MMW IM - Here\'s the instructional material for Mathematics in the Modern World PDF

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for the sole noncommercial use of the Faculty of the Department of Mathematics and Statistics Polytechnic University of the Philippines 2020

Conributors: Abdul, Alsafat Atienza, Jacky Boy Bang-as, Pamela Bernardino, Rhea Cabanig, Sarah Jean Criseno, Regine Dilla, Perlyn Mae Duarte, Rafael Elizon, Katrina Equiza, Cynthia Hernandez, Andrew Isaac, Emelita Lara, Jose Alejandro Constantino Longhas, Paul Ryan Macatangay, Shaina Lyra Malvar, Rolan Nuguid, Kenneth James Saguindan, Ian Sta. Maria, John Patrick

Republic of the Philippines POLYTECHNIC UNIVERSITY OF THE PHILIPPINES COLLEGE OF SCIENCE Department of Mathematics and Statistics

Course Title

:

MATHEMATICS IN THE MODERN WORLD

Course Code

:

GEED 10053

Course Credit

:

3 units

Pre-Requisite

:

GENERAL MATHEMATICS, STATISTICS AND PROBABILITY (SHS)

Course Description : The course deals with the nature of mathematics, appreciation of its practical, intellectual and aesthetic dimensions, and application of mathematical tools in daily life. It also bridges the study of mathematics to other domains of interest like business, finance, social sciences and arts and design. COURSE LEARNING PLAN Week

Topics and Subtopics I. Nature of Mathematics

Mathematics in Nature Week 1

1. Patterns and Numbers in Nature 2. Fibonacci Sequence 3. Mathematics for Our World

Language of Mathematics Week 2

1. Propositions and Logical Connectives 2. Sets, Operations and Venn Diagrams

Problem Solving Week 3

1. Inductive and Deductive Reasoning 2. Polya’s Guidelines for Problem Solving 3. Mathematical Problems involving Patterns

Week 4

Problem Set 1 II. Mathematics as a Tool: Statistics and Data Management

Week 5

Week 6

Data Gatheing and Sampling Techniques 1. Steps in Statistical Investigation 2. Sampling Techniques, Sample Size Considerations, Methods of Data Collection 3. Levels of Measurement

Data Presentation 1. Tabular Presentations: Frequency Distributions and Crosstabulations 2. Graphical Presentations: Graphs, Charts, Time Series Plots

Descriptive Measures Week 7

1. Measures of Central Tendency 2. Measures of Dispersion or Variability

Week 8

Problem Set 2 III. Special Topics Financial Mathematics

Week 9-10

1. 2. 3. 4.

Simple and Compound Interest Ordinary Annuities Paying Off a Debt or Loan Repayment Other Applications of Financial Mathematics

Mathematics of Voting and Apportionment Week 11-12

Week 13 Week 14

1. Voting Methods 2. Apportionment Problem Set 3 Final Exam

*Note: Financial Mathematics and Mathematics of Voting and Apportionment are required special topics for the programs under the following colleges: Accountancy and Finance (CAF), Arts and Letters (CAL), Business Administration (CBA), Communication (COC), Education (CoED), Human Kinetics (CHK), Political Science and Public Administration (CPSPA), Social Sciences and Development (CSSD) and Tourism, Hospitality and Transportation Management (CTHTM).

Reference Materials: • Smith, Karl J. The Nature of Mathematics. 12ed. Cengage Learning. 2012 • Angel, Abbott, Runde. Survey of Mathematics with Applications. 10ed. Pearson. 2016 • Lippman, David. Mathematics in Society. 2ed. 2017 • Thomas, Christopher. Schaum’s Outline of Mathematics for the Liberal Arts. McGrawHill. 2009

Contents

1

2

3

4

5

Mathematics in Our World . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.1

Overview: What is mathematics? . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2

Patterns and Numbers in Nature . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.3

Fibonacci Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.4

Mathematics for Our World . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

Logic and Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.1

Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.2

Compound Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.3

Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

3.1

Inductive and Deductive Reasoning . . . . . . . . . . . . . . . . . . . . . . . .

34

3.2

George Polya’s Guidelines for Problem Solving . . . . . . . . . . . . . . . . . .

37

Statistics and Data Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

4.1

Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

4.2

Steps in Statistical Investigation . . . . . . . . . . . . . . . . . . . . . . . . . .

45

4.3

Sampling and Sampling Techniques . . . . . . . . . . . . . . . . . . . . . . . .

45

4.4

Sample Size Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

4.5

Methods of Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

4.6

Levels of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

4.7

Presentation of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

4.8

Measures of Central Tendency . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

4.9

Measures of Dispersion or Variability . . . . . . . . . . . . . . . . . . . . . . .

58

Financial Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

5.1

Simple and Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . .

64

5.2

Ordinary Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

5.3

Loan Repayment or Paying Off a Debt . . . . . . . . . . . . . . . . . . . . . .

72

Lesson 0

4 5.4

6

Other Applications of Financial Mathematics . . . . . . . . . . . . . . . . . . .

78

Voting Methods and Apportionment . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

6.1

Voting Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

6.2

Apportionment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

All Rights Reserved. 2020

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Lesson 1

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Lesson 1: Mathematics in Our World Learning Outcomes At the end of the lesson, the students are able to: 1. identify patterns in nature in the world; 2. articulate the importance of mathematics in one’s life; 3. argue about the nature of mathematics, what it is how it is expressed, represented and used; 4. enumerate and discuss the role of mathematics in various disciplines; 5. express appreciation for mathematics as a human endeavor.

1.1

Overview: What is mathematics?

Mathematics can be defined in many ways. For some people, Mathematics is just the study of numbers. For others, it is a set of problem-solving tools, a language, a process of thinking, and a study of patterns among others. Whatever point of view is taken, there is no denying the reality that mathematics is everywhere. Individuals from around the world use math in their daily lives. Mathematics has various applications in the world. However, Mathematics is not only concerned with everyday problems, but also with using imagination, intuition and reasoning to find new ideas and to solve puzzling problems. Mathematics is a branch of science, which deals with numbers and their operations. It involves calculation, computation, solving of problems etc. Its dictionary meaning states that, ‘Mathematics is the science of numbers and space’ or ‘Mathematics is the science of measurement, quantity and magnitude.’ It is exact, precise, systematic and a logical subject. Mathematics helps us to organize and systemize our ideas about patterns; in so doing, not only can we admire and enjoy these patterns, we can also use them to infer some of the underlying principles that govern the world of nature. In this lesson, attention will be focused on the nature of mathematics, patterns and numbers in nature and the world and the uses of mathematics.

1.2

Patterns and Numbers in Nature

What are patterns anyway? We usually think of it as anything that repeats again and again. A pattern is an arrangement which helps observers anticipate what they might see or what happens next. A pattern also shows what may have come before. A pattern organizes information so that it becomes more useful. All Rights Reserved. 2020

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The human mind is programmed to make sense of data or to bring order where there is disorder. It seeks to discover relationships and connections between seemingly unrelated bits of information. In doing so, it sees patterns.

According to the National Council of Teachers of Mathematics (1991) defines the nature of mathematics as follows: Mathematics is a study of patterns and relationship, a way of thinking, an art, a language, and a tool. It is about patterns and relationships. Numbers are just a way to express those patterns and relationships. patterns

Patterns are everywhere. They are deeply embedded all around us. You can observe patterns- things like colors, shapes, actions, line or curves of building, pathways or even in the grocery store where boxes of various items are lined up. Number patterns such as 2,4,6,8 and 5,10,15,20 are among the first patterns encountered in younger years. As we advance, we encounter more patterns and discover that number patterns are not restricted to a few types. They could be ascending, descending, multiples of a certain number. We learned patterns through the concept of functions and sequences like arithmetic and geometric sequences. Number patterns, logic patterns, geometric patterns and word patterns are examples of the various patterns we learned in school. However, patterns are not limited to these types. One can observe patterns in nature, art, architecture, human behavior, anywhere. On this section, we will discuss the different patterns in nature, arts and architecture. Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modeled mathematically. Natural patterns include symmetries, fractals, spirals, meanders, waves, foams, tessellations, cracks, and stripes. Studying patterns allows one to watch, guess, create, and discover. The present mathematics is considerably more than arithmetic, algebra, and geometry. The method of doing it has advanced from simply performing computations or derivations into observing patterns, testing guesses, and evaluating results. Let us focus on the different types of symmetric patterns, analyze and observe the similarities as well as the differences and give examples of these types of patterns as seen in nature, arts, architecture and mathematics. All Rights Reserved. 2020

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which is known as the Fibonacci sequence. The Fibonacci sequence was invented by the Italian Leonardo Pisano Bigollo (1180-1250), who is known in mathematical history by several names: Leonardo of Pisa (Pisano means “from Pisa”) and Fibonacci (which means “son of Bonacci”). To formally, define the Fibonacci sequence, we start by defining F1 = 1 and F2 = 1. For n > 2, we define Fn := Fn−1 + Fn−2 : The sequence F1 ; F2 ; F3 ; : : : is then the Fibonacci sequence. Such a definition is called a recursive definition because it starts by defining some initial values and defines the next term as a function of the previous terms. If we take the ratio of Fn to Fn−1 for n ≥ 1, n

Fn

Fn =Fn−1

n

Fn

Fn =Fn−1

1

1

-

8

21

1:61538 : : :

2

1

1

9

34

1:61904 : : :

3

2

2

10

55

1:61764 : : :

4

3

1.5

11

89

1:61818 : : :

5

5

1:666 : : :

12

144

1:61797 : : :

6

8

1.6

13

233

1:61805 : : :

7

13

1.625

14

377

1:618025 : : :

we see that as n gets larger and larger, the ratio gets closer and closer to a value denoted by ’. The number ’ is called as the golden ratio and can be formally defined as ’ := lim

n−→∞

Fn : Fn−1

The symbol n−→∞ lim means ‘the limit as n approaches infinity’ which is usually studied in a calculus course. It can be calculated that the exact value of ’ is

√ 1+ 5 ≈ 1:6180339887 : : : : ’= 2 √ 1− 5 If we denote by ’ := , we can write the nth Fibonacci number explicitly using the formula 2 Fn =

’n − ’n √ : 5

This is known as the Binet Formula. All Rights Reserved. 2020

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2 3 1

1

8

5

Do you see the Fibonacci Numbers? The red curve is known as the Fibonacci Spiral. A rectangle whose side ratio (length:width) equals ’ is called a golden rectangle. George Dvorsky (2013) emphasized that the Fibonacci sequence has captivated mathematicians, scientists, artists and designers for centuries. It is a sequence with many interesting properties. Among these is its visibility in nature. Most, if not all, natureâĂŹs most beautiful patterns contain Fibonacci numbers. The Fibonacci numbers appear in nature in various places. These numbers are evident at the flower head of a sunflower or daisy. Spirals are also easier to see and to count on pineapples and pine cones. Fibonacci numbers are there on broccoli florets and flowers and on the arrangement of leaves around stems on many plants too. • Pinecones, Speed Heads, Vegetables and Fruits

Spiral patterns curving from left and right can be seen at the array of seeds in the center of a sunflower. The sum of these spirals when counted will be a Fibonacci number. You will get two consecutive Fibonacci numbers if you divide the spirals into those pointed left and right. The seed pods on a pinecone are also arranged in a spiral pattern. Each cone consists of a pair of spirals, each one spiraling upwards in opposing directions. Spiral patterns can also be deciphered in cauliflower and pineapples. Fibonacci sequence appears on these fruits and vegetables.

• Flowers and Branches

Most flowers express the Fibonacci sequence if you count the number of petals on these flowers. For example, lilies and irises have three petals, roses and buttercups have five, delphiniums have eight petals and so on. Some plants also exhibit the Fibonacci sequence in their growth points, on the places where tree branches form or split. A trunk grows until it produces a branch, resulting

All Rights Reserved. 2020

Abdul, Atienza, et. al.

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Earth scientists have relied in the past on statistical methods to forecast natural hazard events. However, Benoit Mandelbrot, a professor of mathematical sciences at Yale University described how he has been using fractals to find order within complex systems in nature, such as the natural shape of a coastline. As a result of his research, earth scientists are taking Mandelbrot’s fractal approach one step further and are measuring past events and making probability forecasts about the size, location, and timing of future natural disasters. Mathematics helps control nature and occurrences in the world for our own good. Mathematical modelling and control theory can be used. By mathematical modeling we see the inputs to events and their most likely outcomes. Knowing these inputs and seeing their consequences and establishing their relationship defined quantitatively, we can prepare for calamities or natural disasters, or better yet, we can probably stop them from happening. Control theory is defined as a field of applied mathematics that is relevant to the control of certain physical processes and systems. As long as human culture has existed, control has meant some kind of power over the environment and control theory may be viewed as the science of modifying that environment, in the physical, biological, or even social sense. Control theory played a major role in many technological advances in the second half of the 20th century. Mathematics has applications in many human endeavors making it indispensable. Mathematics existed since the beginning of time, written or unwritten. Its unwritten history is carved in all things found in cosmos , found in the patterns created in nature, appreciated in the juxtaposition of the heavens and the earth, contrast between darkness and light , made sense in the harmony created not just by a well-known orchestra but even by the rain drops falling on offshore wind-turbines. Its language, though considered by many as abstract is in fact easy to grasp when the logic and formula that govern it are understood by the inquisitive minds of students, bakers, chemists , carpenters and appreciated by the receptive hearts of the musicians - drummers, guitarists, pianists and composers; dance choreographers, gymnasts and marathon runners. Mathematics permeates every area of man’s life , leaving every man convinced of its value. As a tool, mathematics is indispensable. It is needed by all people in honing their logical thinking and reasoning, in making wise financial decisions - in budgeting or making both ends meet when financial resources are scarce. It is needed in choosing the best interior and outdoor designs of houses , offices and business sites. It is useful in determining traveling time and calculating the amount of fuel needed to get to the destination. It is not just needed in the classrooms but also at home when doing the mundane baking or preparing foods for breakfast , dinner or lunch; calculating steps when performing simple to complex acrobatic stance; determining speed in a short distance or marathon run, preparing chemical solutions in All Rights Reserved. 2020

Abdul, Atienza, et. al.

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a biological or chemical laboratory and the like. Indeed, its application and use are uncountable and the list of uses it offers is unending. As it is valuable and integral in the life of man, mathematics as a discipline that Introduces students with the wide array of pos...