Nature of Mathematics PDF

1 NATURE OF MATHEMATICS experiment, discover, and recreate. On the other Mathematics is a formal system of thought that hand, mathematics is an art and a process of was gradually developed in the human mind thinking. For it involves reas...

1 NATURE OF MATHEMATICS experiment, discover, and recreate. On the other Mathematics is a formal system of thought that hand, mathematics is an art and a process of was gradually developed in the human mind thinking. For it involves reasoning, which can be and evolved in the human culture. inductive or deductive, and it applies methods of proof both in fashion that is conventional and From their realization, a system of thought further unventional. advanced their knowledge into understanding measures. They were able to gradually develop MATHEMATICS IS EVERYWHERE the science of measures and gained the ability We use mathematics in our daily tasks and to count, gauge, assess, quantify, and size almost activities. It is our important tool in the field of everything. sciences, humanities, literature, medicine, and even in music and arts; it is in the rhythm of our MATHEMATICS AS A TOOL daily activities, operational in our communities, Mathematics, as a tool, is immensely useful, and a default system of our culture. practical, and powerful. It is all about forming new ways to see problems so we can understand There is mathematics wherever we go. It helps us them by combining insights with imagination. It cook delicious meals by exacting our ability to also allows us to perceive realities in different measure and moderately control heat. It also contexts that would otherwise be intangible to helps us to shop wisely, read maps, use the us. It can be likened to our sense of sight and computer, remodel a home with constrained touch. budget with utmost economy Mathematics is our sense to decipher patterns, ESSENTIAL ROLE OF MATHEMATICS relationships, and logical connections. It is our It allows us to learn and understand the natural whole new way to see and understand the order of the world modern world. It is asserting that mathematics is a powerful tool in decision making and it is a way Mathematics helps us to take the complex of life processes that are naturally occuring in the world around us and it represents them by utilizing logic MATHEMATICS IS A/AN: to make things more organized and more Study of patterns efficient. Art Language MATHEMATICAL LANDSCAPE Set of problem solving tools The human mind and culture developed a Process of thinking conceptual landscape for mathematical thoughts and ideas to flourish and propagate. It provides answers to existing questions and presents solutions to occurring problems. It has The wind in this landscape is unpredictable, that the power to unveil the reasons behind the rate of change of the rate of change of occurrences and it offers explanations weather is known as calculus. And beneath the surface of this mathematical landscape are Moreover, mathematics, as a study of patterns, firmly woven proofs, theorems, definitions, and allows people to observe, hypothesize, axioms which are intricately “fertilized” by reasoning, analytical, critical thinking and mathematical pattern of a storm to avoid or germicide by mathematical logic that made prevent catastrophes. We want to know the them precise, exact and powerful mathematical concept behind the contagion of the virus to control its spread. We want to The numbers in mathematics is not a "thing" but a understand the unpredictability of cancer cells to process. combat it before it even exists HOW MATHEMATICS IS DONE? 1.2 MATHEMATICS IN OUR WORLD Math is a way of thinking, and it is undeniably DIFFERENT KINDS OF PATTERN important to see how that thinking is going to be developed rather than just merely see face value 1 PATTERN OF VISUAL - are often unpredictable, of the results. never quite repeatable, and often contain fractals. For most, mathematics is just nothing but something to survive, rather than to learn - can be seen from the seeds and pinecones to the branches and leaves. They are also visible in self-similar MATHEMATICS IS FOR EVERYONE replication of trees, ferns, and plants There is mathematics we call pure and applied, throughout nature. as there are scientists we call social and natural. There is mathematics for engineers to build, 2 PATTERN OF FLOW - The flow of liquids provides mathematics for commerce and finance, an inexhaustible supply of nature’s patterns. mathematics for weather forecasting, Patterns of flow are usually found in the water, mathematics that is related to health, and stone, and even in the growth of trees. There is mathematics to harness energy for utilization. To also a flow pattern present in meandering rivers simply put it, everyone uses mathematics in with the repetition of undulating lines. different degrees and levels. 3 PATTERN OF MOVEMENT - In the human walk, Everyone uses mathematics, whoever they are, the feet strike the ground in a regular rhythm: the wherever they are, and whenever they need to. left-right-left-right-left rhythm. When a horse, a From mathematicians to scientists, from four-legged creature walks, there is more of a professionals to ordinary people, they all use complex but equally rhythmic pattern. This mathematics. It helps us become better persons prevalence of pattern in locomotion extends to and helps make the world a better place to live the scuttling of insects, the flights of birds, the in. pulsations of jellyfish, and also the wave-like movements of fish, worms, and snakes. IMPORTANCE OF MATHEMATICS Mathematical training is vital to decipher the 4 PATTERN OF RHYTHM - is conceivably the most clues provided by nature. But the role of basic pattern in nature. Our hearts and lungs mathematics goes clues and it goes beyond follow a regular repeated pattern of sounds or prediction. Once we understand how the system movement whose timing is adapted to our works, our goal is to control it to make it do what body’s needs. Many of nature’s rhythms are most we want. We want to understand the likely similar to a heartbeat, while others are like breathing. The beating of the heart, as well as SPIRAL - The spiral patterns exist on the scale of breathing, have a default pattern. the cosmos to the minuscule forms of microscopic animals on Earth. The Milky Way that 5 PATTERN OF TEXTURE - A texture is a quality of a contains our Solar System is a barred spiral galaxy certain object that we sense through touch. It with a band of bright stars emerging from the exists as a literal surface that we can feel, see, center and running across the middle of it. Spiral and imagine. Textures are of many kinds. It can patterns are also common and noticeable be bristly, and rough, but it can also be smooth, among plants and some animals. Spirals appear cold, and hard in many plants such as pine cones, pineapples, and sunflowers. On the other hand, animals like 6 GEOMETRIC PATTERN - is a kind of pattern which ram and kudu also have spiral patterns on their consists of a series of shapes that are typically horns. repeated. These are regularities in the natural world that are repeated in a predictable SYMMETRIES - In mathematics, if a figure can be manner. Geometrical patterns are usually visible folded or divided into two with two halves that on cacti and succulents.. are the same, such a figure is called a symmetric figure. Used to classify and organize information PATTERNS FOUND IN NATURE about patterns by classifying the motion or deformation of both pattern structures and WAVES AND DUNES - A wave is any form of processes. disturbance that carries energy as it moves. Waves are of different kinds: mechanical waves KIND OF SYMMETRY which propagate through a medium ---- air or 1. REFLECTION - also called line symmetry or water, making it oscillate as waves pass by. Wind mirror symmetry. Left pattern is same as waves, on the other hand, are surface waves the right half rotation that create the chaotic patterns of the sea. 2. Translations - Translational symmetry Similarly, water waves are created by energy exists in patterns that we see in nature passing through water causing it to move in a and in man-made objects. Translations circular motion. Likewise, ripple patterns and acquire symmetries when units are dunes are formed by sand wind as they pass repeated and turn out having identical over the sand. figures, like the bees’ honeycomb with hexagonal tiles. SPOTS AND STRIPES - We can see patterns like 3. Rotation - also called rotational spots on the skin of a giraffe. On the other hand, symmetry. Object still looks the same stripes are visible on the skin of a zebra. Patterns after rotation like spots and stripes that are commonly present in different organisms are results of a The degree of rotational symmetry of an object is reaction-diffusion system (Turing, 1952). The size recognized by the number of distinct orientations and the shape of the pattern depend on how in which it looks the same for each rotation. fast the chemicals diffuse and how strongly they interact. Ex. 1. 7- fold symmetry = 51.43° 2. 13-fold symmetry = 27. 69° SYMMETRIES IN NATURE refers to an ordered list of numbers called terms 1. Bilateral - It can be divided into two identical that may have repeated values. The halves. arrangement of these terms is set by a definite 2. Radial – Divided into many parts rule. 3. Symmetry of Motion - The symmetry of motion is present in animal movements. E.g. 4. Wallpaper Symmetry - This kind of symmetry is A, C, E, G, I created when a pattern is repeated until it 3, 1, -1, -3 5, 10, 20, 40 covers a plane. 1, 1/2, 1/3, 1/4 3, 6, 12, 24, 48, 96 Snow Flakes – 6-fold radial symmetry 1, 10, 100,1000 100,10,1,0.1 Star fish – 5 fold radial Symmetry 144,24,4,⅔ 2 FIBONACCI SEQUENCE AND OTHER TYPE OF SEQUENCE THESE SEQUENCES ARE OBSERVABLE IN SOME FLOWER PETALS, ON THE SPIRALS OF SOME SHELLS AND EVEN ON SUNFLOWER SEEDS. SEQUENCE DIFFERENT KIND OF SEQUENCE KIND OF SEQUENCE DESCRIPTION EXAMPLE FORMULA ARITHMETIC SEQUENCE A sequence of numbers that 2, 4, 6, 8, 10, an=a1+(n−1)d follows a definite pattern. 12, 14, 16, 18, 20…. an - nth term Check the difference between two a1 - First term consecutive terms. If common n - nth term Common difference is observed, then d - common difference: +2 definitely arithmetic sequence difference governed the pattern GEOMETRIC SEQUENCE If in the arithmetic sequence we an​=a1​(r)(n−1) need to check for the common difference, in the geometric an - nth term sequence we need to look for the a1 - First term common ratio. r - common ratio n - nth term  HARMONIC SEQUENCE In the sequence, the reciprocal of 1/5, 1/10, the terms behaved in a manner 1/15, 1/20 like an arithmetic sequence. The pattern of Harmonic sequence is the reciprocal, which appears like an arithmetic sequence moon difference; in geometric sequence we need to look for the common ratio. FIBONACCI SEQUENCE Perfect illustration of nature’s beauty Standard for beauty and aesthetics because it seems to appear often in nature ( some petals and leaves of plants), but cannot be considered as universal property. The ratio of its preceding term to previous term will lead to the GOLDEN RATIO It was first tackled in the ELEMENTS of Euclid, 2300 years ago Also called as Golden Mean, Golden Section, Divine Proportion One of the famous irrational number This ratio appear in natures and in various form of art The golden Ratio could be the key to unlocking the divine secrets of the universe Some naturally occuring example that exhibits the golden ratio includes: SUNFLOWER SEA SHELLS HURRICANE GALAXIES HUMAN EAR The amazing grandeur of the Fibonacci sequence was also discovered in the structure of GOLDEN RECTANGLE. The Golden Rectangle is made up of squares whose sizes, surprisingly, also behave similar to the Fibonacci sequence. Frequently found in art and architecture as a rectangular shape that seems “RIGHT” to the eye.  SOME APPLICATION OF THE GOLDEN RATIO: MONA LISA VITRUVIAN MAN (Proportion of the Human body by Leonardo Davinci) EIFFEL TOWER PARTHENON PYRAMID OF EGYPT FORMULAS AND SEQUENCE BINET'S FORMULA FIBONACCI SEQUENCE F(n) = F (n-1) + F (n-2) LIST OF FIBONACCI NUMBERS https://miniwebtool.com/list-of-fibonacci-num bers/ 3 MATHEMATICS AS A LANGUAGE symbols. It is a system used to communicate Mathematics is a language itself, It is very mathematical ideas. The language of essential in communicating important ideas. But mathematics makes it easy to express the kind of most mathematical language is in the form of thoughts that mathematicians like to express. CHARACTERISTICS OF LANGUAGE OF MATHEMATICS EXPRESSION PRECISE - It can be stated clearly; able make Sum of two numbers: x + y very fine distinctions - a group of number or variable with or without mathematical operation Set of positive numbers: {1, 2, 3, 4, 5,...} - All do not state complete thought EQUATION CONCISE - It can be stated briefly Sum of two numbers is 8: x + y = 8 - a group of number or variable with or The sum of of 2 and a number (x+2) “A regular without mathematical operation pentagon is a polygon with 5 equal sides and separated by an equal sign angles” MATHEMATICAL SENTENCE is the analog of an English sentence; it is a POWERFUL - It is capable of expressing complex correct arrangement of mathematical symbols ideas into simpler forms; It can express complex that states a complete thought. It makes sense to thoughts with relative ease ask about the TRUTH of a sentence: Is it true? Is it false? Is it sometimes true/sometimes false? E.g. PARTS OF 1. The capital of the Philippines is Manila 2. Rizal Park is in Cebu SPEECH 3. 5 + 3 = 8 4.5 + 3 = 9 FOR MATHEMATICS Numbers CONVENTION IN MATHEMATICAL LANGUAGE First symbols that can be used to represent The following are standard symbols used as quantity. These are nouns (objects) in the English conventions in mathematics: language. Operation symbols +,,^, and v can act as connectives in a SYMBOL NAME SYMBOL NAME mathematical sentence. Composition Union Relation symbols ∘ of function ∪ =, ≤, and are used for comparison and act as = Equal verbs in the mathematical language sign ∩ Intersection Grouping symbols (),{}, (), {}, and [] are used to associate groups of numbers and operators. ≠ Not equal to ∈ Element Variables > Greater Not an Letters that represent quantities and act as than ∉ element of pronouns < Less PHRASE - A group of words that expresses a than {} A set of concept ≥ Greater Subset SENTENCE - A group of words that are put than or ⊆ together to mean something. equal to ≤ Less Not a l Such ≡ Congruence/Equ than or ⊄ subset of that ivalent equal to End of a, b, Variables ∎ *First part of * Binary … Ellipses Proof c,.... z English Alphabet operatio uses as fixed variable* n (lower case) *Middle part of English alphabet Therefor ^ Conjunction ∴ e use as subscript and subscript variable* Composition Disjunction ∘ of function ∨ *Last part of an ENglish alphabet uses as unknown variable* = Equal sign ~ Negation ≠ Not equal to → If-then statements 4 FOUR BASIC CONCEPTS IN MATHEMATICS CLASSIFICATION OF NUMBERS: > Greater RATIONAL NUMBERS: 3.64, ⅝, 0.42 ↔ If and only than If IRRATIONAL NUMBERS: √2 √5 π INTEGERS 20, -8, 1, 0, 2, -17 < Less For all than ∀ WHOLE NUMBERS: 3, 0, 4, 11 NATURAL NUMBERS: 1, 12, 7, 10 COMPLEX NUMBERS: 3-4i, 7+2i, -i√4, √3-5I ≥ Greater There exist than or ∃ equal to 1 SET AND ITS BASIC OPERATIONS BASIC IDEA OF SETS ≤ Less than or * Binary operation SET - is a collection of objects which are clearly equal to defined as belonging to a well defined group. ELEMENT - Each object in a setELEMENT - Each N0 No natural N1 natural numbers/w object in a set numbers hole 2 SET NOTATION - way of describing a set /whole numbers numbers set (without TWO MAIN METHODS OF SET NOTATION set (with zero) 1. Rule Method or Set Builder Notation zero) 2. Roster Method or Listing Method Z integer numbers Q real numbers 1 RULE METHOD A = {X|X is a counting number from 1 - 5} B = {X|X is a month that starts with set set letter A} C = {Prime factor of 15} R rational numbers C complex numbers 2 Roster or Listing Method A = {1, 2, 3, 4, 5} B = set set {April, August} C = {3, 5} e.g. 1. A = { X|X is a letter in the word subtract} A = { The Complement of a Set S, U, B, T, R, A, C, T} 2. B = {X|X is a counting number greater than 8} B = {9, 10, 11, 12,..} 4 SYMBOLS INVOLVING SETS OTHER TERMINOLOGIES INVOLVING SETS UNIVERSAL SET- a set that contains everything or all things NULL OR EMPTY SET- a set with no elements Symbolized by A' is the set of all elements in the FINITE SET- sets having finite or exact list of universal set that are not in A. This idea can be elements expressed in set-builder notation as follows: A'= {x INFINITE SET- set where elements canot be | x ∈ U and x & A) counted (...) The cardinality of a set is defined as the number The shaded region represents the complement of of elements in a mathematical set. It can be set A. finite or infinite. E.g. For example, the cardinality of the set A = {1, 2, 3, 4, 5, 6} is equal to 6 because set A has six elements. SPECIAL SET PROPER SUBSET- subset contain missing elements IMPROPER SUBSET- subset is the set itself EQUAL SET- if each member of one set is also a member of other A = B EQUIVALENT SET - If two sets have same member of elements |A| = |B| OPERATION OF SET UNIVERSAL SET AND VENN DIAGRAM The universal set is a general set that contains all elements under discussion. Universal set is represented by a rectangle Subsets within the universal set are depicted by circles, or sometimes ovals or other shapes LONG QUIZ ITEM PROBLEM ANSWER 1 Find the 11th term of the geometric sequence with the first term of 9 and the 87 890 625 common ratio of 25 2 X= {m, a, t, h, s} {} Y = {m, o, d, e, r, n} What is X ∩ Y ∩ Z? 3 Golden Ratio What do you call to the ratios of successive Fibonacci numbers approach the number 1.618? 4 A-B 5 What is the next two terms in the sequence AZ, CX, EV, GT, _____, _____? IR, KP 6 Viola Makes gift basket for Valentine’s Day, She has 13 baskets Left over from last 193 Baskets year, and she plans to make 12 more each day. If there are 15 working days until the day she began to sell the baskets, How many baskets will she have to sell? 7 He is an Italian mathematician who was better known by his nickname Fibonacci. Leonardo of Pisa He is also the proponent of Fibonacci sequence. 8 X= {m, a, t, h, s} {a, t, h, s, w, l} Y = {m, o, d, e, r, n} Z = {w, o, r, l, d} What is (X ∪ Z) – Y? 9 Given that set A = {3, 5, 7, 9}, B = {2, 4, 6, 8, 10, 12} and C = {2, 4, 8, 12}, which of the C is a subset of B following is correct? 10 The human mind is the place where mathematical thoughts and ideas are flourish Mathematical Landscape and propagate. It is capable of constructing and discerning the deepest insights being perceived from the natural world. 11 Given A = {1, 3, 6, 8, 9, 12, 15} and B = {6, 9, 12}, which of the following is TRUE? B is a subset of A 12 This is a symmetry where in if you rotate an image, you can still achieve the same Rotational Symmetry appearance as the original position. 13 B' 14 {1, 2, 3, 5, 7, 9} U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {4, 6, 8, 10} B = {1, 3, 5, 7, 9} Which of the following sets is a complement of set A? 15 The correct mathematical symbol translation for “The product of the sum and ( a + b) (a - b) = a² difference of two numbers is the difference of the square of two numbers” is: - b² 16 Determine the angle of rotation of an object with 18-fold symmetry. 20 degrees 17 Which of the following is the correct translation to an English expression for a The sum of the cube of two mathematical expression: x ³ + y ³ ? numbers 18 Which term of the sequence 5, 3, 1, -1,…. Is -29? 18th 19 Mathematics can provides answers to existing questions and presents solutions to Mathematics as a tool occurring problem. In can help in decision making and a way of life 20 Find the 50th term of the sequence 5, -2, -9, -16, …… -338 21 What is the next term in the sequence A5, D25, G125, J625, M3125? P15625 22 Identify the domain of the function. f(x) = 2/x+3 x = -3 23 All women are mortal. Deductive Gregoria De Jesus is a woman. Therefore, Gregoria De Jesus is a mortal. What kind of reasoning is shown in the given statements? 24 Nature of mathematics as a language that pertains to the ability to say things Concise briefly. 25 How many degrees are between the hands of a clock at 3:40? 130° 26 Which of the following will give the same value as a(b+c-d)? ab+ac-ad 27 Particular order in which related numbers follow each other. Sequence 28 Translate the expression x - 3^2 into an English sentence. Square of three less than a number 29 Nature of mathematics as a language that pertains to express complex thoughts Powerful with relative ease/into simpler forms. 30 Which of the following is a relation but not a function? {(1,2), (1,3), (1,4), (1,5)} 31 32 Solve for x. A number greater than three is five. 2 33 For the relation {(8,11),(3,5),(6,17),(x,22)} to be a function, which value of x can be 22 used? 34 Translate the sentence “Half of a number is the sum of five and that number.” into x/2 = 5+x a mathematical equation. 35 The translation of sentences to mathematical equations shows what nature of Concise mathematics as language. 36 Let f(x) = 4x – 1 and g(x) = x – 3. f[g(x)] = -5 Perform f(2) o g(2). 37 Mathematical Symbol for the sum / summation of numbers ∑ 38 Which of the following is CLOSEST to the square root of 4000? 63 39 What is the least common multiple of 5,2 and 7? 70 40 Repeated arrangement of numbers, shapes, colors and so on Pattern 41 Nature of mathematics as a language that pertains to the ability to make very fine Precise distinctions. 42 Solve for x. Twice a number increased by eight is greater than ten. x>1 43 The commutative property is shown in _______. 2+5 = 5+2 44 What comes next based on the given name sequence? DranneL ErnestO, OtsenrE, IsraeL, LearsI, LennarD, ______ 45 Through inspection, what would be the next number on the sequence 1, 14, 51, 679 124, 245, 426…? 46 Any set of ordered pair. Relation 47 It is the process of reaching a general conclusion by examining specific examples. Inductive Reasoning 48 What word comes next; are, era, was, saw, war, _____? raw