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Math in modern world - Lecture notes 1
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Mathematics in the
Modern World
A Worktext for GEC
Academic Year 2020 – 2021
Mindanao State University – Main Campus, Marawi City
Mathematics in the
Modern World
▪ Mahid M. Mangontarum ▪ Mark P. Laurente ▪ Salma L. Naga-Marohombsar ▪ Norlailah M. Madid ▪ Aslayn H. Datu-dacula ▪ Rodelito M. Aldema ▪ Ednelyn Ranas-Cantallopez ▪ Janet B. Macoy ▪ Charles B. Montero ▪ Redeemtor R. Sacayan ▪ Gloria A. Rosalejos ▪ Erneta A. Intod ▪ Merry Jane S. Benitez ▪ Amerkhan G. Cabaro ▪ Alyssa Fatmah S. Mastura ▪ Raicah R. Cayongcat- Rakim
Mathematics Department College of Natural Sciences and Mathematics
Preface
Mathematics in the Modern World is the worktext for the course GEC104 – Mathematics in the Modern World for the First Semester of the Academic Year 2020
- 2021 in Mindanao State University (MSU) – Main Campus, Marawi City. It was prepared by a committee composed of select faculty members from the Mathematics Department of the MSU Main Campus by virtue of Office Order No. 2, Series of 2020, issued by the Chairperson of the Department.
GEC104 is one of the nine (9) General Education courses listed in CHED Memorandum Order No. 20, Series of 2013, which is mandatory for all undergraduate students to take regardless of their major. This emanated from the implementation of the K to 12 Curriculum in the Philippines per Republic Act No. 10533, otherwise known as the “Enhanced Basic Education Act or 2013”.
The worktext Mathematics in the Modern World is also in line with the Implementing Rules and Regulations for the Flexible Learning Options (FLO) of the Mindanao State University – Main Campus as approved by the MSUS President Habib W. Macaayong, DPA, on September 4, 2020. The purpose is to ensure continued delivery of quality education to the university constituents amidst the COVID- 19 pandemic in accordance with the guidelines of the Inter-Agency Task Force for the Management of Emerging Infectious Diseases (IATF) and advisories from the Commission on Higher Education (CHED).
The main intent of the text is to promote interest in and simplify the study of Mathematics, especially to undergraduate students. Hence, it provides discussions on the relevance and importance of mathematics, and presents methods in solving practical problems in the real world. In order to fully appreciate this worktext, as a prerequisite, a thorough understanding of basic algebra and statistics–including operations, sets, data collection and sampling techniques is of great use and importance.
####### MAHID M. MANGONTARUM
Co-Chairperson, GEC104 Committee
Message from the Chairperson
Greetings of Peace MSU students. Today, the world of education is undergoing massive changes brought about by the global pandemic. Teachers, students and parents are now struggling to face more of life’s challenges during these unprecedented times. Because of the so-called “new normal”, the faculty members of the Mindanao State University – Main Campus, the Mathematics Department in particular, will be engaging the students using the virtual platform and other flexible learning options. The road may be difficult but it is still an exciting one since this is an opportunity to for everyone to learn new things.
It has been two years now since the planning of a worktext for GEC (Mathematics in the Modern World). We have been dreaming of this moment since the day we had our training for the teaching of the new General Education Curriculum back in 2018. I would like to express my heartfelt congratulations to the working committee who were able to transform that humble dream into a reality. This is a job well-done. The introduction of this GEC104 worktext is very timely given the current situation. I strongly believe that every instructor, every students and even parents who will be given a chance to read this text will have the chance to dive into the complex but rich world of Mathematics, and might develop new ideas that will be beneficial to to each in the future. Mathematics plays a vital role in our lives that is why we should learn and continue to appreciate mathematics.
I encourage all to work together and contribute in making this First Semester, AY 2020 – 2021, a fruitful and meaningful life journey for everyone.
Thank you very much!!!
MARK P. LAURENTE, PhD, CSPE, LPT Department Chairperson
Chapter 0. Classroom Orientation
Brief Description of the Course
The Course Code is “GEC104” and the Course Description is “Mathematics in the Modern World”. This is a three-unit course that deals with the nature of mathematics, appreciation of its practical, intellectual, and aesthetic dimensions, and application of mathematics in daily life.
Learning Outcomes
At the end of the course, the students should be able to:
- Discuss and argue about the nature of mathematics, what it is, how it is expressed, represented, and used;
- Use different types of reasoning to justify statements and arguments made about mathematics and mathematical concepts;
- Discuss the language and symbols of mathematics;
- Use a variety of statistical tools to process and manage numerical data;
- Analyze codes and coding schemes used for identification, privacy and security purposes;
- Use mathematics and math tools in other areas such as business and finance, health and medicine, arts and design, and recreation;
- Appreciate the nature uses of mathematics in everyday life; and affirm honesty and integrity in the application of mathematics to various human endeavors.
The Mindanao State University
Mandate
As enshrined in its Charter (RA 1387), MSU was established in Marawi City on September 1, 1961 to achieve the following mandate:
- Educate the youth of Mindanao, Sulu and Palawan (MINSUPALA) by offering degree programs in various fields of learning;
- Support businesses and industries in the region by providing their manpower requirements; and
- Integrate Muslims and other cultural minorities into the mainstream of national
life.
Philosophy
The MSUS is committed to the total development of man and to the search for truth, virtue and academic excellence.
Vision
The MSUS aspires to be the Premier Supra Regional University in the MINSUPALA region.
Mission
Committed to the attainment of peace and sustainable development in the MINSUPALA region, the MSUS will set the standards of excellence in Science, Arts, Technology and other fields; accelerate the economic, cultural, socio-political and agro- industrial development of the Muslim and other cultural groups, thereby facilitating their integration into the national community; preserve and promote the cultural heritage of the region and conserve its rich natural resources; and infuse moral and spiritual values.
For collaborative efforts, for diplomatic relations and for international recognition as a leading institution of higher learning, the MSUS will pursue vigorously linkages with foreign agencies.
Chapter 1. Mathematics in our World
The field of Mathematics is a diverse discipline that deals not only with arithmetic and geometry, but also with data, measurements, and observations from science; with inference, deduction, and proof; and with mathematical models of natural phenomena, of human behavior, and of social systems.
Mathematics reveals hidden patterns that help us understand nature, the world around us, and the universe in general. Through the years, many scientists and mathematicians have discovered mathematical concepts and patterns in nature which are also observed in various phenomena. In this chapter, we will learn more of what is mathematics.
Getting to know more about Mathematics
What is Mathematics?
To help us understand the world of mathematics, we need to understand first the meaning of mathematics. The following are some common definitions of mathematics:
- Mathematics is all about the unbelievable patterns of numbers formed by nature and of the universe;
- Mathematics is all about language expressed in different forms like patterns, shapes, and music, among others;
- Mathematics is all about what our eyes can see, what our ears can hear and even what we can perceive in our physical environment;
- Mathematics is a language we understand.
Aside from these definitions, British mathematician and writer Ian Nicholas Stewart, also defines mathematics in his book “Nature’s numbers: The Unreal Reality of Mathematical Imagination” as a formal system of thought for recognizing and exploiting patterns. The book takes readers on an exciting, lucid voyage of discovery as Stewart investigates patterns of form, number, shape and movement in the world around us.
Summing up these ideas lead to the discovery of the great secret uncovered by mathematics: Nature’s patterns are not just there to be admired, they are vital clues to the rules that govern natural processes.
Most factory workers nowadays need to understand basic algebra and trigonometry to operate complex manufacturing electronic equipment. 4. In people and communities. Mathematics plays an important role in people’s life. For instance, mathematics is very helpful in understanding various features of the human body starting from body structures to cellular formations. Populations and livelihood of different communities also make use of mathematics. 5. In events. Mathematics helps us understand past and present events so that we may be able to predict a possible recurrence. This is particularly important in the case of preventing calamities.
What is Mathematics for?
Mathematics is on the list of underappreciated disciplines by most people. Many are still unaware of the need for every individual to know and appreciate the role that mathematics plays, some of which are listed as follows.
- To help us unravel the puzzles of nature, and provides a useful way to think about nature. Mathematics help provide solutions to anything concerning the nature. This helps human beings build a better connection to what surrounds them.
- Organize patterns and regularities as well as irregularities. With mathematics, we are able to make arrangements and understand patterns of various behaviors.
- To be able to predict. By examining previous occurrences and patterns, we are able to predict future outcomes using various mathematical tools.
- To help us control weather and epidemics. Meteorologists make use of mathematical concepts in studying and understanding the behavior of the atmosphere in order to prevent a disaster from happening. Also, mathematics help provide effective solutions in preventing the widespread of epidemics.
- Provides tool for calculations. Mathematics is best-known for calculation.
- Provides new questions to think about. Mathematics gives precise information about anything. Thus, a single idea may lead to a series of questions which can develop into further research.
What is Mathematics all about?
Mathematics one broad discipline. The following list enumerate some of the things which are usually associated with mathematics.
Numbers, symbols, notations. When people are asked what mathematics is all about, they usually associate mathematics with numbers, symbols and notations.
Operations, equations, functions. People recognize mathematics when looking at expressions involving operations, or mathematical equations and functions.
Processes and “thingification” of processes (that are abstractions). There are a few people with adequate knowledge in mathematics who may say that mathematics is all about the study of certain processes, and giving concrete representations to things that are usually abstract in form.
Proof – a story rather than a sequence of statements. Mathematics also serves to clarify and render precise information which can be used as a proof in testing the credibility of an idea being studied.
How is Mathematics done?
For us to understand the world of mathematics, there is a need for us to understand how it is done. The following is a list of guidelines that may help one to study mathematics:
- With curiosity. All mathematical study are motivated by the author’s curiosity. The need to find answers to questions pushes mathematicians to dig deeper and conduct more studies.
- With a penchant for seeking patterns and generalities (inquisitiveness). Mathematicians do not only depend on a piece of idea. They in fact seek patterns and combine pieces of information to make generalizations of particular cases.
- With a desire to know the truth. One major factor for the continuous development of mathematics is the desire of individuals in seeking explanations to various phenomena.
- With trial and error. Like in any scientific discovery or invention, there are times when mathematics requires us to undergo numerous trials and experience failures before arriving at a successful end.
- Without fear of facing more questions and problems to solve. When studying mathematics, one must be fearless. Fearless in the sense that when one person attempts to acquire more knowledge about something, he usually faces more questions and more problems in the process.
Who uses Mathematics?
- Mathematicians who specializes in either Pure or Applied fields.
- Scientists studying either natural or social sciences.
- Practically everyone uses mathematics in dealing with everyday life activities.
Every individual uses different mathematics at different times for different purposes using different tools with different attitudes (diversity and universality).
Fibonacci Sequence
Fibonacci (1170–1250), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano (which means Leonardo the Traveller from Pisa) is one of the best-known mathematicians of medieval Europe. In 1202, after a trip that took him to several Arab and Eastern countries, Fibonacci wrote the book Liber Abaci. In this book, he introduced the so- called modus Indorum (method of the Indians), today known as the Hindu–Arabic numeral system, and explained why the Hindu-Arabic numeral system that he had learned about during his travels was a more sophisticated and efficient system than the Roman numeral system. Moreover, in the said book, Fibonacci solved a problem that he himself created. The problem is concerned with the birth rate of rabbits based on idealized assumptions. Here is a statement of Fibonacci’s rabbit problem.
“At the beginning of a month, you are given a pair of newborn rabbits. After a month the rabbits have produced no offspring; however, every month thereafter, the pair of rabbits produces another pair of rabbits. The offspring reproduce in exactly the same manner. If none of the rabbits dies, how many pairs of rabbits will there be at the start of each succeeding month?” The solution of this problem is a sequence of numbers that we now call the Fibonacci sequence. This solution can be represented by the following figure that shows the numbers of pairs of rabbits on the first day of each of the first six months. The larger rabbits represent mature rabbits that produce another pair of rabbits each month. The numbers in the blue region—1, 1, 2, 3, 5, 8—are the first six terms of the Fibonacci sequence.
totallyhistory/fibonacci/
In this diagram, it can be seen that the number of pairs of rabbits for any month after the first two months can be determined by adding the numbers of pairs of rabbits in each of the two previous months. For instance, the number of pairs of rabbits at the start of the sixth month is 3+5=8.
This can be used as a basis in giving a formal definition to the sequence. A recursive definition for any sequence is one in which each successive term of the sequence is defined by using some of the preceding terms. In this case, if we use the notation 퐹푛 to represent 푛푡ℎ Fibonacci number, then the sequence of numbers 퐹푛, where 푛 = 1,2, ... , is defined by the linear recurrence equation
퐹푛= 퐹푛−1+ 퐹푛−2,
for 푛 ≥ 3 and with 퐹 1 = 퐹 2 = 1. As a convention, we set 퐹 0 = 0.
Example. Use the definition of the Fibonacci numbers to find the 7th and 8th Fibonacci numbers.
Solution. Recall that the first six terms in the Fibonacci sequence are 1, 1, 2, 3, 5, and 8. Using the recursive definition of 퐹푛, we have
퐹 7 = 퐹 6 + 퐹 5 = 8 + 5 = 13
and
####### 퐹 40 =
####### 1
####### √ 5
####### [(
####### 1 +√ 5
####### 2
####### )
40 −(
####### 1 −√ 5
####### 2
####### )
40 ]= 102334155.
####### EXERCISE
- Use the Binet’s formula to find the 8th Fibonacci number without using a calculator.
####### REFERENCES
R. N. Aufmann, J. S. Lockdown, R. D. Nation, & D. K. Clegg, Mathematical Excursions, 3 rd Edition, Brooks/Cole Cengage Learning, 2013.
E. Kilic, The Binet formula, sums and representations of generalized Fibonacci p- numbers, European J. Combin. 29 (3), ( 2008 ), 701–711.
I. Stewart, Nature’s Number: The Unreal Reality of Mathematics Imagination, BasicBook: A Division of Harper Collins Publishers, 1995.
artofproblemsolving/wiki/index.php/Binet%27s_Formula
Math in modern world - Lecture notes 1
Course: bs nursing
University: Southwestern University PHINMA
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