Nature of Mathematics PDF

NATURE OF MATHEMATICS Created by: Engr. Rey Angeles Manalo Lyceum of the Philippines - Cavite | 2024  INTRODUCTION Lyceum of the Philippines University - Cavite | 2024...

NATURE OF MATHEMATICS Created by: Engr. Rey Angeles Manalo Lyceum of the Philippines - Cavite | 2024  INTRODUCTION Lyceum of the Philippines University - Cavite | 2024 Mathematics is a broad field of study that involves the exploration and understanding of abstract concepts related to numbers, quantities, shapes, structures, and patterns. It is both a science and a language that helps describe and analyze the world around us. CORE AREAS OF MATHEMATICS 1. Arithmetic 2. Algebra 3. Geometry 4. Calculus 5. Statistics and Probability 6. Mathematical Logic 1  OBJECTIVES Lyceum of the Philippines University - Cavite | 2024 At the end of the topic, students will be able to: 1 Articulate the importance of mathematics in one’s life; 2 Identify patterns in nature and regularities in the world; Argue about the nature of mathematics, what it is, how it is expressed, represented 3 and used; and Express appreciation of mathematics as a human endeavor. 4 2  PATTERNS AND NUMBERS IN NATURE AND THE WORLD Lyceum of the Philippines University - Cavite | 2024 3  PATTERNS Lyceum of the Philippines University - Cavite | 2024 Patterns are regularities or repeated arrangements of elements that can be observed in various contexts. They can be found in many different domains, from nature and art to mathematics and everyday life. NATURAL PATTERNS SPIRAL SYMMETRIES STRIPES DOTS 4 GREEK PHILOSOPHERS WHO STUDIED PATTERNS “I would teach children music, physics and philosophy; but most importantly music for the patterns in music and all the arts are keys to learning.” -- PLATO-- Lyceum of the Philippines University - Cavite | 2024 5 GREEK PHILOSOPHERS WHO STUDIED PATTERNS “There is geometry in the humming of the strings, there is music in the spacing of the spheres.” -- PYTHAGORAS-- Lyceum of the Philippines University - Cavite | 2024 6 GREEK PHILOSOPHERS WHO STUDIED PATTERNS “The nature of God is a circle of which the center is everywhere and the circumference is nowhere.” -- EMPEDOCLES-- Lyceum of the Philippines University - Cavite | 2024 7 OTHER PEOPLE WHO STUDIED PATTERNS Joseph Plateau examined soap films, leading him to formulate the concept of a minimal surface. Lyceum of the Philippines University - Cavite | 2024 8 OTHER PEOPLE WHO STUDIED PATTERNS Ernst Haeckel painted hundreds of marine organisms to emphasize their symmetry. Lyceum of the Philippines University - Cavite | 2024 9 OTHER PEOPLE WHO STUDIED PATTERNS D’Arcy Thompson pioneered the study of growth patterns in both plants and animals, showing that simple equations could explain spiral growth. Lyceum of the Philippines University - Cavite | 2024 10 OTHER PEOPLE WHO STUDIED PATTERNS Alan Turing predicted mechanisms of morphogenesis which give rise to patterns of spots and stripes. Lyceum of the Philippines University - Cavite | 2024 11 OTHER PEOPLE WHO STUDIED PATTERNS Aristid Lindenmayer and Benoît Madelbrot showed how the mathematics of fractals could create plant growth patterns. Lyceum of the Philippines University - Cavite | 2024 12 OTHER PEOPLE WHO STUDIED PATTERNS A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Lyceum of the Philippines University - Cavite | 2024 13 OTHER PEOPLE WHO STUDIED PATTERNS W. Gary Smith adopts eight patterns in his landscape work, namely: scattered, fractured, mosaic, naturalistic drift, serpentine, spiral, radial, and dendritic. These patterns occur in plants, animals, rocks formation, river flow, stars or in human creations. Lyceum of the Philippines University - Cavite | 2024 14 THE FIBONACCI SEQUENCE Lyceum of the Philippines University - Cavite | 2024 15  WHAT IS FIBONACCI SEQUENCE? Fibonacci sequence is a series of numbers that follow a unique integer sequence. These numbers generate mathematical patterns that can be found in all aspects of life. The patterns can be seen in everything from the human body to the physiology of plants and animals. The fibonacci sequence is derived from the fibonacci numbers. Lyceum of the Philippines University - Cavite | 2024 16  HOW ARE THESE FIBONACCI NUMBERS OBTAINED? These numbers are obtained from the formula- Fn=Fn-1+Fn-2 These numbers are obtained by adding the two previous numbers in the sequence to obtain the next higher number. Also, we can use the formula: Lyceum of the Philippines University - Cavite | 2024 17  HOW DOES THE FIBONACCI SEQUENCE WORK? The Fibonacci sequence is as follows: 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610..and so on. As a rule, the first 2 numbers in the sequence has to be 0 and 1. All other numbers follow the rule of adding the two previous numbers in the sequence. EXAMPLE: 1+1=2, 2+3=5, 5+3=8 Every third number in the sequence is even. Lyceum of the Philippines University - Cavite | 2024 18  WHAT IS THE HISTORY OF THE FIBONACCI SEQUENCE? The exact date of origin of the Fibonacci sequence is unknown. It is believed that contributions to the theory began in 200 BC by Indian mathematicians whose studies were based in the language of Sanskrit. The sequence was introduced to Western European mathematicians in 1202 by Aka Fibonacci (famous as the Leonardo of Pisa). Lyceum of the Philippines University - Cavite | 2024 19  WHO IS “FIBONACCI”? Fibonacci’s real name is Leonardo Pisano Bigollo. He lived between 1170 and 1250 in Italy. His nickname, “Fibonacci” roughly means “Son of Bonacci”. His father is Guglielmo Bonacci. His study of the sequence began with the breeding patterns of rabbits. In which, he found rabbit generations duplicated in accordance with the Fibonacci numbers. Lyceum of the Philippines University - Cavite | 2024 20  WHO IS “FIBONACCI”? He helped spread Hindu Arabic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) through Europe in place of Roman numerals which is part of the sequence, which he developed. The Fibonacci Sequence was recognized as the Golden Ratio. Lyceum of the Philippines University - Cavite | 2024 21  GOLDEN RATIO, FIBONACCI RECTANGLE AND SPIRAL The Golden Ratio can be generated by ‘golden rectangle’ or ‘Fibonacci rectangle’. The Fibonacci rectangle is a rectangle which is further divided into squares whose lengths are the consecutive numbers of the Fibonacci sequence. Lyceum of the Philippines University - Cavite | 2024 22  GOLDEN RATIO, FIBONACCI RECTANGLE AND SPIRAL This spiral is created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci rectangle. The numbers form what are called as Fibonacci rectangles. These rectangles are unique because each rectangle has length of sides equal to the magnitude of the Fibonacci numbers. Lyceum of the Philippines University - Cavite | 2024 23  FIBONACCI IN NATURE Fibonacci sequence in petal patterns The Fibonacci sequence can be seen in most petal patterns. For example, most daisies have 34, 55or 89 petals and most common flowers have 5, 8 or 13 petals. Lyceum of the Philippines University - Cavite | 2024 24  FIBONACCI IN NATURE Fibonacci sequence in sunflowers The Fibonacci sequence can be found in a sunflower heads seed arrangement. The arrangement of seeds corresponds to Fibonacci spiral and they are arranged in an angle of 137.5 degrees which is also called the ‘golden angle’. Lyceum of the Philippines University - Cavite | 2024 25  FIBONACCI IN NATURE Fibonacci sequence in pinecones and seashells Lyceum of the Philippines University - Cavite | 2024 26  FIBONACCI IN NATURE Fibonacci sequence in human body Lyceum of the Philippines University - Cavite | 2024 27 PHENOMENA IN THE WORLD AS PREDICTED BY MATHEMATICS Lyceum of the Philippines University - Cavite | 2024 28  NATURAL PHENOMENA Mathematics serves as a universal language that bridges the gap between abstract concepts and real-world observations, allowing scientists and researchers to describe, understand, and predict the complexities of the natural world. Lyceum of the Philippines University - Cavite | 2024 29 ECONOMIC AND SOCIAL SYSTEMS Mathematics provides a structured and rigorous approach to defining, analyzing, and understanding economic and social systems. Through modeling, optimization, statistical analysis, and various mathematical frameworks, it helps to make sense of complex interactions, predict outcomes, and inform policy decisions in these fields. Lyceum of the Philippines University - Cavite | 2024 30 ENGINEERING AND TECHNOLOGY Mathematics defines engineering and technology through modeling, optimization, analysis, and simulation. It provides the foundation for designing, understanding, and improving systems and processes in these fields, making it an indispensable tool for engineers and technologists. Lyceum of the Philippines University - Cavite | 2024 31 FINANCE AND INVESTMENT Mathematics provides the tools for valuing financial assets, managing risks, optimizing portfolios, pricing derivatives, and forecasting financial trends. Its rigorous and quantitative approach enables finance and investment professionals to make informed decisions, analyze complex systems, and develop effective strategies. Lyceum of the Philippines University - Cavite | 2024 32  PATTERNS IN NATURE Mathematics defines patterns in nature through symmetry, fractals, geometric patterns, chaos theory, mathematical biology, statistics, differential equations, and topology. These mathematical frameworks allow us to capture, analyze, and understand the regularities and complexities found in natural phenomena. Lyceum of the Philippines University - Cavite | 2024 33 ASTRONOMY AND COSMOLOGY Mathematics is integral to both astronomy and cosmology, providing the tools and models needed to understand the universe’s vast and intricate structures and phenomena. It allows scientists to translate observations into theoretical frameworks, make predictions, and test hypotheses about the nature of our cosmos. Lyceum of the Philippines University - Cavite | 2024 34 THANK YOU Lyceum of the Philippines University - Cavite | 2024