Mathematics In Modern World Instructional Materials PDF

INSTRUCTIONAL MATERIALS /PUP FOR MATHEMATICS IN MODERN WORLD GEED 004 1ST SEMESTER SY 2024-2025 COMPILED BY: ENGR. JER ANTHONY F. PALO PART-TIME FACULTY Mathematics in Modern World Overview...

INSTRUCTIONAL MATERIALS /PUP FOR MATHEMATICS IN MODERN WORLD GEED 004 1ST SEMESTER SY 2024-2025 COMPILED BY: ENGR. JER ANTHONY F. PALO PART-TIME FACULTY Mathematics in Modern World Overview Mathematics is the science that deals with the logic of shapes, quantity and arrangement. It exists literally everywhere. From the speed of light, to temperatures and even to the sizing of clothes. It is all around us, in everything we do. It is the building block of everything in our daily lives. Including mobile devices, architecture, art, money, engineering and even sports. It puts our life in order and prevents chaos. Certain qualities that are nurtured by mathematics are power of reasoning, creativity, abstract or spatial thinking, critical thinking, problem-solving ability and effective communication skills. PART I: The Nature of Mathematics Introduction § Mathematics exists everywhere and it is applied in the most useful phenomenon. Even by just looking at the ordinary part of the house, the room, the street, mathematics is there. § The origin of mathematics can be traced to the history and significance of patterns and numbers. It deals with ideas translated to objects and concepts created by humans. They are invented to link the meaning of pattern which result experiences associated with the counting, sequences, and regularities. § Mathematics is not just about numbers. Much of it is problem solving and reasoning-inductive and deductive. It also discusses intuition, proof and certainty. It utilizes Polya’s 4-steps in problem solving, varied problem solving strategies, mathematical problems involving patterns and recreational problems using mathematics. “There is geometry in the humming of strings, there is music in the spacing of spheres. Friends are as companions on a journey, who ought to aid each other to persevere in the road to a happier life. Do not say a little in many words but a great deal in a few. Above the cloud with its shadow is the star with its light. Above all the things reverence thyself. Rest satisfied with doing well, and leave others to talk of you as they please. Silence is better than unmeaning words. Choose rather to be strong of soul than strong of body. The oldest, shortest words – “yes” and “no” – are those which Pythagoras require the most thought. Above all things, reverence yourself. Strength of mind rests in sobriety; for this keeps your reason unclouded by passion”. Mathematics in Our World CHAPTER 1 Chapter Overview At the end of this chapter, the students should be able to: § Identify patterns in nature and regularities in the world § Articulate the importance of mathematics in one’s life § Argue about the nature of mathematics, what it is, how it is expressed, represented and used § Discuss the role of mathematics in some disciplines § Express appreciation for mathematics as human endeavor READ: pgs. 2-3 of “Mathematics in the Modern World” by Romeo M. Daligdig PATTERNS IN NATURE Symmetry means agreement in dimensions, due proportion and arrangement. In everyday language, it refers to a sense of harmonious and beautiful proportion and balance. In mathematics, “symmetry” means that an object is invariant to any of various transformations , including reflection, rotation or scaling. (Wikipedia) PATTERNS IN NATURE Spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Cutaway of nautilus shell shows the chambers arranged in an approximately logarithmic spiral. PATTERNS IN NATURE Meander is one of a series of regular sinuous curves, bends, loops, turns , or wind i n g s i n t h e channel of a river, stream or other watercourse. It is produced by a stream or river swinging from side to side as it flows across its floodplain or shifts its channels within a valley. A meander is produced by a stream or river as it erodes the sediment downstream on an inner, convex bank which is typically a point bar. PATTERNS IN NATURE Wave is a disturbance that transfers energy through matter or space, with little associated mass transport. Waves consists oscillation and vibrations of physical medium or a field, around relatively fixed locations. Surface waves in water show water ripple. PATTERNS IN NATURE Foam is a substance formed by trapping pockets of gas in a liquid or solid. A bath sponge and the head on the glass of beer are examples of foams. In most foams, the volume of gas is large, with thin films of liquid or solid separating the regions of gas. Soap foams are also known as suds. PATTERNS IN NATURE Tessellation tilling of a plane using one or more geometric shape, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries. PATTERNS IN NATURE Fracture or Crack is the separation of an object or material into two or more pieces under the action of stress. The fracture of a solid usually occurs due to the development of certain displacement discontinuity surfaces within the solid. If displacement develops perpendicular to the surface of displacement, it is called a normal tensile or crack; if a displacement develops tangentially to the surface of displacement, it is called a shear crack, slip band or dislocation. PATTERNS IN NATURE Stripes are made by a series of bands or strips, often of the same width and color along the length. PATTERNS IN NATURE Fractal never-ending pattern. It is an infinitely complex patterns that are self-similaracross different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of chaos. Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar, since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc. PATTERNS IN NATURE Fractal never-ending pattern. It is an infinitely complex patterns that are self-similaracross different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of chaos. Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar, since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc.  PATTERNS IN NATURE Affine Transformations Is a type of geometric transformation which preserves collinearity and th e r a ti os of distances between points in a line. It is a process of rotation, reflection and scaling. Many plant forms utilize these processes to generate their structure. WATCH: “Affine Transformations” https://www.youtube.com/watch?v=il6Z5LCykZk Fibonacci Sequence ØNature’s numbering system – Appears everywhere in nature, from the leaf arrangement in plants, to the pattern of the florets in a flower, the bracts of a pinecone, or the scales of a pineapple. – Applicable to the growth of every living thing including a single cell, a grain of wheat, a hive of bees and even all of mankind. ØN u m b e r s i n t h e f o l l o w i n g i n t e g e r s e q u e n c e characterized by the fact that every number after the first two is the sum of the two preceding ones. WATCH: “The Fibonacci Sequence: Nature’s Code” https://www.youtube.com/watch?v=wTlw7fNcO-0 Fibonacci Sequence The sequence Fn of Fibonacci numbers is defined by the recurrence relation: = −1 + −2 with seed values 1 = 1, 2 = 1 1 = 1, 3 = 2 The first 6 Fibonacci numbers Fn for n=0, 1, 2,…, 6 The Fibonacci Sequence is also known as The Golden Ratio Fibonacci, also known as Leonardo Bonacci, Leonardo of Pisa or Leonardo Bigollo Pisano, was an Italian mathematician from the Republic of Pisa, considered to be “the most talented Western mathematician of the Middle Ages”. (Wikipedia) Leonardo Fibonacci came up with the sequence when calculating the ideal expression pairs of rabbits over the course of one year. Today, its emergent patterns and ratios (phi = 1.61803…) can be seen from the microscale to the macroscale, and right through to biological systems and inanimate objects. Leonardo While the Golden Ratio doesn’t account for every structure or pattern in the Fibonacci universe, it’s certainly a major player. Seed heads Pine cones The head of a flower is also Similarly, the seed pods on a subject to Fibonaccian pinecone are arranged in a spiral processes. Typically, seeds pattern. Each cone consists of a are produced at the center, pair of spirals, each one and then migrate towards spiraling upwards in opposing the outside to fill all the directions. The number of steps space. Sunflowers provide a sill almost always match a pair great example of these of consecutive Fibonacci spiraling patterns. numbers. Tree branches Shells The Fibonacci sequence can The unique properties of the also be seen in the way a tree golden rectangle provide another example. This shape, a branches for a split. A main rectangle in which the ratio of trunk will grow until it the sides a/b is equal to the produces a branch which golden mean (phi), can result in a creates two growth points. nesting process that can be Then one of the new stems repeated into infinity - and branches into two, while the which takes on form of spiral. other lies dormant. It’s called the logarithmic spiral, and it abounds in nature. Spiral Galaxies and Hurricane The milky way has several spiral arms, each of them a logarithmic spiral of about 12 degrees. As an interesting aside spiral galaxies appear to defy Newtonian physics. “Pure mathematics is, in its way, the poetry of logical ideas.” Albert Einstein Importance of Mathematics in Life According to Katie Lim (2015), Math is a subject that make students either jump for joy or rip their hair out. However, math is inescapable as you become an adult in the real world. From calculating complicated algorithms to counting down the days till the next Game of Thrones episode, math is versatile and important, no matter how hard it is to admit. 1. Restaurant Tipping 2. Netflix film viewing 3. Calculating Bills 4. Computing Test Scores 5. Tracking Career 6. Doing Exercise 7. Handling Money READ:Countdowns 8. Making pgs. 9-10 of “Mathematics in the Modern World” 9. Baking and Cooking by Romeo M. Daligdig Nature of Mathematics Mathematics relies on both logic and creativity, and is pursued both for a variety of practical purposes and for its intrinsic interest. The essence of mathematics lies in its beauty and its intellectual challenge 1. Patterns and Relationships 2. Mathematics, Science and technology 3. Mathematical Inquiry 4. Abstraction and Symbolic Representations 5. Manipulating Mathematical Statements 6. Application Nature of Mathematics 1. Patterns and Relationships Mathematics explores the possible relationships among abstractions without concern for whether those abstractions have counterparts in the real world. 2. Mathematics, Science and Technology Mathematics is universal in a sense that other fields of human thought are not. It finds useful applications in business, industry, music, historical scholarship, politics, sports, medicine, agriculture, engineering, and the social and natural, sciences. 3. Mathematical Inquiry Normally, people are confronted with problem. In order to live at peace, these problems must be solved using mathematics. Nature of Mathematics 4. Abstraction and Symbolic Representations Mathematical thinking often begins with the process of abstraction - that is, noticing a similarity between two or more objects or events. Aspects that they have in common, whether concrete or hypothetical, can be represented by symbols such as numbers, letter, other marks diagrams, geometrical constructions, or even words. Abstractions are made not only from concrete objects or processes; they can also be made from other abstractions such as kind of numbers. 5. Manipulating Mathematical Statements After abstractions have been made and symbolic representations of them have been selected, those symbols can be combined and recombined in various ways according to precisely defined rules. Typically, strings of symbols are combined into statements that express ideas or propositions. Nature of Mathematics 6. Application Mathematical processes can lead to a kind of model of a thing, from which insights can be gained about the thing itself any mathematical relationships arrived at by manipulating abstract statements may or may not convey something truthful about the thing being modeled. The role of Mathematics in Some Discipline Mathematics is offered in any college course. It is found in every curriculum because its theories and applications are needed in any workplace. Every second of the day needs mathematical knowledge and skills to perform academic activities and office routines. Mathematics is not only number work or computation, but is more about forming generalization, seeing relationship, and developing logical thinking and reasoning. 1. Mathematics in Physical Sciences In Physics, every rule and principle take the mathematical form ultimately. Mathematics gives a final shape of to the rules of physics. It presents them in a workable form. Mathematical calculations occur at every step in Physics. 2. Mathematics in Chemistry Math is extremely important in physical Science especially in advanced topics such as quantum or statistical mechanics. 3. Mathematics in Biological Sciences Biomathematics is a rich fertile field with open, challenging and fascination problems in the areas of mathematical genetics, mathematical ecology, mathematical neuron - physiology, development of computer software for special biological and medical problems, mathematical theory of epidemics, use of mathematical programming and reliability theory in biosciences and mathematical problems in biomechanics, bioengineering and bioelectronics. 4. Mathematics in Engineering and Technology The use of mathematics in engineering is very well known. It is considered to be the foundation of engineering. Engineering deals with surveying, levelling, designing, estimating, construction, etc. In all these processes, application of mathematics is very important. 5. Mathematics and Agriculture Agriculture as a science depends extensively on mathematics. It needs a directs application of mathematics, such as measurement of land or area average investment expenditure, average return or income, production per unit are, cost of labor, time and work, seed rate etc. 6. Mathematics and Economics The social sciences are also beginning to draw heavily upon mathematics. Mathematical language and methods are used frequently in describing economic phenomena. The whole economic situation is regarded as a game between consumers, distributors and producers, each group trying to optimize its profits. 7. Mathematics and Psychology Now, experimental psychology has become highly mathematical due to its concern with such factors as intelligence quotients, standard deviation, mean, median, mode, correlation, coefficients, and probable errors. Statistical analysis is the only reliable method of attacking social and psychological phenomena. Until mathematicians, entered into the field of psychology, it was nothing but a flight of imagination. 8. Mathematics and Actuarial Science, Insurance and Finance Actuaries use mathematics and statistics to make financial sense of future. For example, if an organization embarks on a large project, an actuary may analyze the project, assess the financial risks involved, model the future financial outcomes and advise the organization on the decisions to be made. 9. Mathematics and Archaeology Archaeologists use a variety of mathematical and statistical techniques to present the data from archaeological surveys and try to distinguish patterns in their results that shed light on past human behavior. Statistical measures are used during excavation to monitor which pits are most successful and decide on further excavation. 10. Mathematics and Logic D’ Alembert says, ”Geometry is a practical logic, because in it, rules of reasoning are applied in the simplest and sensible manner. C.J.Keyser - ”Symbolic logic is mathematics, mathematics is symbolic logic”. The symbols and methods used in the investigation of the foundation of mathematics can be transferred to the study of logic. They help in the development and formulation of logical laws. 11. Mathematics in Music Leibnitz, the great mathematician said, - ”Music is a hidden exercise in arithmetic of a mind unconscious of dealing with numbers.” Calculations are the root of all sorts of advancement in different disciplines. Music theorists often use mathematics to understand musical structure and communicate new ways of hearing music. Music scholars have also used mathematics to understand musical scales, and some composers have incorporated the Golden Ratio and Fibonacci numbers into their work. 12. Mathematics in Arts Mathematics and art are just two different languages that can be used to express the same ideas.” It is considered that the universe is written in the language of mathematics, and its characters are triangles, circles and other geometric figures. Artists who strive and seek to study nature must therefore first fully understand mathematics. Appreciation of rhythm proportion, balance and symmetry postulates a mathematical mind. 13. Mathematics in Philosophy Mathematics occupies a central place between natural philosophy and mental philosophy. It was in their search of distinction between fact and fiction that Plato and other thinkers came under the influence of mathematics. 14. Mathematics in Social Networks Graph theory, text analysis, multidimensional scaling and cluster analysis and a variety of special models are some mathematical techniques used in analyzing data on a variety of social networks. 15. Mathematics in Political Science In Mathematical Political Science, we analyze past election results to see changes in voting patterns and the influence of various factors on voting behavior, on switching of votes among political parties and mathematical models for conflict resolution. 16. Mathematics in Linguistics The concepts of structure and transformation are as important for linguistic as they area for mathematics. Development of machine languages are comparison with natural and artificial language require a high degree of mathematical ability. Information theory, mathematical biology, mathematical psychology etc. are all needed in the study of linguistics. 17. Mathematics in Management Mathematics in management is a great challenge to imaginative minds. It is not meant for the routine thinkers. Different mathematical models are being used to discuss management problems of hospitals, public health, pollution, education planning and administration and similar other problems of social decisions. 18. Mathematics in Computers An important area of applications of mathematics is in the development of formal mathematical theories related to the development of computer science. Now most of applications of mathematics to science and technology today are via computers. The foundation of computer science is based only on mathematics. 19. Mathematics in Geography Geography is nothing but a scientific and mathematical description of our earth and its universe. The dimension and magnitude of earth, its situation and position in the universe, the formation of days and nights, lunar and solar eclipses, latitude and longitude, maximum and minimum rainfall, etc. are some of the numerous learning areas of geography which need the application of mathematics.  Appreciating Mathematics as a Human Endeavor In order to appreciate mathematics much better, every person should have the thorough understanding of the discipline as a human endeavor. Mathematics brings impact to the life of a learner, worker, or an ordinary man in society. The influences of mathematics affects everyone for a lifetime. Mathematics works in the life of all professionals. Accountants assist businesses by working on their taxes and planning for upcoming years. They work with tax codes and forms, use formula for circulating interest, and spend a considerable amount of energy organizing paperwork. Agriculturists determine the proper amounts of fertilizers pesticides and water to produce bountiful amounts of foods. They must be familiar with chemistry and mixture problems. Architects design buildings for structural integrity and beauty. They must know how to calculate loads for finding acceptable materials in design which involve calculus. Biologists study nature to act in concert with it since we are very closely tied to nature. They use proportions to count animals as well as use statistics and probability. Chemists find ways to use chemicals to assist people in purifying water, dealing with waste management researching superconductors, analyzing crime scenes making food products and in working with biologists to study the human body. Computer Programmers create complicated sets of instructions called programs or software to help us use computers to solve problems. They must have a strong sense of logic and have critical thinking and problem solving skills. Engineers build products, structures, systems like automobiles, buildings, computers, machines, and planes, to name just a few examples. They cannot escape the frequent use of variety of calculus. Geologists use mathematical models to find oil and study earthquakes. Lawyers argue cases using complicated lines of reason. That skill is nurtured by high level math co u r ses. Th ey also spend a lo t o f time researching cases, which means learning relevant codes, laws and ordinances. Managers maintain schedules, regulate worker performance, and analyze productivity. Medical Doctors must understand the dynamic systems of the human body. They research illnesses, carefully administer the proper amounts of medicine, read charts or tables, and organize their workload and manage the duties nurses and technicians. Meteorologists forecast the weather for agriculturists pilots, vacationers, and those who are marine-dependent. They read maps, work with computer models, and understand the mathematical laws of Physics. Military Personnel carry out the detailed instructions that the doctor gave them. They adjust intravenous drip rates, take vitals, dispense medicine, and even assist in operations. Politicians help solve the social problems of our time by making complicated decisions within the confines of the law, public opinion, and budgetary restraints. Sales People typically work on commission and operate under a buy low, sell high profit model. Their job requires good interpersonal skills and the ability to estimate basic math problems without the need of paper or pencil. Technicians repair and maintain the technical gadgets we depend on like computers, televisions, DVDs, cars, refrigerators. They always read measuring devices, referring to manuals, and diagnosing system problems. Tradesmen estimate job costs and use technical math skills specific to their field. They deal with slopes, areas volumes, distances and must have an excellent foundation in math.