Mathematics in Modern World (PDF)

LECTURE 1.1 MATHEMATICS IN OUR - It has the power to unveil the MODERN WORLD reasons behind occurrences and it offers explanations Stewart’s Nature’s Number...

LECTURE 1.1 MATHEMATICS IN OUR - It has the power to unveil the MODERN WORLD reasons behind occurrences and it offers explanations Stewart’s Nature’s Number - Mathematics, as a study of patterns, - stated that mathematics is a formal allows people to observe, system of thought that was gradually hypothesize, experiment, discover, developed in the human mind and and recreate. evolved in the human culture. - mathematics is an art and a process - Suggest a relationship between of thinking. For it involves numbers growth and natural patterns. reasoning, which can be inductive or o Pattern of Nature - Nature deductive, and it applies methods of often exhibits patterns that can proof both in fashion that is be described mathematically, conventional and unconventional. such as the Fibonacci sequence Mathematics is Everywhere in flowers or the golden ratio in - It is our important tool in the field of shells. sciences, humanities, literature, o Mathematical Relationships - medicine, and even in music and Stewart explores how these arts; it is in the rhythm of our daily natural patterns can be activities, operational in our expressed with numbers, communities, and a default system highlighting connections of our culture. between mathematics and the natural world. THE MATHEMATICAL LANDSCAPE o Applications - Understanding these relationships helps in 1. Mathematical Research and Theory fields like biology, architecture, Development and art, where mathematical 2. Visualization and Representation principles can enhance design 3. Theories and Concept and understanding of growth 4. Computational and Applied processes. Mathematics 5. Abstract Structures HOW MATHEMATICS IS DONE Math is a way of thinking, and it is undeniably important to see how that thinking is going to be developed rather than just merely see face value of the results. WHAT IS MATHEMATICS MATHEMATICS IS FOR EVERYONE Mathematics is a tool There is mathematics we call pure and - allows us to perceive realities in applied, as there are scientists we call social different contexts that would and natural. There is mathematics for otherwise be intangible to us. engineers to build, mathematics for - Math deals with the logic of shape, commerce and finance, mathematics for quantity, and arrangement. weather forecasting, mathematics that is - It provides answers to existing related to health, and mathematics to questions and presents solutions to harness energy for utilization. occurring problems. THE IMPORTANCE OF KNOWING AND 3. Patterns of Movement LEARNING MATHEMATICS - human walk: left-right-left-right-left rhythm. It is beneficial in generating conclusions and - Four-legged creature walks in predicting events. It is because they - pattern in locomotion extends to the provide clues. The clues that make us scuttling of insects, the flights of realize that interference in the motion of birds, the pulsations of jellyfish, and heavenly bodies can predict lunar eclipse, also the wave-like movements of solar eclipse as well as comets’ fish, worms, and snakes. appearances. That the position of the sun 4. Patterns of Rhythm and the moon relative to the earth can - most basic pattern in nature. predict high tide and low tide events - hearts and lungs follow a regular affecting human activities. And that human repeated pattern of sounds or activities need clues for the human culture to movement whose timing is adapted meaningfully work. to our body’s needs. - Heartbeat and breathing 5. Patterns of Texture LECTURE 1.2 MATHEMATICS IN OUR - A texture is a quality of a certain MODERN WORLD object that we sense through touch. It exists as a literal surface that we Pattern can feel, see, and imagine. Textures - structure, form, or design that is are of many kinds. It can be bristly, regular, consistent, or recurring. and rough, but it can also be - can be found in nature, in human- smooth, cold, and hard. made designs, or in abstract ideas. 6. Geometric Patterns They occur in different contexts and - series of shapes that are typically various forms. repeated. - The investigation of nature’s - regularities in the natural world that patterns is an investigation of are repeated in a predictable nature’s numbers. manner. - visible on cacti and succulents. DIFFERENT KINDS OF PATTERN PATTERNS FOUND IN NATURE 1. Patterns of Visuals - often unpredictable, never quite 1. Waves and Dunes repeatable, and often contain - A wave is any form of disturbance fractals. that carries energy as it moves. - can be seen from the seeds and - TWO KINDS OF WAVES pinecones to the branches and o Mechanical Waves - leaves propagate through a - visible in self-similar replication of medium - air or water, trees, ferns, and plants throughout making it oscillate as waves nature. pass by. 2. Patterns of Flow o Wind Waves - surface - flow of liquids provides waves that create the - found in the water, stone, and even chaotic patterns of the sea. in the growth of trees. Similarly, water waves are - Flow of meandering rivers with the created by energy passing repetition of undulating lines. through water causing it to move in a circular motion.  Likewise, ripple patterns o captures symmetries when and dunes are formed by it still looks the same after sand wind as they pass some rotation (of less than over the sand. one full turn). 2. Spots and Stripes o The degree of rotational - spots on the skin of a giraffe. symmetry of an object is - stripes visible on the skin of a zebra. recognized by the number - commonly present in different of distinct orientations in organisms are results of a reaction- which it looks the same for diffusion system (Turing, 1952). each rotation - size and the shape of the pattern c. Translations depend on how fast the chemicals o exists in patterns that we diffuse and how strongly they see in nature and in man- interact. made objects. 3. Spirals o when units are repeated - exist on the scale of the cosmos to and turn out having identical the minuscule forms of microscopic figures, like the bees’ animals on earth. honeycomb with hexagonal - The Milky Way that contains our tiles Solar System is a barred spiral SYMMETRIES IN NATURE galaxy with a band of bright stars emerging from the center running 1. Human Body - Our body exhibits across the middle of it. bilateral symmetry. - appear in many plants such as 2. Animal Movement - symmetry of pinecones, pineapples, and motion is present sunflowers. 3. Sunflower - contains both radial - animals like ram and kudu also have and bilateral symmetry. Ray florets spiral patterns on their horns. are bilaterally symmetrical. The dark 4. Symmetry inner ring of the sunflower is a - If a figure can be folded or divided cluster of radially symmetrical disk into two with two halves which are florets. the same, 4. Snowflakes - have six-fold radial - has a vital role in pattern formation. symmetry. - used to classify and organize 5. Honeycombs/Beehive - wallpaper information about patterns by symmetry; created when a pattern is classifying the motion or repeated until it covers a plane. deformation of both pattern 6. Starfish - have a radial fivefold structures and processes. symmetry - Types: reflections, rotations, and translations are less formally called FIBONACCI IN NATURE flips, turns, and slides. - classic five-petal flowers like - THREE KINDS OF SYMMETRY buttercup, columbine, and hibiscus. a. Reflection Symmetry - eight-petal flowers like clematis and o line symmetry or mirror delphinium symmetry - ragwort and marigold have thirteen o captures symmetries when - occurs in nautilus shells with a the left half of a pattern is logarithmic spiral growth. the same as the right half - present in pineapples and red b. Rotational Symmetry cabbages. E. CONVENTIONS IN MATHEMATICS, SOME COMMONLY USED SYMBOLS, ITS MEANING AND AN EXAMPLE a. Sets and Logic b. Basic Operations and Relational Symbols c. Set of Numbers