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MATHEMATICS IN THE MODERN WORLD MATHEMATICS IN THE MODERN WORLD MODULE ONE...

MATHEMATICS IN THE MODERN WORLD MATHEMATICS IN THE MODERN WORLD MODULE ONE THE NATURE OF MATHEMATICS BATANGAS STATE UNIVERSITY GENERAL EDUCATION COURSE 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. MATHEMATICS IN THE Learning Outcome: MODERN WORLD 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: Learning Modules Lesson 1.1 Mathematics of Our World by Lesson 1.2 Mathematics in Our World Lesson 1.3 Mathematics of Sequence Jose Alejandro R. Belen Neil M. Mame  Israel P. Piñero Time Allotment: Four lecture hours 2020 1 7 MATHEMATICS IN THE MODERN WORLD 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 Lesson surroundings. From their realization, a system of thought further advanced their 1.1 The Mathematics of Our World 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. Specific Objectives 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 1. To understand the mathematics of the modern world. to build human culture: an important ingredient for a civilization to flourish. From 2. To revisit and appreciate the mathematical landscape. then, man realized that the natural world is embedded in a magnanimously 3. To realize the importance of mathematics as a utility. mathematical realm of patterns----and that natural order efficiently utilizes all 4. To gain awareness of the role of mathematics as well as our role mathematical patterns to its advantage. As a result, we made use of mathematics in mathematics. as a brilliant way to understand the nature by comprehending the structure of its underlying patterns and regularities. Lesson 1.1 does not only attempt to explain the essence of mathematics, it serves Mathematics is present in everything we do; it is all around us and it is the building also as a hindsight of the entire course. The backbone of this lesson draws from the block of our daily activities. It has been at the forefront of each and every period of Stewart’s ideas embodied in his book entitled Nature’s Numbers. The lesson our development, and as our civilized societies advanced, our needs of provides new perspective to understand the irregularity and chaos of our world as mathematics pioneering arose on the frontier of our course as we prepare our we move through the landscape of regularity and order. It poses some thought- human species to traverse the cosmic shore. 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. Mathematics is a Tool Mathematics, as a tool, is immensely useful, practical, and powerful. It is not about Discussions crunching numbers, formulas, and symbols but rather, it is all about forming new The Nature of Mathematics 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 In the book of Stewart, Nature’s Number, he that mathematics is a formal system otherwise be intangible to us. It can be likened to our sense of sight and touch. of thought that was gradually developed in the human mind and evolved in the Mathematics is our sense to decipher patterns, relationships, and logical connections. It is our whole new way to see and understand the modern world. 8 9  MATHEMATICS IN THE MODERN WORLD MATHEMATICS IN THE MODERN WORLD observe, hypothesize, experiment, discover, and recreate. On the other hand, Mathematics, being a broad and deep discipline, deals with the logic of shape, mathematics is an art and a process of thinking. For it involves reasoning, which quantity, and arrangement. Once, it was perceived merely a collective thoughts can be inductive or deductive, and it applies methods of proof both in fashion that dealing with counting numbers, but it is now being understood as a universal is conventional and unventional. 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 Mathematics is Everywhere of life. 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. 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 Source: Space Telescope Science/NASA tools. It provides answers to existing questions and presents solutions to occurring The Universe problems. It has the power to unveil the reasons behind occurrences and it offers Figure 1.2 explanations. Moreover, mathematics, as a study of patterns, allows people to 10 11 MATHEMATICS IN THE MODERN WORLD MATHEMATICS IN THE MODERN WORLD Even the cosmic perspective, the patterns in the firmament are always presented diseases can now be predicted and controlled. Scientists and researchers use as a mystery waiting to be uncovered by us-the sentient being. In order to applied mathematics in doing or performing researches to solve social, scientific, unearthed this mystery, we are challenged to investigate and deeply examine its medical, or even political crises. structure and rules to the infinitesimal level. The intertwined governing powers of cosmic mystery can only be decoded by seriously observing and studying their It is a common fact that mathematics plays an important role in many sciences. It regularities, and patiently waiting for the signature of some kind interference. It is is and it provides tools for calculations. We use of calculations in other disciplines only by observing the abundance of patterns scattered everywhere that this whenever we are underrating some kind of research or experiment. The use of irregularities will beg to be noticed. Some of them are boldly exposed in a simple mathematical calculations is indispensable method in scientifically approaching and obvious manner while others are hidden in ways that is impossible to perceive most of the problems. In a similar way, mathematics, provides new questions to by easy to discern. While our ancestors were able to discover the presence of think about. Indeed, in learning and doing mathematics, there will always be new mathematics in everything, it took the descendants, us, a long time to gradually questions to answer, new problems to solve, and new things to think about (Vistru- notice the impact of these patterns in the persistence of our species to rightfully Yu PPT presentation). exist. The Mathematical Landscape The Essential Roles of Mathematics Mathematics has countless hidden uses and applications. It is not only something The human mind and culture developed a conceptual landscape for mathematical that delights our mind but it also allows us to learn and understand the natural thoughts and ideas to flourish and propagate. There is a region in the human mind order of the world. This discipline was and is often studied as a pure science but it that is capable of constructing and discerning the deepest insights being perceived also finds its place in other areas of perpetuating knowledge. Perhaps, science from the natural world. In this region, the mathematical landscape exists- wherein would definitely agree that, when it comes to discovering and unveiling the truth concepts of numbers, symbols, equations, operations calculations, abstractions, behind the inherent secrets and occurrences of the universe, nothing visual, verbal, and proofs are the inhabitants as well as the constructs of the impenetrable or aural come close to matching the accuracy, economy, power and elegance of vastness of its unchartered territories. In this landscape, a number is not simply a mathematics. Mathematics helps us to take the complex processes that is naturally mathematical tree of counting. Also, infinite variables can be encapsulate to finite. occuring in the world around us and it represents them by utilizing logic to make Even those something that is hard to express in decimal form can be expressed in things more organized and more efficient. terms of fractions. Those things that seemed eternal ℤ can further be exploited using mathematical operations. This landscape claimed complex numbers as the Further, mathematics also facilitate not only to weather, but also to control the firmament and even asserted that imaginary numbers also exist. To the low state weather ---- be it social, natural, statistical, political, or medical. Applied negative numbers relentlessly enjoying recognition as existent beings. The wind in mathematics, which once only used for solving problems in physics, and it is also this landscape is unpredictable that the rate of change of the rate of change of becoming a useful tool in biological sciences: for instance, the spread of various weather is known as calculus. And beneath the surface of this mathematical landscape are firmly-woven proofs, theorems, definitions, and axioms which are 12 13  MATHEMATICS IN THE MODERN WORLD MATHEMATICS IN THE MODERN WORLD intricately “fertilized” by reasoning, analytical, critical thinking and germicide by components or the operations that it wants you to carry out, and everything mathematical logic that made them precise, exact and powerful. 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 With this landscape, the mathematician's instinct and curiosity entice to explore know the truth, with trial and error, and without fear of facing more questions and further the vast tranquil lakes of functions and impassable crevasse of the problems to solve. (Vistru-Yu) 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 Mathematics is for Everyone 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" The relationship of the mathematical landscape in the human mind with the natural but a process. Conventionally, we are just simply made ourselves comfortable on world is so strange that in the long run, the good math provides utilization and the “thingification” of those processes and we forgot that 1+1 is not a noun but a usefulness in the order of things. Perhaps, for most people, they simply need to verb. 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, How Mathematics is Done humanity needs, beyond rudiment of mathematics. To safeguard our existence, we already have delegated the functions of mathematics across all disciplines. There is Math is a way of thinking, and it is undeniably important to see how that thinking mathematics we call pure and applied, as there are scientists we call social and is going to be developed rather than just merely see face value of the results. For natural. There is mathematics for engineers to build, mathematics for commerce some people, few math theorems can bring up as much remembered pain and and finance, mathematics for weather forecasting, mathematics that is related to anxiety. For others, this discipline is so complex and they have to understand the health, and mathematics to harness energy for utilization. To simply put it, confusing symbols, the difficult procedures, and the dreaded graphs and charts. For everyone uses mathematics in different degrees and levels. Everyone uses most, mathematics is just nothing but something to survive, rather than to learn. mathematics, whoever they are, wherever they are, and whenever they need to. From mathematicians to scientists, from professionals to ordinary people, they all To the untrained eye, doing mathematics is quite difficult and challenging. It is use mathematics. For mathematics puts order amidst disorder. It helps us become ambiguous, for it follows a set of patterns, formulas, and sequences that make it better persons and helps make the world a better place to live in. (Vistru-Yu). 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. The Importance of Knowing and Learning Mathematics Mathematics builds upon itself. More complex concepts are built upon simpler Why do we want to observe and describe patterns and regularities? Why do we concepts, and if you do not have a strong grasp of the fundamental principles, then want to understand the physical phenomena governing our world? Why do we a more complex problem is more likely going to stump you. If you come across a want to dig out rules and structures that lie behind patterns of the natural order? mathematical problem that you cannot solve, the first thing to do is to identify the It is because those rules and structures explain what is going on. It is because they 14 15 MATHEMATICS IN THE MODERN WORLD 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 Learning Activity 1.1 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 Answer one of the following questions (15-20 lines) and submit your answer to your course facilitator. activities need clues for the human culture to meaningfully work. 1. What are the new things that you learned about the nature of mathematics? Mathematical training is vital to decipher the clues provided by nature. But the role 2. What aspect of the lesson significant changed your view about mathematics? of mathematics goes clues and it goes beyond prediction. Once we understand how 3. What is the most important contribution of mathematics in humankind? 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 Observe the following this format: control its spread. We want to understand the unpredictability of cancer cells to Paper Font Font All Line Page Substance Margin Orientation Paper Size Number combat it before it even exists. Finally, we want to understand the butterfly effect (if printed) Type Size Spacing 20 Normal Portrait 8.5 x 13 Arial 12 1.5 Page x of x as much as we are so curious to know why the “die” of the physical world play god. Justified “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 17  MATHEMATICS IN THE MODERN WORLD MATHEMATICS IN THE MODERN WORLD Discussions Lesson 1.2 The Mathematics in Our World 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 Specific Objective intriguing. Paradoxically, it seemed that everything in the world follows a pattern : 1. To develop one’s understanding about patterns; of their own and tamed by the same time pattern of their own. 2. To identify different patterns in nature; Patterns of Visuals. Visual patterns are often unpredictable, never quite 3. To recognize different symmetries in nature; and repeatable, and often contain fractals. These patterns are can be seen from the 4. To explain the presence of Fibonacci numbers in nature seeds and pinecones to the branches and leaves. They are also visible in self-similar replication of trees, ferns, and plants throughout nature. The mathematics in our world is rooted in patterns. Patterns are all around us. Patterns of Flow. The flow of liquids provides an inexhaustible supply of nature’s Finding and understanding patterns give us great power to play like god. With patterns. Patterns of flow are usually found in the water, stone, and even in the patterns, we can discover and understand new things; we learn to predict and growth of trees. There is also a flow pattern present in meandering rivers with the ultimately control the future for our own advantage. repetition of undulating lines. A pattern is a structure, form, or design that is regular, consistent, or recurring. Patterns of Movement. In the human walk, the feet strike the ground in a regular Patterns can be found in nature, in human-made designs, or in abstract ideas. They rhythm: the left-right-left-right-left rhythm. When a horse, a four-legged creature occur in different contexts and various forms. Because patterns are repetitive and walks, there is more of a complex but equally rhythmic pattern. This prevalence of duplicative, their underlying structure regularities can be modelled pattern in locomotion extends to the scuttling of insects, the flights of birds, the mathematically. In general sense, any regularity that can be explained pulsations of jellyfish, and also the wave-like movements of fish, worms, and mathematically is a pattern. Thus, an investigation of nature’s patterns is an snakes. investigation of nature’s numbers. This means that the relationships can be observed, that logical connections can be established, that generalizations can be Patterns of Rhythm. Rhythm is conceivably the most basic pattern in nature. Our inferred, that future events can be predicted, and that control can possibly be hearts and lungs follow a regular repeated pattern of sounds or movement whose possible. timing is adapted to our body’s needs. Many of nature’s rhythms are most likely 18 19 MATHEMATICS IN THE MODERN WORLD MATHEMATICS IN THE MODERN WORLD similar to a heartbeat, while others are like breathing. The beating of the heart, as surface waves that create the chaotic patterns of the sea. Similarly, water waves well as breathing, have a default pattern. 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 Patterns of Texture. A texture is a quality of a certain object that we sense through sand. 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 Spots and Stripes 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 We can see patterns like spots on the skin of a giraffe. On the other hand, stripes plants, flowers, animals, and even at our bodies. These common patterns are all are visible on the skin of a zebra. Patterns like spots and stripes that are commonly incorporated in many natural things. 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 Waves and Dunes diffuse and how strongly they interact. Spirals 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 Jean Beaufort has released this “Spiral water, making it oscillate as waves pass by. Wind waves, on the other hand, are Galaxy” image under Public Domain license 20 21  MATHEMATICS IN THE MODERN WORLD 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 Rotations, also known as rotational symmetry, captures symmetries when it still pattern formation. It is used to classify and organize information about patterns by looks the same after some rotation (of less than one full turn). The degree of classifying the motion or deformation of both pattern structures and processes. rotational symmetry of an object is recognized by the number of distinct There are many kinds of symmetry, and the most important ones are reflections, orientations in which it looks the same for each rotation. 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 23 MATHEMATICS IN THE MODERN WORLD 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. Translations. This is another type of symmetry. Translational symmetry exists in identical halves. 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 25  MATHEMATICS IN THE MODERN WORLD MATHEMATICS IN THE MODERN WORLD Sunflower Snowflakes Starfish One of the most interesting things about Snowflakes have six-fold radial Starfish have a radial fivefold symmetry. Each arm portion of the starfish is a sunflower is that it contains both radial symmetry. The ice crystals that identical to each of the other regions. 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. 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. Honeycombs/Beehive In describing the amazing variety of phenomena in nature we stumble to discover Honeycombs or beehives are examples the existence of Fibonacci numbers. It turns out that the Fibonacci numbers appear of wallpaper symmetry. This kind of from the smallest up to the biggest objects in the natural world. This presence of symmetry is created when a pattern is Fibonacci numbers in nature, which was once existed realm mathematician’s repeated until it covers a plane. Beehives curiously, is considered as one of the biggest mysteries why the some patterns in are made of walls with each side having nature is Fibonacci. But one thing is definitely made certain, and that what seemed the same size enclosed with small solely mathematical is also natural. hexagonal cells. Inside these cells, honey and pollen are stored and bees are For instance, many flowers display figures adorned with numbers of petals that are raised. in the Fibonacci sequence. The classic five-petal flowers are said to be the most 26 27 MATHEMATICS IN THE MODERN WORLD MATHEMATICS IN THE MODERN WORLD Paper Page common among them. These include the buttercup, columbine, and hibiscus. Aside Substance Margin Orientation Paper Size Font Font All Line Number Type Size Spacing (if printed) from those flowers with five petals, eight-petal flowers like clematis and 20 Normal Portrait 8.5 x 13 Arial 12 1.5 Page x of x Justified delphinium also have the Fibonacci numbers, while ragwort and marigold have thirteen. These numbers are all Fibonacci numbers. Please bear in mind the following criteria for grading your work. Apart from the counts of flower petals, the Fibonacci also occurs in nautilus shells 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 with a logarithmic spiral growth. Multiple Fibonacci spirals are also present in understanding of these. pineapples and red cabbages. The patterns are all consistent and natural. 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. Learning Activity 1.2 Lesson Synthesis 1.3 The Fibonacci Sequence 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. Specific Objectives III. It is highly recommended that at the outset, an outline must be made. 1. To define sequence and its types 2. To differentiate Fibonacci sequence from other types of sequence IV. Please ensure that topic sentence can be clearly understood. 3. To discover golden ratio and golden rectangle; and -Your topic sentence must be supported by at least three arguments. 4. To learn how to compute for the nth term in the Fibonacci Sequence V. Your synthesis must be around 1400-1500 words. VI. Rules on referencing and citation must be strictly observed. As we have discussed in the preceding lesson, human mind is capable of VII. You may use either MLA or APA system. identifying and organizing patterns. We were also to realized that there are VIII. The last page must contain the references or bibliography. structures and patterns in nature that we don’t usually draw attention to. IX. Write your name, student number, email address at the last page Likewise, we arrived at a position that in nature, some things follow X. Please observe the following format: 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 28 29  MATHEMATICS IN THE MODERN WORLD 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. Notice in the given example above, the common difference between two Discussion 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 sequence. Sequence refers to an ordered list of numbers called terms, that may have Geometric sequence. If in the arithmetic sequence we need to check for repeated values. The arrangement of these terms is set by a definite rule. the common difference, in geometric sequence we need to look for the (Mathematics in the Modern World, 14th Edition, Aufmann, RN. et al.). common ratio. The illustrated in the example below, geometric sequence is Cosider the given below example: 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 1, 3, 5, 7, … (1stterm) (2nd term) (3rd term) (4th term) 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,… 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 2, 8, 32, 128, … end of the visible patterns means that the sequence is infinite. 4 4 4 There are different types of sequence and the most common are the arithmetic sequence, geometric sequence, harmonic sequence, and Harmonic Sequence. In the sequence, the reciprocal of the terms behaved Fibonacci sequence. in a manner like arithmetic sequence. Consider the example below and notice an interesting pattern in the series. With this pattern, the reciprocal Arithmetic sequence. It is a sequence of numbers that follows a definite appears like arithmetic sequence. Only in recognizing the appearance that pattern. To determine if the series of numbers follow an arithmetic sequence, we can finally decode the sequencing the govern the series. check the difference between two consecutive terms. If common difference is observed, then definitely arithmetic sequence governed the pattern. To 1 1 1 1 1 , , , , , … clearly illustrate the arrangement, consider the example below: 2 4 6 8 10 Fibonacci Sequence. This specific sequence was named after an Italian 2, 4, 6, 8, 10, 12 … mathematician Leonardo Pisano Bigollo (1170 - 1250). He discovered the sequence while he was studying rabbits. The Fibonacci sequence is a series 2 2 2 2 2 30 31 MATHEMATICS IN THE MODERN WORLD 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 Where: numbers. Xn stands for the Fibonacci number we’re looking for N stands for the position of the number in the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, … Φ stands for the value of the golden ratio Let us try for example: What is the 5th Fibonacci number? By using the formula 0+1=1 0, 1, 1 we’ll get: 1+1=2 0, 1, 1, 2 X5 = (1.618)5 − (1−1.618)5 1+2=3 0, 1, 1, 2, 3 √5 2+3=5 0, 1, 1, 2, 3, 5 X5= 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 The amazing grandeur of Fibonacci sequence was also discovered in the mathematical and mental exercise but later on the it was observed that the structure of Golden rectangle. The golden rectangle is made up of squares ownership of this pattern was also being claimed by some species of flowers, whose sizes, surprisingly is also behaving similar to the Fibonacci sequence. petals, pineapple, pine cone, cabbages and some shells. Take a serious look at the figure: 1, 1, 2, 3, 5, 8, 13, 21, … The Golden Ratio 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 2 term in the sequence which is “1”. And Fib(6) refers nd 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 33  MATHEMATICS IN THE MODERN WORLD 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 Learning Activity 1.3 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 I. Identify what type of sequence is the one below and supply the sequence with the next around from infinite to infinitesimal. 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) ? 34 35 MATHEMATICS IN THE MODERN WORLD MATHEMATICS IN THE MODERN WORLD A. Arithmetic Sequence C. Geometric Sequence B. Fibonacci Sequence D. Harmonic Sequence 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. ____________7. What is the sum of Fib (10) + Fib(5) ? A. 58 C. 60 ____________ 1. What is said to be the most basic pattern in nature? B. 59 D. 61 A. Pattern of Flow C. Pattern of Rhythm B. Pattern of Movement D. Pattern of Visuals ____________8. What is Fib (12) ? A. 144 C. 377 ____________2. This kind of pattern is unpredictable and it often contains fractals. B. 233 D. 89 A. Geometric Patterns C. Pattern of Movement B. Pattern of Forms D. Pattern of Visuals ____________9. What are the next two terms of the sequence, 8, 17, 26, 35? A. 49, 58 C. 44, 53 ____________3. What kind of pattern is a series of shapes that are repeating? B. 39, 48 D. 54, 63 A. Geometric Pattern C. Pattern of Texture B. Pattern of Flows D. Pattern of Visuals ____________10. What type of sequence is 5, 8, 13, 21, 34, 55, … ? A. Fibbonacci Sequence C. Fibonacci Sequence ____________4. Among the following, what is not a type of symmetry? B. Fibonaacii Sequence D. Fibonacii Sequence 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? ANSWER KEY 36 37  MATHEMATICS IN THE MODERN WORLD MATHEMATICS IN THE MODERN WORLD 1. C 6. C MODULE TWO 2. D 7. C 3. A 8. B MATHEMATICAL LANGUAGE AND SYMBOLS 4. C 9. C 5. B 10. C CORE IDEA Like any language, mathematics has its own symbols, syntax and rules. Learning Outcome: References 1. Discuss the language, symbols, and conventions of mathematics. Akshay, A. (n.d.). 13 Reasons Why Math is Important. Https://Lifehacks.Io/. Retrieved from 2. Explain the nature of mathematics as a language https://lifehacks.io/reasons-why-math-is-important/ 3. Perform operations on mathematical expressions correctly. 4. Acknowledge that mathematics is a useful language. A. (2019, September 12). An Ode to Math, Mathematics in Nature. Https://Www.Minuteschool.Com/. Retrieved from  Time Allotment: Four (4) lecture hours 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- Lesson Characteristics and Conventions in the world-through-math 3.1 Mathematical Language 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. Specific Objective Discoverycube.Org/Blog. Retrieved from https://www.discoverycube.org/blog/moment- science-patterns-in-nature/ At the end of this lesson, the student should be able to: Fibonacci Number Patterns. (n.d.). Http://Gofiguremath.Org. Retrieved from 1. Understand what mathematical language is. http://gofiguremath.org/natures-favorite-math/fibonacci-numbers/fibonacci-number- 2. Name different characteristics of mathematics. patterns/ 3. Compare and differentiate natural language into a Grant, S. (2013, April 21). 10 Beautiful Examples of Symmetry In Nature. Https://Listverse.Com/. mathematical language and expressions into sentences. Retrieved from https://listverse.com/2013/04/21/10-beautiful-examples-of-symmetry-in- 4. Familiarize and name common symbols use in mathematical nature/ expressions and sentences. 5. Translate a sentence into a mathematical symbol. H., E. J. (2013, August 16). What is Mathematics? Https://Www.Livescience.Com/. Retrieved from https://www.livescience.com/38936-mathematics.html Introduction: Irish Times. (2018, October 18). Who Uses Maths? Almost Everyone! Https://Thatsmaths.Com/. Have you read about one of the story in the bible known as “The Tower of Babel?” Retrieved from https://thatsmaths.com/2018/10/18/who-uses-maths-almost-everyone/ This story is about constructing a tower in able to reach its top to heaven; the Kingdom of God. 38 40 MATHEMATICS IN THE MODERN WORLD MATHEMATICS IN THE MODERN WORLD At first, the construction of a tower is smoothly being done since all of the workers have only one very important that you must learn first on how to read and understand different symbols in and only one language. But God disrupted the work of the people by making their language mathematics which used in mathematical 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. A. Characteristics of Mathematical Language Based on the story, what was the most important thing that people should have in The language of mathematics makes it easy to express the kinds of thoughts that order to accomplish a certain task? Yes, a “language”. Language is one of the most important mathematicians like to express. 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 It is: mathematical relative on what you have learned in your English subject. 1. precise (able to make very fine distinction) Discussion: 2. concise (able to say things briefly); and 3. powerful (able to express complex thoughts with relative cases). 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 B. Vocabulary vs. Sentences 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 Every language has its vocabulary (the words), and its rules for combining of communication used by people in a particular country or type of work. 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 Did you know that mathematics is a language in itself? Since it is a language also, ‘nouns’ of mathematics (used to name mathematical objects of interest) and the ‘sentences’ of mathematics is very essential in communicating important ideas. But most mathematical language mathematics (which state complete mathematical thoughts)’ 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 You must study the Mathematics Vocabulary! 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 Student must learn on how to use correctly the language of Mathematics, when and where symbols: to use and figuring out the incorrect uses. Students must show the relationship or connections the mathematics language with the 𝑓(𝑥) = 𝐿 natural language. Students must look backward or study the history of Mathematics in order to understand ∀𝜀 > 0, ∃ 𝛿 > 0 → |𝑥 − 𝑎| < 𝛿, |𝑓(𝑥) − 𝐿| < 𝜀, 𝑥𝜖𝑅 more deeply why Mathematics is important in their daily lives. Did you understand what these symbols are? This mathematical sentence is a complex idea; Importance of Mathematical Language 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 Major contributor to overall comprehension Vital for the development of Mathematics proficiency to understand and appreciate its grandeur can never be attained. Mathematics, being a language Enables both the teacher and the students to communicate mathematical knowledge with in itself, may appear complex and difficult to understand simply because it uses a different precision kind of alphabet and grammar structure. It uses a kind of language that has been historically C. Comparison of Natural Language into Mathematical Language proven effective in communicating and transmitting mathematical realities. The language of The table below is an illustration on the comparison of a natural language mathematics, like any other languages, can be learned; once learned, it allows us to see (expression or sentence) to a mathematical language. 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 42  MATHEMATICS IN THE MODERN WORLD MATHEMATICS IN THE MODERN WORLD English Mathematics Expressions Connectives Name given to an Noun such as person, 2 A question commonly encountered, when presenting the sentence example 1 + 2 = 3 is object of interest. place and things and that; pronouns 3–2 If = is the verb, then what is + ? Example: 3x The answer is the symbol + is what we called a connective which is used to connect objects a) Ernesto of a given type to get a ‘compound’ object of the same type. Here, the numbers 1 and 2 are b) Batangas City 3x + 2 connected to give the new number 1 + 2. c) Book d) He ax + by + c In English, this is the connector “and”. Cat is a noun, dog is a noun, cat and dog is a ‘compound’ noun. Sentence It has a complete Group of words that Mathematical Sentence thought express a statement, 3+2=5 Mathematical sentence is the analogue of an English sentence; it is a correct arrangement question or command. of mathematical symbols that states a complete thought. It makes sense to as about the TRUTH a+b=c of a sentence: Is it true? Is it false? Is it sometimes true/sometimes false? Example: a) Ernesto is a boy. ax + by + c = 0 Example: b) He lives in Batangas City. c) Allan loves to read book. 1. The capital of Philippines is Manila. d) Run! (x + y)2 = x2 + 2xy + y2 2. Rizal park is in Cebu. e) Do you love me? 3. 5+3=8 4. 5+3=9 D. Expressions versus Sentences Truth of Sentences Ideas regarding 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 Ideas regarding sentences are explored. Just as English sentences have verbs, so do read the book. mathematical sentences. In the mathematical sentence; Conventions in Languages 3+4=7 Languages have conventions. In English, for example, it is conventional to capitalize name the verb is =. If you read the sentence as ‘three plus four is equal to seven, then it’s easy to hear (like Israel and Manila). This convention makes it easy for a reader to distinguish between a the verb. Indeed, the equal sign = is one of the most popular mathematical verb. 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 Example: mathematical expression. 1. The capital of Philippines is Manila. Expression 2. Rizal park is in Cebu. 3. 5+3=8 An expression is the mathematical analogue of an English noun; it is a correct arrangement 4. 5+3=9 of mathematical symbols used to represent a mathematical object of interest. 43 44 MATHEMATICS IN THE MODERN WORLD MATHEMATICS IN THE MODERN WORLD | Such that x such that y x|y An expression does NOT state a complete thought; in particular, it does not make sense to ask if an expression is true or false. End of proof  Congruence / equivalent A is equivalent to B A B E. Conventions in mathematics, some commonly used symbols, its meaning and an example a) Sets and Logic a is congruent to b a  b mod n modulo n SYMBOL NAME MEANING EXAMPLE a, b, c, …, z Variables  Union Union of set A and set AB *First part of English Alphabet B uses as fixed variable* (lower case)  Intersection Intersection of set A A B *Middle part of English alphabet and set B use as subscript and superscript (axo)p (5x2)6  Element x is an element of A xA variable*  Not an element of x is not an element of x A *Last part of an English alphabet set A uses as unknown variable* { } A set of.. A set of an element {a, b, c} b) Basic Operations and Relational Symbols  Subset A is a subset of B AB SYMBOL NAME MEANING EXAMPLE  Not a subset of A is not a subset of B AB + Addition; Plus sign a plus b … Ellipses There are still other a, b, c, … items to follow a added by b 3+2 a increased by b a + b + c + …. Subtraction; minus sign a subtracted by  Conjunction A and B AB b  Disjunction A or B A B a minus b - 3-2  Negation Not A A a diminished by b → Implies (If-then statement) If A, then B A→B Multiplication sign a multiply by b 4 3  If and only if A if and only if B AB () *we do not use x as a symbol for a times b (4)(3)  For all For all x x multiplication in our discussion since its use as a variable*  There exist There exist an x   or | Division sign; divides ab 10  5  Therefore Therefore C C b|a 5 | 10 45 46  MATHEMATICS IN THE MODERN WORLD MATHEMATICS IN THE MODERN WORLD  Composition of function f of g of x f  g(x) Examples: = Equal sign a=a 5=5 Let x be a number. Translate each phrase or sentence into a mathematical expression or equation. a+b=b+a 3+2=2+3 1.Twelve more than a number.  Not equal to ab 34  Greater than ab 10  5 Ans.: 12+x  Less than b a 5  10 2.Eight minus a number.  Greater than or equal to ab 10  5 Ans.: 8−x  Less than or equal to ba 5  10 3.An unknown quantity less fourteen.  Binary operation ab a * b = a + 17b Ans.: x−14 c) Set of Numbers 4.Six times a number is fifty-four. Ans.: 6x=54 SYMBOL NAME MEANING EXAMPLE 5. Two ninths of a number is eleven. natural numbers / whole ℕ0 0 = {0,1,2,3,4,...} 0∈ 0 numbers set (with zero) Ans.: 2/9x=11 natural numbers / whole ℕ1 6. Three more than seven times a number is nine more than five times the number. numbers set (without 1 = {1,2,3,4,5,...} 6∈ 1 zero) Ans.: 3+7x=9+5x ℤ integer numbers set = {...-3,-2,-1,0,1,2,3,...} -6 ∈ ℚ rational numbers set 7. Twice a number less eight is equal to one more than three times the number. = {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. Example. Consider the binary operation * on R given by a*b = ab/2. Show CLOSED that a*b = b*c. 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. Solution: Illustrative examples: 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 1) Is S = { ±1, ±3, ±5, ±7, …} is closed under usual addition? that b*a = ab/2. Hence a*b = b*a Solution: 71 72 MATHEMATICS IN THE MODERN WORLD MATHEMATICS IN THE MODERN WORLD Definition: Let  be a binary operation of a set S. Then; = a + b + c – ab – ac – bc + abc (a)  is associative if for all a, b, c  S, (ab)c = a (bc) For a  (b  c) (b)  is commutative if for all a, b  S, ab = b a a  (b  c) = a  (b + c – bc) = a + (b + c – bc) – (a)(b + c – bc) (c) An element e  S is called a left identity element if for all a  S, we have e a=a = a + b + c – bc – ab – ac + abc (d) An element e  S is called a right identity element if for all a  S, we have Hence  is associative on S  Z+. ae=a (b) Commutative (e) An element e  S is called an identity element if for all a  S, we have a  ab=ba e = a and e  a = a. a + b – ab = b + a – ba a + b – ab = a + b – ab (f) Let e be an identity element is S and a  S, then b is called an inverse of the Hence  is commutative on S  Z+. element “a” if a  b = e and b  a = e. (c) Identity Note that a  b = b  a = e or a * a-1 = a-1 * a = e a*e=a e*a=a If a  S, then the inverse of “a” is denoted by a-1. Here -1 is not an a + e – ae = a e + a – ea = a exponent of a. e – ae = a – a e – ea = a – a Example: Let S = Z+ as define  on S by a  b = a + b – ab. Show the e(1 – a) = 0 e(1 – a) = 0 associativity and the commutativity of S in a binary operation. Find also its identity and inverse if any. e=0 e=0 (a) Associativity hence, the identity exist except when a = 1. Let a, b, c  Z+. Then; (a  b)  c = a  (b  c) (d) Inverse For (a  b)  c a * a-1 = e a-1 * a = e (a  b)  c = (a + b – ab)  c 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? = (a + b – ab) + c - (a + b – ab)c 73 74  MATHEMATICS IN THE MODERN WORLD MATHEMATICS IN THE MODERN WORLD (a) Commutative ab=ba Self - Learning Activity a2 + ab + b2 = b2 + ba + a2 Directions. Do as indicated. a2 + ab + b2 = a2 + ab + b2 A. Define a relation C from R to R as follows: For any (x,y) R x R, Hence, the operation  is commutative. (x,y)  C meaning that x2 + y2 = 1. (b) Associative