Mathematics in the Modern World PDF Learning Modules 2020
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Batangas State University
2020
Jose Alejandro R. Belen, Neil M. Mame, Israel P. Piñero
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This document is a learning module for an undergraduate course in Mathematics in the Modern World at Batangas State University. The module covers topics about the nature of mathematics, mathematical language, problem solving, and reasoning.
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MATHEMATICS IN THE MODERN WORLD BATANGAS STATE UNIVERSITY GENERAL EDUCATION COURSE MATHEMATICS IN THE MODERN WORLD Learning Modules by Jose Alejandro R. Belen Neil M. Mame Israel P. Piñero...
MATHEMATICS IN THE MODERN WORLD BATANGAS STATE UNIVERSITY GENERAL EDUCATION COURSE MATHEMATICS IN THE MODERN WORLD Learning Modules by Jose Alejandro R. Belen Neil M. Mame Israel P. Piñero 2020 1 MATHEMATICS IN THE MODERN WORLD LEARNING MODULES IN MATHEMATICS IN THE MODERN WORLD 2 MATHEMATICS IN THE MODERN WORLD FOREWORD Mathematics in the Modern World is a general education course that deals with the nature of mathematics. It is concerned with the appreciation of its practical, intellectual, aesthetic, and application of mathematical tools in daily life. Its primary objective is to educate and secondarily to train learners in this area of discipline. On educative side, it aims to equip students with necessary intelligence to become leaders and partakers in nation building. While on the training side, it provides them necessary and sufficient skills that they can harness in order to combat the challenges of daily living. This learning material being guided by CHED CMO No. 20, series of 2013, is divided into several modules encompassing both the nature of mathematics and the utility of mathematics in the modern world. Each module is subdivided into a number of lessons designed to introduce each topic pedagogically in a management fashion intended for independent learning. Learning activities as well as Chapter Tests are provided in compliance with the learning plan suggested by the Commission on Higher Education and of the Institution. Module 1 is concerned with the mathematics in our world. It provides a new way of looking at mathematics as a science of patterns. Basically, it encapsulates the entirety of the course by providing insights that mathematical structure is embedded in the structure of the natural world. Module 2 is focused on mathematical language. It expounded the idea that like any language, mathematics has its own symbols, syntax and rules. It explained the conventions and usefulness of mathematics as a language. Module 3 discussed about problem solving and reasoning. It asserted that mathematics is not just about numbers and much of it is problem solving and 3 MATHEMATICS IN THE MODERN WORLD reasoning. It recalibrated the learner’s problem solving skills, by providing new understanding on the relevance of the Polya’s method in solving mathematical problems. Module 4 introduced the mathematical system. It was a recreational mathematics that finally put into the limelight at the dawn of the modern world. Its utilization served as a backbone of commerce in the information age. It is an indispensable tool of the modern time. Module 5 is entitled Data Management. It explained that statistical tools derived from mathematics are useful in processing and managing numerical data in order to describe a phenomenon and predict values. It is intended to go beyond the typical understanding as merely set of formulas but as tools that can decode nature’s numbers. Module 6 explored the very fabric that woven of mathematical landscape. It is the art and science of correct thinking and reasoning: Logic. It disciplined learner’s understanding by exploring the application of formal logic to mathematics. Module 7 is an innovative mathematical concept concerning network and connectivity. It basically concerned on how networks can be encoded and solved economically. It challenges the mind of the learner to metaphorically settle the Konigsberg Bridge Problem of the modern world. Over-all the modules comprising this book are eclectically treated in such a way that both interpretative and applicative dimensions of learning become an integral part of the lesson presented, and the learning activities as well. 4 MATHEMATICS IN THE MODERN WORLD Table of Contents FOREWORD................................................................................................................................................. 3 MODULE 1: THE NATURE OF MATHEMATICS............................................................................................. 7 The Mathematics of Our World.............................................................................................................. 8 The Mathematics in Our World............................................................................................................ 18 The Fibonacci Sequence........................................................................................................................ 29 Chapter Test 1....................................................................................................................................... 36 MODULE 2: MATHEMATICAL LANGUAGE AND SYMBOLS......................................................................... 40 Characteristics and Conventions in the Mathematical Language......................................................... 40 Four Basic Concepts.............................................................................................................................. 51 Logic and Formality............................................................................................................................... 78 Chapter Test 2....................................................................................................................................... 85 MODULE 3:PROBLEM SOLVING AND REASONING.................................................................................... 96 Inductive and Deductive Reasoning...................................................................................................... 96 Intuition, Proof and Certainty............................................................................................................. 108 Polya’s Four Steps in Problem Solving................................................................................................ 128 Mathematical Problems Involving Patterns........................................................................................ 136 Recreational Problems Using Mathematics........................................................................................ 145 Chapter Test 3..................................................................................................................................... 152 MODULE 4: MATHEMATICAL SYSTEM.................................................................................................... 158 Modular Arithmetic............................................................................................................................ 158 Operations on Modular Arithmetic..................................................................................................... 170 Applications of Modular Arithmetic.................................................................................................... 177 Group Theory...................................................................................................................................... 193 Chapter Test 4..................................................................................................................................... 206 MODULE 5:DATA MANAGEMENT........................................................................................................... 215 The Data.............................................................................................................................................. 216 Measures of Central Tendency........................................................................................................... 226 Measures of Dispersion...................................................................................................................... 235 5 MATHEMATICS IN THE MODERN WORLD Measures of Relative Position............................................................................................................. 242 The Normal Distributions.................................................................................................................... 254 The Linear Correlation: Pearson r....................................................................................................... 266 The Least-Squares Regression Line..................................................................................................... 273 Chapter Test 5..................................................................................................................................... 279 MODULE 6: LOGIC.................................................................................................................................. 288 Logic Statements and Quantifiers....................................................................................................... 289 Truth Tables, Equivalent Statements, and Tautology.......................................................................... 298 Switching Networks and Logic Gates.................................................................................................. 312 The Conditional and Related Statements............................................................................................ 234 Symbolic Argument............................................................................................................................. 328 Chapter Test 6..................................................................................................................................... 330 MODULE 7: MATHEMATICS OF GRAPHS................................................................................................. 335 Graphs and Eulerian Circuit................................................................................................................ 336 Graph Coloring.................................................................................................................................... 361 Chapter Test 7..................................................................................................................................... 367 6 MATHEMATICS IN THE MODERN WORLD MODULE ONE THE NATURE OF MATHEMATICS CORE IDEA Module One is an introduction to the nature of mathematics as an exploration of patterns. It is a useful way to think about nature and our world. Learning Outcome: 1. To identify patterns in nature and regularities in the world. 2. To articulate importance of mathematics in one’s life. 3. To argue about the nature of mathematics, what it is, how it is expressed, represented, and used. 4. Express appreciation of mathematics as a human endeavor. Unit Lessons: Lesson 1.1 Mathematics of Our World Lesson 1.2 Mathematics in Our World Lesson 1.3 Mathematics of Sequence Time Allotment: Four lecture hours 7 MATHEMATICS IN THE MODERN WORLD Lesson 1.1 The Mathematics of Our World Specific Objectives 1. To understand the mathematics of the modern world. 2. To revisit and appreciate the mathematical landscape. 3. To realize the importance of mathematics as a utility. 4. To gain awareness of the role of mathematics as well as our role in mathematics. Lesson 1.1 does not only attempt to explain the essence of mathematics, it serves also as a hindsight of the entire course. The backbone of this lesson draws from the Stewart’s ideas embodied in his book entitled Nature’s Numbers. The lesson provides new perspective to understand the irregularity and chaos of our world as we move through the landscape of regularity and order. It poses some thought- provoking questions to draw one’s innate mathematical intelligence by making one curious, not so much to seek answers, but to ask more right questions. Discussions The Nature of Mathematics In the book of Stewart, Nature’s Number, he that mathematics is a formal system of thought that was gradually developed in the human mind and evolved in the 8 MATHEMATICS IN THE MODERN WORLD human culture. Thus, in the long course of human history, our ancestors at a certain point were endowed with insight to realize the existence of “form” in their surroundings. From their realization, a system of thought further advanced their knowledge into understanding measures. They were able to gradually develop the science of measures and gained the ability to count, gauge, assess, quantify, and size almost everything. From our ancestor’s realization of measures, they were able to notice and recognize some rudiment hints about patterns. Thus, the concept of recognizing shapes made its course towards classifying contour and finally using those designs to build human culture: an important ingredient for a civilization to flourish. From then, man realized that the natural world is embedded in a magnanimously mathematical realm of patterns----and that natural order efficiently utilizes all mathematical patterns to its advantage. As a result, we made use of mathematics as a brilliant way to understand the nature by comprehending the structure of its underlying patterns and regularities. Mathematics is present in everything we do; it is all around us and it is the building block of our daily activities. It has been at the forefront of each and every period of our development, and as our civilized societies advanced, our needs of mathematics pioneering arose on the frontier of our course as we prepare our human species to traverse the cosmic shore. Mathematics is a Tool Mathematics, as a tool, is immensely useful, practical, and powerful. It is not about crunching numbers, formulas, and symbols but rather, it is all about forming new ways to see problems so we can understand them by combining insights with imagination. It also allows us to perceive realities in different contexts that would otherwise be intangible to us. It can be likened to our sense of sight and touch. Mathematics is our sense to decipher patterns, relationships, and logical connections. It is our whole new way to see and understand the modern world. 9 MATHEMATICS IN THE MODERN WORLD Mathematics, being a broad and deep discipline, deals with the logic of shape, quantity, and arrangement. Once, it was perceived merely a collective thoughts dealing with counting numbers, but it is now being understood as a universal language dealing with symbols, arts, equations, geometric shapes and patterns. It is asserting that mathematics is a powerful tool in decision-making and it is a way of life. The nature of mathematics Figure 1.1 In the Figure 1.1 illustrated by Nocon and Nocon, it portrays the function of mathematics. As shown, it is stated that mathematics is a set of problem-solving tools. It provides answers to existing questions and presents solutions to occurring problems. It has the power to unveil the reasons behind occurrences and it offers explanations. Moreover, mathematics, as a study of patterns, allows people to 10 MATHEMATICS IN THE MODERN WORLD observe, hypothesize, experiment, discover, and recreate. On the other hand, mathematics is an art and a process of thinking. For it involves reasoning, which can be inductive or deductive, and it applies methods of proof both in fashion that is conventional and unventional. Mathematics is Everywhere We use mathematics in their daily tasks and activities. It is our important tool in the field of sciences, humanities, literature, medicine, and even in music and arts; it is in the rhythm of our daily activities, operational in our communities, and a default system of our culture. There is mathematics wherever we go. It helps us cook delicious meals by exacting our ability to measure and moderately control of heat. It also helps us to shop wisely, read maps, use the computer, remodel a home with constrained budget with utmost economy. Source: Space Telescope Science/NASA The Universe Figure 1.2 11 MATHEMATICS IN THE MODERN WORLD Even the cosmic perspective, the patterns in the firmament are always presented as a mystery waiting to be uncovered by us-the sentient being. In order to unearthed this mystery, we are challenged to investigate and deeply examine its structure and rules to the infinitesimal level. The intertwined governing powers of cosmic mystery can only be decoded by seriously observing and studying their regularities, and patiently waiting for the signature of some kind interference. It is only by observing the abundance of patterns scattered everywhere that this irregularities will beg to be noticed. Some of them are boldly exposed in a simple and obvious manner while others are hidden in ways that is impossible to perceive by easy to discern. While our ancestors were able to discover the presence of mathematics in everything, it took the descendants, us, a long time to gradually notice the impact of these patterns in the persistence of our species to rightfully exist. The Essential Roles of Mathematics Mathematics has countless hidden uses and applications. It is not only something that delights our mind but it also allows us to learn and understand the natural order of the world. This discipline was and is often studied as a pure science but it also finds its place in other areas of perpetuating knowledge. Perhaps, science would definitely agree that, when it comes to discovering and unveiling the truth behind the inherent secrets and occurrences of the universe, nothing visual, verbal, or aural come close to matching the accuracy, economy, power and elegance of mathematics. Mathematics helps us to take the complex processes that is naturally occuring in the world around us and it represents them by utilizing logic to make things more organized and more efficient. Further, mathematics also facilitate not only to weather, but also to control the weather ---- be it social, natural, statistical, political, or medical. Applied mathematics, which once only used for solving problems in physics, and it is also becoming a useful tool in biological sciences: for instance, the spread of various 12 MATHEMATICS IN THE MODERN WORLD diseases can now be predicted and controlled. Scientists and researchers use applied mathematics in doing or performing researches to solve social, scientific, medical, or even political crises. It is a common fact that mathematics plays an important role in many sciences. It is and it provides tools for calculations. We use of calculations in other disciplines whenever we are underrating some kind of research or experiment. The use of mathematical calculations is indispensable method in scientifically approaching most of the problems. In a similar way, mathematics, provides new questions to think about. Indeed, in learning and doing mathematics, there will always be new questions to answer, new problems to solve, and new things to think about (Vistru- Yu PPT presentation). The Mathematical Landscape The human mind and culture developed a conceptual landscape for mathematical thoughts and ideas to flourish and propagate. There is a region in the human mind that is capable of constructing and discerning the deepest insights being perceived from the natural world. In this region, the mathematical landscape exists- wherein concepts of numbers, symbols, equations, operations calculations, abstractions, and proofs are the inhabitants as well as the constructs of the impenetrable vastness of its unchartered territories. In this landscape, a number is not simply a mathematical tree of counting. Also, infinite variables can be encapsulate to finite. Even those something that is hard to express in decimal form can be expressed in terms of fractions. Those things that seemed eternal ℤ can further be exploited using mathematical operations. This landscape claimed complex numbers as the firmament and even asserted that imaginary numbers also exist. To the low state negative numbers relentlessly enjoying recognition as existent beings. The wind in this landscape is unpredictable that the rate of change of the rate of change of weather is known as calculus. And beneath the surface of this mathematical landscape are firmly-woven proofs, theorems, definitions, and axioms which are 13 MATHEMATICS IN THE MODERN WORLD intricately “fertilized” by reasoning, analytical, critical thinking and germicide by mathematical logic that made them precise, exact and powerful. With this landscape, the mathematician's instinct and curiosity entice to explore further the vast tranquil lakes of functions and impassable crevasse of the unchartered territories of abstract algebra. For to claim ownership is to understand the ebb and flow of prime numbers. To predict the behavior of its Fibonacci weather, to be amazed with awe and wonder the patternless chaos of fractal clouds, and to rediscover that after all, the numbers in mathematics is not a "thing" but a process. Conventionally, we are just simply made ourselves comfortable on the “thingification” of those processes and we forgot that 1+1 is not a noun but a verb. How Mathematics is Done Math is a way of thinking, and it is undeniably important to see how that thinking is going to be developed rather than just merely see face value of the results. For some people, few math theorems can bring up as much remembered pain and anxiety. For others, this discipline is so complex and they have to understand the confusing symbols, the difficult procedures, and the dreaded graphs and charts. For most, mathematics is just nothing but something to survive, rather than to learn. To the untrained eye, doing mathematics is quite difficult and challenging. It is ambiguous, for it follows a set of patterns, formulas, and sequences that make it more demanding to do and to learn. It is abstract and complex ---- and for these reasons, a lot of people adopt the belief that they are not math people. Mathematics builds upon itself. More complex concepts are built upon simpler concepts, and if you do not have a strong grasp of the fundamental principles, then a more complex problem is more likely going to stump you. If you come across a mathematical problem that you cannot solve, the first thing to do is to identify the 14 MATHEMATICS IN THE MODERN WORLD components or the operations that it wants you to carry out, and everything follows. Doing and performing mathematics is not that simple. It is done with curiosity, with a penchant for seeking patterns and generalities, with a desire to know the truth, with trial and error, and without fear of facing more questions and problems to solve. (Vistru-Yu) Mathematics is for Everyone The relationship of the mathematical landscape in the human mind with the natural world is so strange that in the long run, the good math provides utilization and usefulness in the order of things. Perhaps, for most people, they simply need to know the basics of the mathematical operations in order to survive daily tasks; but for the human society to survive and for the human species to persistently exist, humanity needs, beyond rudiment of mathematics. To safeguard our existence, we already have delegated the functions of mathematics across all disciplines. There is mathematics we call pure and applied, as there are scientists we call social and natural. There is mathematics for engineers to build, mathematics for commerce and finance, mathematics for weather forecasting, mathematics that is related to health, and mathematics to harness energy for utilization. To simply put it, everyone uses mathematics in different degrees and levels. Everyone uses mathematics, whoever they are, wherever they are, and whenever they need to. From mathematicians to scientists, from professionals to ordinary people, they all use mathematics. For mathematics puts order amidst disorder. It helps us become better persons and helps make the world a better place to live in. (Vistru-Yu). The Importance of Knowing and Learning Mathematics Why do we want to observe and describe patterns and regularities? Why do we want to understand the physical phenomena governing our world? Why do we want to dig out rules and structures that lie behind patterns of the natural order? It is because those rules and structures explain what is going on. It is because they 15 MATHEMATICS IN THE MODERN WORLD are beneficial in generating conclusions and in predicting events. It is because they provide clues. The clues that make us realize that interference in the motion of heavenly bodies can predict lunar eclipse, solar eclipse as well as comets’ appearances. That the position of the sun and the moon relative to the earth can predict high tide and low tide events affecting human activities. And that human activities need clues for the human culture to meaningfully work. Mathematical training is vital to decipher the clues provided by nature. But the role of mathematics goes clues and it goes beyond prediction. Once we understand how the system works, our goal is to control it to make it do what we want. We want to understand the mathematical pattern of a storm to avoid or prevent catastrophes. We want to know the mathematical concept behind the contagion of the virus to control its spread. We want to understand the unpredictability of cancer cells to combat it before it even exists. Finally, we want to understand the butterfly effect as much as we are so curious to know why the “die” of the physical world play god. “Whatever the reasons, mathematics is a useful way to think about nature. What does it want to tell us about the patterns we observe? There are many answers. We want to understand how they happen; to understand why they happen, which is different; to organize the underlying patterns and regularities in the most satisfying way; to predict how nature will behave; to control nature for our own ends; to make practical use of what we have learned about our world. Mathematics helps us to do all these things, and often, it is indispensable.“ [Stewart] 16 MATHEMATICS IN THE MODERN WORLD Learning Activity 1.1 Answer one of the following questions (15-20 lines) and submit your answer to your course facilitator. 1. What are the new things that you learned about the nature of mathematics? 2. What aspect of the lesson significant changed your view about mathematics? 3. What is the most important contribution of mathematics in humankind? Observe the following this format: Paper Font Font All Line Page Substance Margin Orientation Paper Size Number Type Size Spacing (if printed) 20 Normal Portrait 8.5 x 13 Arial 12 1.5 Page x of x Justified 17 MATHEMATICS IN THE MODERN WORLD Lesson 1.2 The Mathematics in Our World Specific Objective : 1. To develop one’s understanding about patterns; 2. To identify different patterns in nature; 3. To recognize different symmetries in nature; and 4. To explain the presence of Fibonacci numbers in nature The mathematics in our world is rooted in patterns. Patterns are all around us. Finding and understanding patterns give us great power to play like god. With patterns, we can discover and understand new things; we learn to predict and ultimately control the future for our own advantage. A pattern is a structure, form, or design that is regular, consistent, or recurring. Patterns can be found in nature, in human-made designs, or in abstract ideas. They occur in different contexts and various forms. Because patterns are repetitive and duplicative, their underlying structure regularities can be modelled mathematically. In general sense, any regularity that can be explained mathematically is a pattern. Thus, an investigation of nature’s patterns is an investigation of nature’s numbers. This means that the relationships can be observed, that logical connections can be established, that generalizations can be inferred, that future events can be predicted, and that control can possibly be possible. 18 MATHEMATICS IN THE MODERN WORLD Discussions Different Kinds of Pattern As we look at the world around us, we can sense the orchestrating great regularity and diversity of living and non-living things. The symphonies vary from tiny to gigantic, from simple to complex, and from dull to the bright. The kaleidoscope of patterns is everywhere and they make the nature look only fascinating but also intriguing. Paradoxically, it seemed that everything in the world follows a pattern of their own and tamed by the same time pattern of their own. Patterns of Visuals. Visual patterns are often unpredictable, never quite repeatable, and often contain fractals. These patterns are can be seen from the seeds and pinecones to the branches and leaves. They are also visible in self-similar replication of trees, ferns, and plants throughout nature. Patterns of Flow. The flow of liquids provides an inexhaustible supply of nature’s patterns. Patterns of flow are usually found in the water, stone, and even in the growth of trees. There is also a flow pattern present in meandering rivers with the repetition of undulating lines. Patterns of Movement. In the human walk, the feet strike the ground in a regular rhythm: the left-right-left-right-left rhythm. When a horse, a four-legged creature walks, there is more of a complex but equally rhythmic pattern. This prevalence of pattern in locomotion extends to the scuttling of insects, the flights of birds, the pulsations of jellyfish, and also the wave-like movements of fish, worms, and snakes. Patterns of Rhythm. Rhythm is conceivably the most basic pattern in nature. Our hearts and lungs follow a regular repeated pattern of sounds or movement whose timing is adapted to our body’s needs. Many of nature’s rhythms are most likely 19 MATHEMATICS IN THE MODERN WORLD similar to a heartbeat, while others are like breathing. The beating of the heart, as well as breathing, have a default pattern. Patterns of Texture. A texture is a quality of a certain object that we sense through touch. It exists as a literal surface that we can feel, see, and imagine. Textures are of many kinds. It can be bristly, and rough, but it can also be smooth, cold, and hard. Geometric Patterns. A geometric pattern is a kind of pattern which consists of a series of shapes that are typically repeated. These are regularities in the natural world that are repeated in a predictable manner. Geometrical patterns are usually visible on cacti and succulents. Patterns Found in Nature Common patterns appear in nature, just like what we see when we look closely at plants, flowers, animals, and even at our bodies. These common patterns are all incorporated in many natural things. Waves and Dunes A wave is any form of disturbance that carries energy as it moves. Waves are of different kinds: mechanical waves which propagate through a medium ---- air or water, making it oscillate as waves pass by. Wind waves, on the other hand, are 20 MATHEMATICS IN THE MODERN WORLD surface waves that create the chaotic patterns of the sea. Similarly, water waves are created by energy passing through water causing it to move in a circular motion. Likewise, ripple patterns and dunes are formed by sand wind as they pass over the sand. Spots and Stripes We can see patterns like spots on the skin of a giraffe. On the other hand, stripes are visible on the skin of a zebra. Patterns like spots and stripes that are commonly present in different organisms are results of a reaction-diffusion system (Turing, 1952). The size and the shape of the pattern depend on how fast the chemicals diffuse and how strongly they interact. Spirals Jean Beaufort has released this “Spiral Galaxy” image under Public Domain license 21 MATHEMATICS IN THE MODERN WORLD The spiral patterns exist on the scale of the cosmos to the minuscule forms of microscopic animals on earth. The Milky Way that contains our Solar System is a barred spiral galaxy with a band of bright stars emerging from the center running across the middle of it. Spiral patterns are also common and noticeable among plants and some animals. Spirals appear in many plants such as pinecones, pineapples, and sunflowers. On the other hand, animals like ram and kudu also have spiral patterns on their horns. Symmetries In mathematics, if a figure can be folded or divided into two with two halves which are the same, such figure is called a symmetric figure. Symmetry has a vital role in pattern formation. It is used to classify and organize information about patterns by classifying the motion or deformation of both pattern structures and processes. There are many kinds of symmetry, and the most important ones are reflections, rotations, and translations. These kinds of symmetries are less formally called flips, turns, and slides. Reflection symmetry, sometimes called line symmetry or mirror symmetry, captures symmetries when the left half of a pattern is the same as the right half. 22 MATHEMATICS IN THE MODERN WORLD Rotations, also known as rotational symmetry, captures symmetries when it still looks the same after some rotation (of less than one full turn). The degree of rotational symmetry of an object is recognized by the number of distinct orientations in which it looks the same for each rotation. 23 MATHEMATICS IN THE MODERN WORLD Translations. This is another type of symmetry. Translational symmetry exists in patterns that we see in nature and in man-made objects. Translations acquire symmetries when units are repeated and turn out having identical figures, like the bees’ honeycomb with hexagonal tiles. 24 MATHEMATICS IN THE MODERN WORLD Symmetries in Nature From the structure of subatomic particles to that of the entire universe, symmetry is present. The presence of symmetries in nature does not only attract our visual sense, but also plays an integral and prominent role in the way our life works. Human Body Animal Movement The human body is one of the pieces The symmetry of motion is present in of evidence that there is symmetry in animal movements. When animals move, nature. Our body exhibits bilateral we can see that their movements also symmetry. It can be divided into two exhibit symmetry. identical halves. 25 MATHEMATICS IN THE MODERN WORLD Sunflower Snowflakes One of the most interesting things about Snowflakes have six-fold radial a sunflower is that it contains both radial symmetry. The ice crystals that and bilateral symmetry. What appears make-up the snowflakes are to be "petals" in the outer ring are symmetrical or patterned. The actually small flowers also known as ray intricate shape of a single arm of a florets. These small flowers are snowflake is very much similar to bilaterally symmetrical. On the other the other arms. This only proves hand, the dark inner ring of the that symmetry is present in a sunflower is a cluster of radially snowflake. symmetrical disk florets. Honeycombs/Beehive Honeycombs or beehives are examples of wallpaper symmetry. This kind of symmetry is created when a pattern is repeated until it covers a plane. Beehives are made of walls with each side having the same size enclosed with small hexagonal cells. Inside these cells, honey and pollen are stored and bees are raised. 26 MATHEMATICS IN THE MODERN WORLD Starfish Starfish have a radial fivefold symmetry. Each arm portion of the starfish is identical to each of the other regions. Fibonacci in Nature By learning about nature, it becomes gradually evident that the nature is essentially mathematical, and this is one of the reasons why explaining nature is dependent on mathematics. Mathematics has the power to unveil the inherent beauty of the natural world. In describing the amazing variety of phenomena in nature we stumble to discover the existence of Fibonacci numbers. It turns out that the Fibonacci numbers appear from the smallest up to the biggest objects in the natural world. This presence of Fibonacci numbers in nature, which was once existed realm mathematician’s curiously, is considered as one of the biggest mysteries why the some patterns in nature is Fibonacci. But one thing is definitely made certain, and that what seemed solely mathematical is also natural. For instance, many flowers display figures adorned with numbers of petals that are in the Fibonacci sequence. The classic five-petal flowers are said to be the most 27 MATHEMATICS IN THE MODERN WORLD common among them. These include the buttercup, columbine, and hibiscus. Aside from those flowers with five petals, eight-petal flowers like clematis and delphinium also have the Fibonacci numbers, while ragwort and marigold have thirteen. These numbers are all Fibonacci numbers. Apart from the counts of flower petals, the Fibonacci also occurs in nautilus shells with a logarithmic spiral growth. Multiple Fibonacci spirals are also present in pineapples and red cabbages. The patterns are all consistent and natural. Learning Activity 1.2 Synthesis I. Read the entire book entitled Nature’s Numbers by Stewart. II. Write synthesis about all the things that you learned about nature’s numbers. III. It is highly recommended that at the outset, an outline must be made. IV. Please ensure that topic sentence can be clearly understood. -Your topic sentence must be supported by at least three arguments. V. Your synthesis must be around 1400-1500 words. VI. Rules on referencing and citation must be strictly observed. VII. You may use either MLA or APA system. VIII. The last page must contain the references or bibliography. IX. Write your name, student number, email address at the last page X. Please observe the following format: 28 MATHEMATICS IN THE MODERN WORLD Paper Font Font All Line Page Substance Margin Orientation Paper Size Number Type Size Spacing (if printed) 20 Normal Portrait 8.5 x 13 Arial 12 1.5 Page x of x Justified Please bear in mind the following criteria for grading your work. 0 Point : The student unable to elicit the ideas and concepts. 1 Point : The student is able to elicit the ideas and concepts but shows erroneous understanding of these. 2 Points: The student is able to elicit the ideas and concepts and shows correct understanding of these. 3 Points: The students not only elicits the correct ideas but also shows evidence of internalizing these. 4 Points: The student elicits the correct ideas, shows evidence of internalizing these, and consistently contributes additional thoughts to the Core Idea. Lesson 1.3 The Fibonacci Sequence Specific Objectives 1. To define sequence and its types 2. To differentiate Fibonacci sequence from other types of sequence 3. To discover golden ratio and golden rectangle; and 4. To learn how to compute for the nth term in the Fibonacci Sequence As we have discussed in the preceding lesson, human mind is capable of identifying and organizing patterns. We were also to realized that there are structures and patterns in nature that we don’t usually draw attention to. Likewise, we arrived at a position that in nature, some things follow mathematical sequences and one of them follow the Fibonacci sequence. We noticed that these sequences is observable in some flower petals, on the spirals of some shells and even on sunflower seeds. It is amazing to think 29 MATHEMATICS IN THE MODERN WORLD that the Fibonacci sequence is dramatically present in nature and it opens the door to understand seriously the nature of sequence. Discussion Sequence Sequence refers to an ordered list of numbers called terms, that may have repeated values. The arrangement of these terms is set by a definite rule. (Mathematics in the Modern World, 14th Edition, Aufmann, RN. et al.). Cosider the given below example: 1, 3, 5, 7, … (1stterm) (2nd term) (3rd term) (4th term) As shown above, the elements in the sequence are called terms. It is called sequence because the list is ordered and it follows a certain kind of pattern that must be recognized in order to see the expanse. The three dots at the end of the visible patterns means that the sequence is infinite. There are different types of sequence and the most common are the arithmetic sequence, geometric sequence, harmonic sequence, and Fibonacci sequence. Arithmetic sequence. It is a sequence of numbers that follows a definite pattern. To determine if the series of numbers follow an arithmetic sequence, check the difference between two consecutive terms. If common difference is observed, then definitely arithmetic sequence governed the pattern. To clearly illustrate the arrangement, consider the example below: 2, 4, 6, 8, 10, 12 … 2 2 2 2 2 30 MATHEMATICS IN THE MODERN WORLD Notice in the given example above, the common difference between two consecutive terms in the sequence is two. The common difference is the clue that must be figure out in a pattern in order to recognize it as an arithmetic sequence. Geometric sequence. If in the arithmetic sequence we need to check for the common difference, in geometric sequence we need to look for the common ratio. The illustrated in the example below, geometric sequence is not as obvious as the arithmetic sequence. All possibilities must be explored until some patterns of uniformity can intelligently be struck. At first it may seemed like pattern less but only by digging a little bit deeper that we can 2 8 32 finally delve the constancy. That is 8 , 32 , 128, , … generate 4, 4, 4,… 2, 8, 32, 128, … 4 4 4 Harmonic Sequence. In the sequence, the reciprocal of the terms behaved in a manner like arithmetic sequence. Consider the example below and notice an interesting pattern in the series. With this pattern, the reciprocal appears like arithmetic sequence. Only in recognizing the appearance that we can finally decode the sequencing the govern the series. 1 1 1 1 1 , , , , , … 2 4 6 8 10 Fibonacci Sequence. This specific sequence was named after an Italian mathematician Leonardo Pisano Bigollo (1170 - 1250). He discovered the sequence while he was studying rabbits. The Fibonacci sequence is a series 31 MATHEMATICS IN THE MODERN WORLD of numbers governed by some unusual arithmetic rule. The sequence is organized in a way a number can be obtained by adding the two previous numbers. 1, 1, 2, 3, 5, 8, 13, 21, … 0+1=1 0, 1, 1 1+1=2 0, 1, 1, 2 1+2=3 0, 1, 1, 2, 3 2+3=5 0, 1, 1, 2, 3, 5 Notice that the number 2 is actually the sum of 1 and 1. Also the 5th term which is number 5 is based on addition of the two previous terms 2, and 3. That is the kind of pattern being generated by the Fibonacci sequence. It is infinite in expanse and it was once purely maintained claim as a mathematical and mental exercise but later on the it was observed that the ownership of this pattern was also being claimed by some species of flowers, petals, pineapple, pine cone, cabbages and some shells. 1, 1, 2, 3, 5, 8, 13, 21, … To explore a little bit more about the Fibonacci sequence, the location of the term was conventionally tagged as Fib(𝑛). This means that Fib(1)=1, Fib(2)=1, Fib(3)=2 and Fib(4)=3. In this method, the Fib(𝑛) is actually referring to the the 𝑛th term of the sequence. It is also possible to make some sort of addition in this sequence. For instance: Fib (2) + Fib (6) = _?__ Fib(2) refers to the 2nd term in the sequence which is “1”. And Fib(6) refers to the 6th term which is “8”. So, the answer to that equation is simply “9” Formula for computing for the nth term in the Fibonacci Sequence xn = φn − (1−φ)n √5 32 MATHEMATICS IN THE MODERN WORLD Where: Xn stands for the Fibonacci number we’re looking for N stands for the position of the number in the Fibonacci sequence Φ stands for the value of the golden ratio Let us try for example: What is the 5th Fibonacci number? By using the formula we’ll get: X5 = (1.618)5 − (1−1.618)5 √5 X5= 5 The amazing grandeur of Fibonacci sequence was also discovered in the structure of Golden rectangle. The golden rectangle is made up of squares whose sizes, surprisingly is also behaving similar to the Fibonacci sequence. Take a serious look at the figure: The Golden Ratio 33 MATHEMATICS IN THE MODERN WORLD As we can see in the figure, there is no complexity in forming a spiral with the use of the golden rectangle starting from one of the sides of the first Fibonacci square going to the edges of each of the next squares. This golden rectangle shows that the Fibonacci sequence is not only about sequence of numbers of some sort but it is also a geometric sequence observing a rectangle ratio. The spiral line generated by the ratio is generously scattered around from infinite to infinitesimal. 34 MATHEMATICS IN THE MODERN WORLD Learning Activity 1.3 I. Identify what type of sequence is the one below and supply the sequence with the next two terms: 1. 1, 4, 7, 10, _, _? Type of Sequence: ____________________ 2. 80, 40, 20, _, _ ? Type of Sequence: ____________________ 3. 1, 1, 2, 3, 5, 8, _, _ ? Type of Sequence: ____________________ 4. 56, 46, 36, 26, _, _ ? Type of Sequence: ____________________ 5. 2, 20, 200, 2000, _, _ ? Type of Sequence: ____________________ II. Compute for the following Fibonacci numbers and perform the given operation: 1. What if Fib (13) ? 2. What is Fib (20) ? 3. What is Fib (8) + Fib (9) ? 4. What is Fib (1) * Fib (7) + Fib (12) – Fib (6) ? 5. What is the sum of Fib (1) up to Fib (10) ? 35 MATHEMATICS IN THE MODERN WORLD Chapter Test 1 Multiple Choice. Choose the letter of the correct answer and write it on the blank provided at the of the test paper. ____________ 1. What is said to be the most basic pattern in nature? A. Pattern of Flow C. Pattern of Rhythm B. Pattern of Movement D. Pattern of Visuals ____________2. This kind of pattern is unpredictable and it often contains fractals. A. Geometric Patterns C. Pattern of Movement B. Pattern of Forms D. Pattern of Visuals ____________3. What kind of pattern is a series of shapes that are repeating? A. Geometric Pattern C. Pattern of Texture B. Pattern of Flows D. Pattern of Visuals ____________4. Among the following, what is not a type of symmetry? A. Reflection C. Transformation B. Rotation D. Translation ____________5. All of the following statements are correct about Fibonacci except one: A. The logarithmic spiral growth of the Nautilus shell B. The total number of family members correspond to a Fibonacci number. C. Fibonacci numbers are the root of the discovery of the secret behind sunflower seeds. D. The numbers of petals of almost all flowers in the world correspond to the Fibonacci numbers. ____________6. What type of sequence deals with common ratio? 36 MATHEMATICS IN THE MODERN WORLD A. Arithmetic Sequence C. Geometric Sequence B. Fibonacci Sequence D. Harmonic Sequence ____________7. What is the sum of Fib (10) + Fib(5) ? A. 58 C. 60 B. 59 D. 61 ____________8. What is Fib (12) ? A. 144 C. 377 B. 233 D. 89 ____________9. What are the next two terms of the sequence, 8, 17, 26, 35? A. 49, 58 C. 44, 53 B. 39, 48 D. 54, 63 ____________10. What type of sequence is 5, 8, 13, 21, 34, 55, … ? A. Fibbonacci Sequence C. Fibonacci Sequence B. Fibonaacii Sequence D. Fibonacii Sequence ANSWER KEY 37 MATHEMATICS IN THE MODERN WORLD 1. C 6. C 2. D 7. C 3. A 8. B 4. C 9. C 5. B 10. C References Akshay, A. (n.d.). 13 Reasons Why Math is Important. Https://Lifehacks.Io/. Retrieved from https://lifehacks.io/reasons-why-math-is-important/ A. (2019, September 12). An Ode to Math, Mathematics in Nature. Https://Www.Minuteschool.Com/. Retrieved from https://www.minuteschool.com/2019/09/an-ode-to-math-mathematics-in-nature/ ASIASOCIETY.ORG. (n.d.). Understanding the World Through Math | Asia Society. Https://Asiasociety.Org. Retrieved from https://asiasociety.org/education/understanding- world-through-math Coolman, R. (2015, June 5). What is Symmetry? | Live Science. Https://Www.Livescience.Com/. Retrieved from https://www.livescience.com/51100-what-is-symmetry.html Discovery Cube. (2018, February 2). Moment of Science: Patterns in Nature. Discoverycube.Org/Blog. Retrieved from https://www.discoverycube.org/blog/moment- science-patterns-in-nature/ Fibonacci Number Patterns. (n.d.). Http://Gofiguremath.Org. Retrieved from http://gofiguremath.org/natures-favorite-math/fibonacci-numbers/fibonacci-number- patterns/ Grant, S. (2013, April 21). 10 Beautiful Examples of Symmetry In Nature. Https://Listverse.Com/. Retrieved from https://listverse.com/2013/04/21/10-beautiful-examples-of-symmetry-in- nature/ H., E. J. (2013, August 16). What is Mathematics? Https://Www.Livescience.Com/. Retrieved from https://www.livescience.com/38936-mathematics.html Irish Times. (2018, October 18). Who Uses Maths? Almost Everyone! Https://Thatsmaths.Com/. Retrieved from https://thatsmaths.com/2018/10/18/who-uses-maths-almost-everyone/ 38 MATHEMATICS IN THE MODERN WORLD Natural Patterns. (n.d.). Ecstep.Com. Retrieved from https://ecstep.com/natural-patterns/ Symmetry- Reflection and Rotation. (n.d.). Https://Www.Mathsisfun.Com/Index.Htm. Retrieved from https://www.mathsisfun.com/geometry/symmetry.html The Franklin Institute. (n.d.-b). Math Patterns in Nature. Https://Www.Fi.Edu. Retrieved from https://www.fi.edu/math-patterns-nature The numbers of nature- the Fibonacci sequence - Eniscuola. (2016, June 27). Http://Www.Eniscuola.Net/En/. Retrieved from http://www.eniscuola.net/en/2016/06/27/the-numbers-of-nature-the-fibonacci-sequence/ Times Reporter. (2014, April 22). Mathematics is for Everyone. Https://Www.Newtimes.Co.Rw/. Retrieved from https://www.newtimes.co.rw/section/read/74768 Sequence, Mathematics in the Modern World, 14th edition, Aufmann, R, N Et. Al Golden ratio, Math is Fun Retrieved from: https://www.mathsisfun.com/numbers/golden-ratio.html Golden rectangle pictures, Retrieved and adapted from: https://www.google.com/search?q=golden+rectangle&source=lnms&tbm=isch&sa=X&ved=2ahU KEwjtwuCWjJrrAhWuyosBHbQqD7cQ_AUoAXoECBMQAw&biw=1366&bih=657#imgrc=NUu89gD wOnuyNM Basic Statistical Analysis, Sprinthall, Richard C, 4 th Edition, Ally and Bacon 1992 Massachusetts Guilford, J. P. (1956). Fundamental statistics in psychology and Education (3rd. ed.). New York: McGrawHill, p.145. Yount, William R. Research Design & Statistical Analysis in Christian Ministry 4 th Edition. (USA, 2006) Mathematical Excurcions, Aufmann R, N retrieved from http://MATHEMATICAL- EXCURSIONS-Richard-N.-Aufmann-Joanne-Lockwood-Richard-D.N.pdf Mathematics in the Modern World (Calingasan, Martin & Yambao) C & E Publishing, Inc. 2018 (Quezon City) https://www.calculator.net/z-score-calculator.html Turing The chemical basis of morphogenesis Philos. Trans. R. Soc. Lond. Ser. B, 237 (1952), p. 36 39 MATHEMATICS IN THE MODERN WORLD MODULE TWO MATHEMATICAL LANGUAGE AND SYMBOLS CORE IDEA Like any language, mathematics has its own symbols, syntax and rules. Learning Outcome: 1. Discuss the language, symbols, and conventions of mathematics. 2. Explain the nature of mathematics as a language 3. Perform operations on mathematical expressions correctly. 4. Acknowledge that mathematics is a useful language. Time Allotment: Four (4) lecture hours Lesson Characteristics and Conventions in the 3.1 Mathematical Language Specific Objective At the end of this lesson, the student should be able to: 1. Understand what mathematical language is. 2. Name different characteristics of mathematics. 3. Compare and differentiate natural language into a mathematical language and expressions into sentences. 4. Familiarize and name common symbols use in mathematical expressions and sentences. 5. Translate a sentence into a mathematical symbol. Introduction: Have you read about one of the story in the bible known as “The Tower of Babel?” This story is about constructing a tower in able to reach its top to heaven; the Kingdom of God. 40 MATHEMATICS IN THE MODERN WORLD At first, the construction of a tower is smoothly being done since all of the workers have only one and only one language. But God disrupted the work of the people by making their language different from each other. There were a language barrier and the people were confused what the other people are talking about resulting the tower was never finished and the people were spread in all over and different places of the earth. Based on the story, what was the most important thing that people should have in order to accomplish a certain task? Yes, a “language”. Language is one of the most important thing among the people because it has an important role in communication. But the question is, what is language? Why is it so important? In this module, we will be discussing about mathematical relative on what you have learned in your English subject. Discussion: For sure you may be asked what the real meaning of a language is. Perhaps you could say that language is the one we use in able to communicate with each other or this is one of your lessons in English or in your Filipino subject. According to Cambridge English Dictionary, a language is a system of communication consisting of sounds, words and grammar, or the system of communication used by people in a particular country or type of work. Did you know that mathematics is a language in itself? Since it is a language also, mathematics is very essential in communicating important ideas. But most mathematical language is in a form of symbols. When we say that “Five added by three is eight”, we could translate this in symbol as “5 + 3 = 8.” Here, the first statement is in a form of group of words while the translation is in a form of symbol which has the same meaning and if your will be reading this, for sure all of you have a common understanding with this. But let us take a look at this mathematical symbols: 𝑓 (𝑥 ) = 𝐿 ∀𝜀 > 0, ∃ 𝛿 > 0 → |𝑥 − 𝑎| < 𝛿, |𝑓 (𝑥 ) − 𝐿| < 𝜀, 𝑥𝜖𝑅 Did you understand what these symbols are? This mathematical sentence is a complex idea; yet, it is contained and tamed into a concise statement. It may sound or look Greek to some because without any knowledge of the language in which the ideas are expressed, the privilege to understand and appreciate its grandeur can never be attained. Mathematics, being a language in itself, may appear complex and difficult to understand simply because it uses a different kind of alphabet and grammar structure. It uses a kind of language that has been historically proven effective in communicating and transmitting mathematical realities. The language of mathematics, like any other languages, can be learned; once learned, it allows us to see fascinating things and provides us an advantage to comprehend and exploit the beauty of beneath and beyond. Hence, in able to understand better different topics in mathematics, it is 41 MATHEMATICS IN THE MODERN WORLD very important that you must learn first on how to read and understand different symbols in mathematics which used in mathematical language. A. Characteristics of Mathematical Language The language of mathematics makes it easy to express the kinds of thoughts that mathematicians like to express. It is: 1. precise (able to make very fine distinction) 2. concise (able to say things briefly); and 3. powerful (able to express complex thoughts with relative cases). B. Vocabulary vs. Sentences Every language has its vocabulary (the words), and its rules for combining these words into complete thoughts (the sentences). Mathematics is no exception. As a first step in discussing the mathematical language, we will make a very broad classification between the ‘nouns’ of mathematics (used to name mathematical objects of interest) and the ‘sentences’ of mathematics (which state complete mathematical thoughts)’ You must study the Mathematics Vocabulary! Student must learn on how to use correctly the language of Mathematics, when and where to use and figuring out the incorrect uses. Students must show the relationship or connections the mathematics language with the natural language. Students must look backward or study the history of Mathematics in order to understand more deeply why Mathematics is important in their daily lives. Importance of Mathematical Language Major contributor to overall comprehension Vital for the development of Mathematics proficiency Enables both the teacher and the students to communicate mathematical knowledge with precision C. Comparison of Natural Language into Mathematical Language The table below is an illustration on the comparison of a natural language (expression or sentence) to a mathematical language. 42 MATHEMATICS IN THE MODERN WORLD English Mathematics Expressions Name given to an Noun such as person, 2 object of interest. place and things and pronouns 3–2 Example: 3x a) Ernesto b) Batangas City 3x + 2 c) Book d) He ax + by + c Sentence It has a complete Group of words that thought express a statement, 3+2=5 question or command. a+b=c Example: a) Ernesto is a boy. ax + by + c = 0 b) He lives in Batangas City. c) Allan loves to read book. d) Run! (x + y)2 = x2 + 2xy + y2 e) Do you love me? D. Expressions versus Sentences Ideas regarding sentences: Ideas regarding sentences are explored. Just as English sentences have verbs, so do mathematical sentences. In the mathematical sentence; 3+4=7 the verb is =. If you read the sentence as ‘three plus four is equal to seven, then it’s easy to hear the verb. Indeed, the equal sign = is one of the most popular mathematical verb. Example: 1. The capital of Philippines is Manila. 2. Rizal park is in Cebu. 3. 5+3=8 4. 5+3=9 43 MATHEMATICS IN THE MODERN WORLD Connectives A question commonly encountered, when presenting the sentence example 1 + 2 = 3 is that; If = is the verb, then what is + ? The answer is the symbol + is what we called a connective which is used to connect objects of a given type to get a ‘compound’ object of the same type. Here, the numbers 1 and 2 are connected to give the new number 1 + 2. In English, this is the connector “and”. Cat is a noun, dog is a noun, cat and dog is a ‘compound’ noun. Mathematical Sentence Mathematical sentence is the analogue of an English sentence; it is a correct arrangement of mathematical symbols that states a complete thought. It makes sense to as about the TRUTH of a sentence: Is it true? Is it false? Is it sometimes true/sometimes false? Example: 1. The capital of Philippines is Manila. 2. Rizal park is in Cebu. 3. 5+3=8 4. 5+3=9 Truth of Sentences Sentences can be true or false. The notion of “truth” (i.e., the property of being true or false) is a fundamental importance in the mathematical language; this will become apparent as you read the book. Conventions in Languages Languages have conventions. In English, for example, it is conventional to capitalize name (like Israel and Manila). This convention makes it easy for a reader to distinguish between a common noun (carol means Christmas song) and proper noun (Carol i.e. name of a person). Mathematics also has its convention, which help readers distinguish between different types of mathematical expression. Expression An expression is the mathematical analogue of an English noun; it is a correct arrangement of mathematical symbols used to represent a mathematical object of interest. 44 MATHEMATICS IN THE MODERN WORLD An expression does NOT state a complete thought; in particular, it does not make sense to ask if an expression is true or false. E. Conventions in mathematics, some commonly used symbols, its meaning and an example a) Sets and Logic SYMBOL NAME MEANING EXAMPLE Union Union of set A and set AB B Intersection Intersection of set A A B and set B Element x is an element of A xA Not an element of x is not an element of x A set A { } A set of.. A set of an element {a, b, c} Subset A is a subset of B AB Not a subset of A is not a subset of B AB … Ellipses There are still other a, b, c, … items to follow a + b + c + …. Conjunction A and B AB Disjunction A or B A B Negation Not A A → Implies (If-then statement) If A, then B A→B If and only if A if and only if B AB For all For all x x There exist There exist an x Therefore Therefore C C 45 MATHEMATICS IN THE MODERN WORLD | Such that x such that y x|y End of proof Congruence / equivalent A is equivalent to B A B a is congruent to b a b mod n modulo n a, b, c, …, z Variables *First part of English Alphabet uses as fixed variable* (lower case) *Middle part of English alphabet use as subscript and superscript (axo)p (5x2)6 variable* *Last part of an English alphabet uses as unknown variable* b) Basic Operations and Relational Symbols SYMBOL NAME MEANING EXAMPLE + Addition; Plus sign a plus b a added by b 3+2 a increased by b Subtraction; minus sign a subtracted by b a minus b - 3-2 a diminished by b Multiplication sign a multiply by b 4 3 () *we do not use x as a symbol for a times b (4)(3) multiplication in our discussion since its use as a variable* or | Division sign; divides ab 10 5 b|a 5 | 10 46 MATHEMATICS IN THE MODERN WORLD Composition of function f of g of x f g(x) = Equal sign a=a 5=5 a+b=b+a 3+2=2+3 Not equal to ab 34 Greater than ab 10 5 Less than b a 5 10 Greater than or equal to ab 10 5 Less than or equal to ba 5 10 Binary operation ab a * b = a + 17b c) Set of Numbers SYMBOL NAME MEANING EXAMPLE natural numbers / whole ℕ0 0 = {0,1,2,3,4,...} 0∈ 0 numbers set (with zero) natural numbers / whole ℕ1 numbers set (without 1 = {1,2,3,4,5,...} 6∈ 1 zero) ℤ integer numbers set = {...-3,-2,-1,0,1,2,3,...} -6 ∈ ℚ rational numbers set = {x | x=a/b, a,b∈ and b≠0} 2/6 ∈ ℝ real numbers set = {x | -∞ < x -1 and a and b < -1, the result would be Z, hence * is a binary operation. CLOSED Definition: A set is “closed” under operation if the operation assigns to every ordered pair of elements from the set an element of the set. Illustrative examples: 1) Is S = { ±1, ±3, ±5, ±7, …} is closed under usual addition? Solution: 71 MATHEMATICS IN THE MODERN WORLD By giving a counter example, S = { ±1, ±3, ±5, ±7, …} is NOT closed under usual addition. Why? Let us say we are going to a 1 and 3. The sum of 1 and 3 is 4 where 4 is not an element of S. Hence, it is not closed. 2) Let + and be usual binary operations of addition and multiplication of Z and let H = {n2 | n Z+}. Is H closed under addition? Under multiplication? Solution: a. To be able to determine if H is closed under addition, we need to have a counter-example. Let us take two elements in Z, say 1 and 4. If we are going to add this two numbers, the result would be 5 and obviously, 5 n2 or 5 is not a perfect square. Hence, H is not closed under addition. b. Let r H and s H. Using H x H (r, s) = r s. Since r H and s H, that means there must be an integers n and m Z+ such that r = n2 and s = m2. So; (r, s) = r s = n2 m2 = (nm)2 and n, m Z+. It follows that nm Z+, then (nm)2 H. Hence, H is closed under multiplication. Example. Consider the binary operation ∗ on R given by x ∗ y = x + y − 3. (x ∗ y) ∗ z = (x + y − 3) ∗ z = (x + y − 3) + z − 3 = x + y + z − 6, x ∗ (y ∗ z) = x ∗ (y + z − 3) = x + (y + z − 3) − 3 = x + y + z − 6. Therefore, ∗ is associative. Since x ∗ y = x + y − 3 = y + x − 3 = y ∗ x, ∗ is commutative. Example. Consider the binary operation * on R given by a*b = ab/2. Show that a*b = b*c. Solution: Let a*b = ab/2. We need to show that a*b = b*a. In b*a = ba/2. But by commutative properties under multiplication, that is ab = ba, then it follows that b*a = ab/2. Hence a*b = b*a 72 MATHEMATICS IN THE MODERN WORLD Definition: Let be a binary operation of a set S. Then; (a) is associative if for all a, b, c S, (ab)c = a (bc) (b) is commutative if for all a, b S, ab = b a (c) An element e S is called a left identity element if for all a S, we have e a=a (d) An element e S is called a right identity element if for all a S, we have ae=a (e) An element e S is called an identity element if for all a S, we have a e = a and e a = a. (f) Let e be an identity element is S and a S, then b is called an inverse of the element “a” if a b = e and b a = e. Note that a b = b a = e or a * a-1 = a-1 * a = e If a S, then the inverse of “a” is denoted by a-1. Here -1 is not an exponent of a. Example: Let S = Z+ as define on S by a b = a + b – ab. Show the associativity and the commutativity of S in a binary operation. Find also its identity and inverse if any. (a) Associativity Let a, b, c Z+. Then; (a b) c = a (b c) For (a b) c (a b) c = (a + b – ab) c = (a + b – ab) + c - (a + b – ab)c 73 MATHEMATICS IN THE MODERN WORLD = a + b + c – ab – ac – bc + abc For a (b c) a (b c) = a (b + c – bc) = a + (b + c – bc) – (a)(b + c – bc) = a + b + c – bc – ab – ac + abc Hence is associative on S Z+. (b) Commutative ab=ba a + b – ab = b + a – ba a + b – ab = a + b – ab Hence is commutative on S Z+. (c) Identity a*e=a e*a=a a + e – ae = a e + a – ea = a e – ae = a – a e – ea = a – a e(1 – a) = 0 e(1 – a) = 0 e=0 e=0 hence, the identity exist except when a = 1. (d) Inverse a * a-1 = e a-1 * a = e Example: Let S = Z+ as define on S by a b = a2 + ab + b2. Is the operation associative? Commutative? What is its identity? What is its inverse? 74 MATHEMATICS IN THE MODERN WORLD (a) Commutative ab=ba a2 + ab + b2 = b2 + ba + a2 a2 + ab + b2 = a2 + ab + b2 Hence, the operation is commutative. (b) Associative (a b) c = a (b c) (a2 + ab + b2 ) c = a (b2 + bc + c2 ) (a2 + ab + b2 )2 + (a2 + ab + b2 )(c) + c2 a2 + (a)( b2 + bc + c2 ) + (b2 + bc + c2 )2 Hence, the operation is not associative. Cayley Tables A (binary) operation on a finite set can be represented by a table. This is a square grid with one row and one column for each element in the set. The grid is filled in so that the element in the row belonging to x and the column belonging to y is x*y. A binary operation on a finite set (a set with a limited number of elements) is often displayed in a table that demonstrates how the operation is performed. Example: The table below is a table for a binary operation on the set {a, b, c, d} a b C d a a b C d b b c D a c c d A b d d a B c a. Is the commutative? b. Is the associative? c. What is its identity? 75 MATHEMATICS IN THE MODERN WORLD Self - Learning Activity Directions. Do as indicated. A. Define a relation C from R to R as follows: For any (x,y) R x R, (x,y) C meaning that x2 + y2 = 1. a. Is (1,0) C? Is (0,0) C? Is -2 C 0? Is 0 C (-1)? b. What are the domain and the co-domain of C? B. If f(x) = 2x2 and g(x) = 3x + 1, evaluate the following: 𝑓 a. (f + g)(x) b) (f g)(x) c) ( ) (𝑥 ) d) (g f)(x) 𝑔 C. Tell whether the following is a binary operation or not. a) G Z, defined * by a * b = a2 – b2 for all set a, b Z. Explanation: b) G N, defined * by a * b = 2 + 3ab for all set a, b N. Explanation: 76 MATHEMATICS IN THE MODERN WORLD c) G Z-, defined * by a * b = a + ab - b for all set a, b Z-. Explanation: d) G R, defined * by a * b = √𝐚 − 𝐛 for all set a, b R. Explanation: D. Let S = Z+ as define on S by a b = a + b + 1. Show the associativity and the commutativity of S in a binary operation. Determine also the identity if there is. E. Let A = Z – {0} and let S = {f1, f2, f3, f4, f5, f6} be the set of functions as A defined as follows: 1 𝑓1(𝑥) = 𝑥 𝑓4 (𝑥) = 𝑥 1 𝑥 𝑓2 (𝑥) = 𝑓5 (𝑥) = 1−𝑥 𝑥−1 𝑥−1 𝑓3 (𝑥) = 𝑓6 = 1 − 𝑥 𝑥 Show that the composition of mappings is a binary operation by completing the multiplication table for * 77 MATHEMATICS IN THE MODERN WORLD * f1 f2 f3 f4 f5 f6 f1 f2 f3 f4 f5 f6 Present your solution below. Lesson 3.3 Logic and Formality Specific Objective At the end of this lesson, the student should be able to: 1. Define what logic is. 2. Tell whether the statement is formal or non-formal. 3. Show the relationship between grammar in English and logic in Mathematics. Introduction What comes first in your mind when we speak about logic? Do you have any idea what logic is all about? Could we say that if a person thinks correctly, then he has logic? Perhaps until now, there are some people arguing whether a logic is an art or it is a science. Now, whether it is an art or a science, studying logic could be very important not only in the field of mathematics but in other sciences such as natural science and social science. On this module, we will studying the fundamental concept of logic but basically logic as mathematical language. Discussion I. What is logic? 78 MATHEMATICS IN THE MODERN WORLD In this particular module, we are going to talk about logic as a mathematical language but a deeper discussion logic as a science as well as its application will be tackled in module 6. It is very essential to understand better what logic is as a language. But first, let us have a definition in logic. In your social science courses, logic could define as the study of the principles of correct reasoning and it is not a psychology of reasoning. Based on the definition which is logic is the study of the principle of correct reasoning, one of the principles in logic that is very much important to study is on how to determine the validity of ones argument. Studying mathematics is also studying theorems. The proof of the theorem uses the principle of arguments in logic. So, in this case, we could say actually that the language of mathematics is logic. In short, mathematical statement is also a grammar. In English, when we construct a sentence or sentences, we always check if it is grammatically correct but in Mathematics, we check mathematical statement or sentence in a logical structure. Wherever you go, we have a common language in mathematics. In order not to conflict with in an English word, we use appropriate symbols in mathematics so that there will no ambiguity on how to communicate as to the meaning of a mathematical expression or even in mathematical sentences II. Formality As stated by Heylighen F. and Dewaele J-M in the “Formality of Language: Definition and Measurement”, an expression is completely formal when it is context- independent and precise (i.e. non-fuzzy), that is, it represents a clear distinction which is invariant under changes of context. In mathematics, we are always dealing in a formal way. Suppose that somebody asked you that the result of adding 5 to 3 is 8 or let us say that if a variable x is an even number then the square of this variable x would be also an even number, you would agree that both mathematical sentences or statements are true and there is no reason for you to doubt. Those two examples statements are precise and it is also an independent. These are the two characteristics in mathematics that the statement must have to say the mathematical sentence is in a formal manner. Speaking of statement, statement is the main component of logic in mathematics. When we say mathematical logic, it is a statements about mathematical objects that are taken seriously as mathematical objects in their own right. More generally, in mathematical logic we formalize, that is, we formulate in a precise mathematical way its definition, theorem, lemma, conjecture, corollary, propositions and the methods of proof which will be discussed in our next lesson. These are the major part of formality in mathematics. a) Definition 79 MATHEMATICS IN THE MODERN WORLD One of the major parts of formality in mathematics is the definition itself. When we say definition, it is a formal statement of the meaning of a word or group of words and it could stand alone. Example of this is a definition of a right triangle. What is the exact or formal definition of this? A right triangle consists of two legs and a hypotenuse. The two legs meet at a 90° angle and the hypotenuse is the longest side of the right triangle and is the side opposite the right angle. Here, you will see the exactness and the precision of the definition of a right triangle. Now suppose we are going to define “carabao”. Can you give a definition for this? Maybe, some of you will define a carabao is a black and strong animal helps the farmer in plowing the rice field. But, have you noticed that this is not a formal definition? How about the cow and the horse? These are also an animal that could also help the farmers in plowing the field. How about the machine tractor? Are we not consider this machine that could possibly help our farmers in plowing the rice field? So, we cannot say that is a formal definition since it cannot stand alone. b) Theorem Another statement that could we consider as a formal statement is the theorem. You will encounter this word in all books of mathematics especially if it is pure mathematics. In your algebra subject during your high school days, have you studied different laws and principles in mathematics? These are just really theorems that proven true and justified using the concept of mathematical logic and all you need to do is to apply those laws and principles, isn’t it? But what does theorem means? A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments. In general, a theorem is an embodiment of some general principle that makes it part of a larger theory. The process of showing a theorem to be correct is called a proof. An example of a theorem that we all know is the Pythagorean Theorem. This is a very well-known theorem in mathematics. The theorem stated that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. If the hypotenuse (the side opposite the right angle, or the long side) is called c and the other two sides are a and b, then this theorem with the formula a2 + b2 = c2. You will notice that the theorem is precise in a form of if-then statement. The if-then statement is one of the statements in logic. So, a statement could not be considered theorem unless it was proven true using mathematical logic. c) Proof To be able to say that a theorem is true, it should be undergo on the process of proving. But what do we mean by proof or a mathematical proof. Proof is a rigorous 80 MATHEMATICS IN THE MODERN WORLD mathematical argument which unequivocally demonstrates the truth of a given proposition. The different methods on proof are as follows: 1. Deductive 2. Inductive 3. Direct Proof 4. Indirect Proof 5. Proof by Counterexample 6. Proof by Contradiction All of these methods of proof are written together with the correct mathematical logic and precise. Discussion and illustrative examples on these different methods of proof will be tackled in module 3. d) Proposition When we say proposition, it is a declarative statement that is true or false but not both. This statement is another major part of formality since all types of proposition are precise and concise. Different propositions that can be also said as logical connectives are as follows: 1. Negation How does the statement translate into its negation. Say, given any statement P, another statement called the negation of P can be formed by writing “It is false that …” before P, or if possible, by inserting in P the word “not”. For example, the given statement is “Roderick attends Mathematics class”. Translating this into its negation, the new statement would be “Roderick will not attend Mathematics class” or it could be “It is false that Roderick attends Mathematics class”. 2. Conjunction Another logical connective is what we called conjunction. If two statements are combined by the word “and”, then the proposition is called conjunction. In other words, any two statements can be combined by the word “and” to form a composite statement which is called the conjunction of the original statements. An example for this is, let us say the first statement is “Ernesto is handsome” and the second statement is “Ernesto is rich”. The new statement after connecting this two statements by the word “and”, this would become “Ernesto is handsome and Ernesto is rich market”. 3. Disjunction 81 MATHEMATICS IN THE MODERN WORLD Disjunction is another form of proposition. Any two statements can be combined by the word “or” to form a new statement which is called the disjunction of the original two statements. Let us have an example for this kind of proposition. Say, the first statement is “Life is beautiful” while the second statement is “Life is challenging. Now, combining these two statements by the word “or” the new combined statement is “Life is beautiful or life is challenging.” 4. Conditional The fourth type of proposition is that what we called conditional. To be able to easily identify that the proposition is in a form of conditional statement, you will notice of the word “If-then”. Most of mathematical definition is in a form of this statement. So, in other words, it is state that a true statement cannot imply a false statement. In this proposition, the first statement would be a premise and the second statement is the conclusion. Let us have an example for this. Say the premise is “If x is positive, then its square is also positive.” We can show the proposition is true with the use of one of the methods of proving. 5. Biconditional The last type of proposition is the biconditional. Its uses a connector for two statements “if an only if”. If your statement is in this form, then your statement is called biconditional. Here is one of the examples of a biconditional statement. Let us say our first statement is “I will attend mass.” The second statement is “Tomorrow is Sunday.” So, the new statement using biconditional statement would be “I will attend mass if and only if tomorrow is Sunday.” Now, supposedly our statement goes like this. “Let’s go!” Can we considered this as a precise formal statement? Perhaps you will be saying no since you may be asking; Who will be my companion?; What time are we going to go?; Where will we go?. This statement is not precise hence it is not formal. All of these statements can be transformed into symbols. More details and specific lesson about this will be tackled in module 6. e) Corollary What is corollary? When we say corollary in mathematics, it is also a proposition that follows with little or no proof required from one already proven. An example of this is it is a theorem in geometry that the angles opposite two congruent sides of a triangle are also congruent. A corollary to that statement is that an equilateral triangle is also equiangular. 82
