we use groupings of 10 today. For each symbol grouping, the new place is multiplied by 60. For three places: = Example 3 Use the Babylonian numeration system to represent 305, Numeration System Page 3

Also, instead of writing number in a horizontal format, numbers were written in a vertical format.

4 Mayan Numeration System The symbols for the first 20 numerals in the Mayan system are shown below. The Mayan system was based on 20, where greater numbers were based on multiples of 20. Also, instead of writing number in a horizontal format, numbers were written in a vertical format. How would you write 20? Example 4 Write the Mayan number below into an equivalent Hindu-Arabic number. 2.1 Numeration System Page 4

This is where we get IV = 4 and IX = 9 A bar was placed over the Roman Numeral to indicate multiplication by 1000.

5 Roman Numeration System Roman Numerals can be combined by using an additive property. To avoid repeating a symbol more than three times, a subtractive property was introduced in the Middle Ages. This is where we get IV = 4 and IX = 9 A bar was placed over the Roman Numeral to indicate multiplication by If there are two bars, that means multiplication by Three bars means Example 5 Use the Roman numeration system to represent 15, Numeration System Page 5

6 Other Number Base Systems Base 5 Example 6 Convert five to base 10. Base Two Binary system only two digits Base two is especially important because of its use in computers. One of the two digits is represented by the presence of an electrical signal and the other by the absence of an electrical signal. 2.1 Numeration System Page 6

7 Example 7 a. Convert 46 to base two. b. Convert two to base 10. Base Twelve Duodecimal system twelve digits Use T to represent a group of 10. Use E to represent a group of 11. The base-twelve digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, and E. Example 8 Convert E2T twelve to base 10. Example 9 Convert 1446 to base Numeration System Page 7

8 Homework 1. Ben claims that zero is the same as nothing. Explain how you as a teacher would respond to Ben s statement. 2. What are the major drawbacks to each of the following system? a. Egyptian b. Babylonian c. Roman 3. Why are large numbers in the United States written with commas to separate groups of three digits? 2.1 Numeration System Page 8

9 2.2 Describing Sets Set: Elements (members):. Ex: Set of vowels: {a, e, i, o, u} One method of denoting a set is to simply list the elements inside braces and label the set with a capital letter. A = {1, 2, 3, 4, 5} Each element of a set is listed. The order of the elements is. We symbolize that an element belongs to a set using the symbol, and we use to indicate that an element does not belong to a set. Well Defined: being able to clearly determine if an object belongs in a set. Sets must be well defined. Set of Natural Numbers: N = {1, 2, 3, 4, } are called meaning the sequence continues in the same manner indefinitely. Listing Method: expressing the elements as in a list. This is also called the roster method. C = {1, 2, 3, 4} Set-Builder Notation: describes in a standard and accepted format the elements in a set. To describe C above in Set-Builder Notation, we would write the following: C = {x x N and x < 5} 2.2 Describing Sets Page 1

10 Ex 1 a) Write in set-builder notation: {1, 3, 5, 7, 9, } b) Write in set-builder notation {2, 4, 6, 8, 10, } c) Express in listing form: A = {2k+1 k = 3, 4, 5} d) Write the set B = {a 2 + b 2 a = 2 or 3, and b = 2, 3, or 4} by listing its elements. Equal: two sets that contain exactly the same elements. The order of the elements makes no difference. If set A is equal to set B, we write A = B. If A = B, then every element of set A is contained in set B, and every element of set B is contained in set A. If A B, then there is at least one element that is not contained in both sets A and B. One-to-one Correspondence: Two sets in which for each element in the first set there exists exactly one element in the second, and vice-versa. Set {1, 2, 3} can be placed in a one-to-one correspondence with set {a, b, c} by establishing either of the following pairings: 2.2 Describing Sets Page 2

11 One way to illustrate all possible one-to-one correspondences is to use a table; another is a tree. Ex 2 Are the following sets one-to-one? a) {m, a, t, h} and {f, u, n} b) {x x is a letter in the word mathematics } and (1, 2, 3,, 11} Fundamental Counting Principle If event M can occur in m ways and, after it has occurred, event N can occur n ways, then event M followed by event N can occur ways. Ex 3 a. How many one-to-one correspondences are there between the sets A={x,y,z,u, } and set B={1,2,3,4,5}? b. If x must correspond to 5, then how many one-to-one correspondences between set A and set B? c. If x, y, z must correspond to odd numbers, then how many one-to-one correspondences between set A and set B? 2.2 Describing Sets Page 3

12 Equivalent sets: Two sets A and B, written, if and only if there exists a between the sets. Recall if two sets are equal, they have the. Equal sets are equivalent, but equivalent sets are not necessarily equal. Ex4 Given sets A = {p, q, r, s} B = {a, b, c} C = {x, y, z} D = {b, a, c} Determine whether the following statements are true or false. A B A~C B~D C D A= B A = C B D C D Cardinal Numbers, n(a): in a set. The cardinal number for set A is. Ex 4 Find the Cardinal Number of the following sets: a) {1, 3, 5,, 1001} b) { {1, 2, 3}, {4, 5, 6}, {7, 8, 9}, {10, 11, 12} } Empty Set (Null Set): a set that contains no elements. This is symbolized by either or. Finite Set: a set that has a cardinal number of 0 or a cardinal number that is a natural number. 2.2 Describing Sets Page 4

13 Infinite Set: a set that has a cardinal number that cannot be represented by a 0 or a natural number. Universal Set (U): the set that contains all elements being considered in a given discussion. Venn Diagram: a diagram used to illustrate ideas in logic. Complement: all the elements in the universal set that are not in the set discussed. The complement of A is written A. If U = students at CSULA campus and F = set of female students at CSULA campus, then what is F? U How could you illustrate F? F Ex 5 If U is the set of all college students and A is the set of all college students with a straight-a average, describe A. Subset: B is a subset of A, written B A, if and only if every element of B is an element of A. Proper Subset: If B is a subset of A and B is not equal to A. B A. Ex 6 If A = {1, 2, 3, 4, 5, 6, 7, 8}, B = {2, 4, 8}, and P = {x x = 2 n and n N}, a) Which set is a subset of the other? 2.2 Describing Sets Page 5

14 b) Which sets are proper subsets of each other? Less Than: If A and B are finite sets, A has than B if A is equivalent to a proper subset of B. n(a) is less than n(b) and write Greater Than: If A and B are finite sets, A has than B if B is equivalent to a proper subset of A. n(a) is greater than n(b) and write Number of Subsets of a Set: If P = {a}, then P has two subsets: and {a} If Q = {a, b}, then Q has 4 subsets:, {a}, {b}, and {a, b}. If R = {a, b, c}, then how many subsets would it have? The number of subsets of a set with n elements is 2 n. Ex 7 In a Math 110 class with students: Sayra, Clara, Dana, Krista, Summer, and Aimee, how many different 3 member study groups could be made? 2.2 Describing Sets Page 6

15 Homework: 1. Describe the set of even numbers using a list and using set-builder notation. 2. How many subsets are there of the set {A, B, C}? If it helps, give these some meaning, like choose three members of a class, call them Albert, Betty, and Charles. What are all the subsets of those three people? What if I pick four people, how many subsets are there now? Five? Ten? 3. If A ={a,b,c,d,e} a. How many subsets does A have? b. How many proper subsets does A have? c. How many subsets does A have that include the elements a and e? 4. Answer each of the following. If you answer is no, tell why or give a counterexample. a. If A=B, can we always conclude A B? b. If A B, can we always conclude A B? c. If A B, can we always conclude A B? d. If A B, can we always conclude that A=B? 2.2 Describing Sets Page 7

16 2 3 Other Set Operations and Their Properties Set Intersection Consider the following sets: C = {x x is a college graduate} S = {x x earns at least $40,000} Suppose we want to mail a questionnaire to all individuals in a certain city who are college graduates and who earn at least $40,000. Then the individuals to receive the questionnaire include those common to both sets C and S. The intersection of sets C and S is the shaded region. C S The intersection of two sets A and B, written, is the set of all elements common to both A and B. The key word is which implies both conditions must be met. 1 Find A B a. A={1,2,3,4}, B={3,4,5,6} b. A={0,2,4,6 }, B={1,3,5,7 } c. A={2,4,6,8 }, B={1,2,3,4 } 2.3 Other Set Operations and Their Properties Page 1

17 Example 2 If C represents the individuals with a college degree and S represents the individuals earning at least $40,000,we can use a Venn diagram to display information. If we know that 4800 individuals have college degrees, 5400 earn at least $40,000, and 1200 have a college degree and make at least $40,000, then How many people have college degrees but earn less than $40,000? Vann Diagram How many people earn at least $40,000 and do not have a college degree? Set Union Consider the following sets: C = {x x is a college graduate} S = {x x earns at least $40,000} Suppose we want to mail a questionnaire to all individuals in a certain city who are college graduates or who earn at least $40,000, or both. Then the individuals to receive the questionnaire include those who are members of set C or set S. C = {x x is a college graduate} S = {x x earns at least $40,000} The union of sets C and S is the shaded region. C S The intersection of two sets A and B, written, is the set of all elements common to both A and B. The key word is or which implies one or the other or both. 2.3 Other Set Operations and Their Properties Page 2

18 Example 3 Find A B. a. A={1,2,3,4}, B={3,4,5,6} b. A={0,2,4,6 }, B={1,3,5,7 } c. A={2,4,6,8 }, B={1,2,3,4 } Set Difference Consider the following sets: C = {x x is a college graduate} S = {x x earns at least $40,000} The set of individuals earning at least $40,000 but not a college graduate is called the complement of C relative to S, or the set difference of S and C. The complement of A relative to B, written, is the set of all elements in B that are not in A. Note B A that is not read as B minus A. Minus is an operation on numbers and set difference is an operation on sets. Note: B A = B A is the set of all elements in B and not in A. 2.3 Other Set Operations and Their Properties Page 3

order we write the sets when we find the union or the intersection.

19 Example 4 If A={d,e,f}, B={a,b,c,d,e,f}, and C={a,b,c}.Find a. A B b. B A c. B C d. C B Properties of Set Operations Commutative Properties For all sets A and B, A B = B A and A B = B A Since the order of the elements in a set is not important, it does not matter in which order we write the sets when we find the union or the intersection. Distributive Property of Intersection For all sets A and B, Since the order of the elements in a set is not important, it does not matter in which way we associate sets when we take the union or the intersection. Distributive Property of Intersection Over Union For all sets A, B, and C, Example 5 Use set notation to describe the shaded portions of the Venn diagrams. 2.3 Other Set Operations and Their Properties Page 4

Identify the students described by each region in the figure. Region (a) contains all students taking mathematics but not English.

Example 6 In a survey of 110 college freshmen that investigated their high school backgrounds, the following information was gathered: 25 took physics, 45

20 a. b. c. Using Venn Diagrams as a Problem Solving Tool Suppose M is the set of all students taking mathematics and E is the set of all students taking English. Identify the students described by each region in the figure. Region (a) contains all students taking mathematics but not English. Region (b) contains all students taking both mathematics and English. Region (c) contains. Region (d) contains. Example 6 In a survey of 110 college freshmen that investigated their high school backgrounds, the following information was gathered: 25 took physics, 45 took biology, 48 took mathematics, 10 took physics and mathematics, 8 took biology and mathematics, 6 took physics and biology, 5 took all three subjects. a. How many students took biology but neither physics nor mathematics? b. How many took physics, biology, or mathematics? c. How many did not take any of the three subjects? 2.3 Other Set Operations and Their Properties Page 5

21 Cartesian Products One way to produce a set from two sets is by forming pairs of elements of one set with the elements of another set in a specific way. Example 7 Suppose a person has three pairs of pants, P = {blue, green, white}, and two shirts, S = {blue, red}. According to the Fundamental Counting Principle, there are possible ways to pair an element from set P with an element from set S, forming a set of ordered pairs. Because the first component in each pair represents pants and the second component in each pair represents shirts, the order in which the components are written is important. Equality for ordered pairs: (x, y) = (m, n) if, and only if, the first components are equal and the second components are equal. That is. A set consisting of ordered pairs is an example of a Cartesian product. For any sets A and B, the Cartesian product of A and B, written, is the set of all ordered pairs such that the first component of each pair is an element of A and the second component of each pair is an element of B. Note that A B is commonly read as A cross B and should never be read A times B. 2.3 Other Set Operations and Their Properties Page 6

22 Example 8 If A = {a, b, c} and B = {1, 2, 3}, find each of the following: a. A B b. B A c. A A Homework If A = {1, 2, 3},B = {3, 4, 5} C={2, 4, 6}, find (1) A B (2) A B (3) A B (4) A B (5) A (B C) (6) A (B C) (7) A (B C) 2.3 Other Set Operations and Their Properties Page 7

Sets 1. The things in a set are called the elements of it. If x is an element of the set S, we say

Sets 1. The things in a set are called the elements of it. If x is an element of the set S, we say Sets 1 Where does mathematics start? What are the ideas which come first, in a logical sense, and form the foundation for everything else? Can we get a very small number of basic ideas? Can we reduce it