GEMATH+ (Mathematics in the Modern World) Prelims Lesson 1

GEMATH+ (Mathematics in the Modern World) Imelda P. Oruga August 2024  What can you say about the pictures presented on screen? Do you find any significance of the pictures in your program(BS Nursing BSA,BSENTRP)? What is Mathematics? Mathêma “that which is learnt” derived fro...

GEMATH+ (Mathematics in the Modern World) Imelda P. Oruga August 2024  What can you say about the pictures presented on screen? Do you find any significance of the pictures in your program(BS Nursing BSA,BSENTRP)? What is Mathematics? Mathêma “that which is learnt” derived from “manthano”, and modern Greek “mathaino” which means “to learn. ” “The science of structure, order, and relations that has evolved from elemental practices of counting, measuring, and describing the shapes and characteristics of objects”. (Encyclopedia Britannica)  What is Mathematics? Mathematics is the study of pattern and structure. Mathematics is fundamental to the physical and biological sciences, engineering and information technology, to economics and increasingly to the social sciences. Mathematics is a useful way to think about nature and our world. Mathematics is a tool to quantify, organize and control our world, predict phenomena and make life easier for us  Where is Mathematics? Many patterns and occurrences exists in nature,in our world, in our life. Mathematics helps make sense of these patterns and occurrences.  WHAT ROLE DOES MATHEMATICS PLAY IN OURWORLD? Mathematics helps organize patterns and regularities in our world. Mathematics helps predict the behavior of nature and phenomena in the world. Mathematics helps control nature and occurrences in the world for our own ends. Mathematics has numerous applications in theworld making it indispensable  PATTERNS AND NUMBERS IN NATURE AND THEWORLD Patterns can be observed even in stars which move in circles across the sky each day. The weather season cycle each year. All snowflakes contains sixfold symmetry which no two are exactly the same.  PATTERNS AND NUMBERS IN NATURE AND THEWORLD Patterns can be seen in fish patterns likes potted trunkfish, spotted puffer, blue spotted stingray, spotted moral eel, coral grouper, red lion fish, yellow boxfish and angel fish. These animals and fish stripes and spots attest to mathematical regularities in biological growth and form.  PATTERNS AND NUMBERS IN NATURE AND THEWORLD Zebras, tigers, cats and snakes are covered in patterns of stripes; leopards and hyenas are covered in pattern of spots and giraffes are covered in pattern of blotches.  PATTERNS AND NUMBERS IN NATURE AND THEWORLD Natural patterns like the intricate waves across the oceans; sand dunes on deserts; formation of typhoon; water drop with ripple and others. These serves as clues to the rules that govern the flow of water, sand and air. PATTERNS AND NUMBERS IN NATURE AND THEWORLD Other patterns in nature can also be seen in the ball of mackerel, the v-formation of geese in the sky and the tornado formation of starlings. PATTERNS AND REGULARITIES 1. SYMMETRY a sense of harmonious and beautiful proportion of balance or an object is invariant to any various transformations(reflection, rotation or scaling.) a.) Bilateral Symmetry : a symmetry in which the left and right sides of the organism can be divided into approximately mirror image of each other along the midline. Symmetry exists in living things such as in insects, animals, plants, flowers and others. Animals have mainly bilateral or vertical symmetry, even leaves of plants and some flowers such as orchids PATTERNS AND REGULARITIES b.) Radial Symmetry ( or rotational symmetry ) A symmetry around a fixed point known as the center and it can be classified as either cyclic or dihedral Plants often have radial or rotational symmetry, as to flowers and some group of animals. A five-folds ymmetry is found in the echinoderms, the group in which includes starfish (dihedral-D5 symmetry), sea urchins and sea lilies. Radial symmetry suits organism like sea anemones whose adults do not move and jellyfish(dihedral-D4 symmetry). Radial symmetry is also evident in different kinds of flowers PATTERNS AND REGULARITIES FRACTALS a curve or geometric figure, each part of which has the same statistical character as the whole. a never-ending pattern found in nature. The exact same shape is replicated in a process called “self similarity.” The pattern repeats itself over and over again at different scales. For example, a tree grows by repetitive branching. This same kind of branching can be seen in lightning bolts and the veins in your body. Examine a single fern or an aerial view of an entire river system and you’ll see fractal patterns PATTERNS AND REGULARITIES SPIRALS A logarithmic spiral or growth spiral is aself- similar spiral curve which often appears in nature. It was first described by Rene Descartes and was later investigated by Jacob Bernoulli. a curved pattern that focuses on a center point and a series of circular shapes that revolve around it. Examples of spirals are pinecones, pineapples, hurricanes. The reason for why plants use a spiral form is because they are constantly trying to grow but stay secure  Fibonacci: Leonardo Pisano An Italian mathematician from the Republic of Pisa, who is considered to be "the most talented Western mathematician of the Middle Ages". Sequences of Numbers The Fibonacci Sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,... The next number is found by adding up the two numbers before it. The 2 is found by adding the two numbers before it (1+1) The 3 is found by adding the two numbers before it (1+2), And the 5 is (2+3), and so on!  The Fibonacci Sequence 1; 1; 2; 3; 5; 8; 13; 21; 34; 55; 89; 144; 233; 377; 610; 987; 1597; 2584; 4181; 6765; 10946; 17711; 28657; 46368; 75025; 121393; 196418; 317811; 514229; 832040; 1346269; 2178309; 3524578; 5702887; 9227465; 14930352; … The first 2 numbers are 1 and subsequent numbers are obtained by adding the previous two numbers. The Rule xn = xn-1 + xn-2 Where: Xn is term number "n" Xn-1 is the previous term (n-1) Xn-2 is the term before that (n-2) Makes A Spiral Do you see how the When we make squares squares fit neatly with those widths, we get together? a nice spiral: For example 5 and 8 make 13, 8 and 13 make 21, and so on. Honeycomb: a mass of hexagonal wax cells built by honeybees THE GOLDEN RULE Golden Ratio a special number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part. It is often symbolized using phi, after the 21st letter of the Greek alphabet. In an equation form, it looks like this: a/b = (a+b)/a = 1.6180339887498948420 … phi the ratio of the circumference of a circle to its diameter. usually rounded off to 1.618. Golden mean, the Golden section, divine proportion, can be seen in the architecture of many ancient creations, like the Great Pyramids and the Parthenon The Golden ratio also appears in all forms of nature and science. Some unexpected places include:  Flower petals: The number of petals on some flowers follows the Fibonacci sequence. It is believed that in the Darwinian processes, each petal is placed to allow for the best possible exposure to sunlight and other factors.  Seed heads The seeds of a flower are often produced at the center and migrate outward to fill the space. For example, sunflowers follow this pattern. Pinecones The spiral pattern of the seed pods spiral upward in opposite directions. The number of steps the spirals take tend to match Fibonacci numbers. Tree branches The way tree branches form or split is an example of the Fibonacci sequence. Root systems and algae exhibit this formation pattern.  Shells Many shells, including snail shells and nautilus shells, are perfect examples of the Golden spiral. Spiral galaxies The Milky Way has a number of spiral arms, each of which has a logarithmic spiral of roughly 12 degrees. The shape of the spiral is identical to the Golden spiral, and the Golden rectangle can be drawn over any spiral galaxy. Hurricanes Much like shells, hurricanes often display the Golden spiral. DNA molecules A DNA molecule measures 34 angstroms by 21 angstroms at each full cycle of the double helix spiral. In the Fibonacci series, 34 and 21 are successive numbers. Animal bodies The measurement of the human navel to the floor and the top of the head to the navel is the Golden ratio. But we are not the only examples of the Golden ratio in the animal kingdom; dolphins, starfish, sand dollars, sea urchins, ants and honeybees also exhibit the proportion. Fingers The length of our fingers, each section from the tip of the base to the wrist is larger than the preceding one by roughly the ratio of phi. WHAT ROLE DOES MATHEMATICS PLAY IN OUR WORLD? ❑Mathematics helps organize patterns and regularities in our world. ❑Mathematics helps predict the behavior of nature and phenomena in the world. ❑Mathematics helps control nature and occurrences in the world for our own ends. ❑Mathematics has numerous applications in the world making it indispensable  Logical Classification Structure Sequence Characteristics Precision & Generalization of Accuracy Mathematics Mathematical Applicability language & Abstractness symbolism  Logical Classification Structure Sequence Characteristics Precision & Generalization of Accuracy Mathematics Mathematical Applicability language & Abstractness symbolism  Characteristics of Mathematics Classification generates a series of mental relations through which objects are grouped according to similarities and differences depending on specific criteria such as shape, color, size, etc.  Characteristics of Mathematics Logical Sequence - ideas in mathematics need to flow in an order that makes sense. It means that each step can be derived logically from the preceding steps.  Characteristics of Mathematics Structure- the structure on a particular mathematical set will allow mathematicians to study the set further and find its relationship with other objects.  Characteristics of Mathematics Precision and Accuracy Accuracy how close a measured value is to the actual (true) value. It is the degree to which a given quantity is correct and free from error. Precision is how close the measured valued values are to each other.  Characteristics of Mathematics Abstractness- is the process of extracting the underlying essence of a mathematical concept by taking away any dependence on real- world objects.  Characteristics of Mathematics Symbolism – This language is uniquely constructed in such a way that all mathematicians understand symbolic notations and mathematical formulas.  Characteristics of Mathematics Applicability- The applicability of mathematics can lie anywhere on a spectrum from the completely simple (trivial) to the utterly complex (mysterious). Characteristics of Mathematics Generalizations – The process of finding and singling out in a whole class of similar objects.  A science of A tool subject measures An intellectual Nature of An intuitive game Mathematics method The art of A system of drawing logical conclusions procedure  Nature of Mathematics A science of measures- it is an activity that involves interaction with a concrete system with the aim of representing aspects of that system in abstract terms of “concrete” implies “real”.  Nature of Mathematics Intellectual game- mathematics requires visual imagery. Games and mathematics both require a strong dose of patience, restraint, and concentration.  Nature of Mathematics The art of drawing conclusions- Being able to reason is essential to understanding mathematics. Reasoning is a way to use mathematical knowledge and to generate and solidify new mathematical ideas.  Nature of Mathematics A tool subject – mathematics undeniably a highly powerful instrument of investigation. Mathematics is applied in the fields of engineering, life sciences, industry, and business.  Nature of Mathematics A system of logical procedure – Problem solving is a skill which can enhance our logical reasoning.  Nature of Mathematics An intuitive method – mathematics also requires the use of intuition, the ability to see what is reasonable or not and the ability to pu all these together. ROLES of MATHEMATICS IN SOME DISCIPLINE Physical Science Chemistry Biological Science Engineering and Technology Mathematics and Agriculture Economics Psychology ROLES of MATHEMATICS IN SOME DISCIPLINE Actuarial Science, Insurance and Finance Archeology Logic Music Arts Philosophy Social Network ROLES of MATHEMATICS IN SOME DISCIPLINE Political Science Linguistics Management Computer Geography ROLES of MATHEMATICS IN SOME DISCIPLINE Political Science Linguistics Management Computer Geography Some Famous Mathematicians of Modern Time Pythagoras Some Famous Mathematicians of Modern Time Albert Einstein Some Famous Mathematicians of Modern Time Rene Descartes Some Famous Mathematicians of Modern Time Marie Curie Some Famous Mathematicians of Modern Time Johannes Kepler Some Famous Mathematicians of Modern Time Some Famous Mathematicians of Modern Time Leonardo Pisano “Those laws of nature are within the grasp of the human mind; God wanted us to recognize them by creating us after his own image so that we could share his ownthoughts” - Johannes Kepler  Can you share in class the things that you have learned from the session and its application when you are already in the field?  REFERENCES Adina, E. M. (2018). Mathematics in the modern world. Quezon City : C & E Publishing, Inc. – (FIL QA93 Ea76 2018) Almazan, E.V., Cabrera, M.B., Lapig, E.C., Manuel, P.A., Inoncillo, F.A., Lorenzo, S.O., Magpantay, F. F., & Flores, R.L. (2018). Mathematics in the modern world. Malabon City: Jimczyville Publications. – (FIL QA93 M42 2018) Baltazar, E.C., Ragasa, C. & Evangelista, J. (2018). Mathematics in the modern world. Quezon City : C & E Publishing Inc. – (FIL QA93 B21 2018)