Chapter 1: The Nature of Mathematics PDF

 Nowadays, people are increasingly dependent upon the application of science and technology in doing day-to-day tasks, indeed the role of mathematics has been redefined to cater the needs of the fast-changing society. Mathematics is all around us. Almost every next moment we do the simpl...

 Nowadays, people are increasingly dependent upon the application of science and technology in doing day-to-day tasks, indeed the role of mathematics has been redefined to cater the needs of the fast-changing society. Mathematics is all around us. Almost every next moment we do the simple calculations at the back of our mind. 2  Mathematics reveals hidden patterns that helps us understand the world around us. Nature also embraces mathematics completely. We see so much of symmetry-around us and have a deep sense of awareness and appreciation of patterns. Observe any natural thing and find out symmetry or pattern in it. Change of day into night, summer into wet season, etc. These lessons will help the students understand the nature of mathematics and the language it uses. In addition, these will make students appreciate the beauty and the important role of mathematics in today’s time. 3  Since the beginning of recorded history, mathematical development was at the forefront of any civilized society, and was in use even in the most primitive of cultures. The emergence of mathematics is based on the societal desires. The more advance a society is, the more advanced its mathematical needs are. Primitive tribes needed nothing more than counting, but they also relied on mathematics to measure the sun’s location and the physics of hunting. But what is mathematics?  Mathematics is often defined as “a formal system of thought for recognizing, classifying, and exploiting patterns”. - Ian Stewart (1995)  Mathematics is also described as “the science that deals with the logic of shape, quantity and arrangement”. It is also considered as an art, having an aesthetic and creative side. -(Hom 2013).  Perhaps it is impossible to give a good definition of mathematics in a sentence or two. However, all could possibly agree that mathematics in the modern world is a huge body of knowledge and a very diverse area of study. Whatever your view of mathematics, there is no denying that it expresses itself everywhere, in almost every facet of life – in nature all around us, and in the technologies in our hands. This chapter presents a discussions of mathematics in certain points of view. 10  Mathematics plays a fundamental role in counting and understanding quantities. Counting is a basic mathematical skill that forms the foundation for more advanced mathematical concepts. Here are some key mathematical principles and concepts related to counting: 1.Natural Numbers: Counting usually starts with natural numbers (1, 2, 3,...), which are used to represent quantities of objects or events. 2. Cardinality: The cardinality of a set is the number of elements in that set. Counting involves determining the cardinality of a set. 3. Counting Principles:  The Counting Principle: This principle states that if there are 'm' ways to do one thing and 'n' ways to do another, then there are m * n ways to do both.  For example, if you have 3 choices for a shirt and 4 choices for pants, you have 3 * 4 = 12 outfit combinations.  Permutations refer to the number of ways to arrange a set of objects in a particular order.  For example, the number of ways to arrange 3 books on a shelf is 3! (3 factorial), which is equal to 3 * 2 * 1 = 6.  Combinations refer to the number of ways to select a subset of objects from a larger set without considering the order.  The formula for combinations is often denoted as "n choose k" and is calculated as C(n, k) = n! / (k! * (n - k)!).  For example, the number of ways to choose 2 cards from a deck of 52 cards is C(52, 2). 4. Counting with Multiplication and Addition: In more complex counting problems, you may use multiplication and addition together. For instance, when counting possibilities in a multi-stage process, you often multiply the number of choices at each stage and then add up these possibilities to get the total. 5. Counting in Probability: Counting is crucial in probability theory. The probability of an event occurring often involves counting favorable outcomes over the total number of possible outcomes. This is known as the "probability of an event" For example, let's say we want to calculate the probability of rolling a 6 on a standard die. There are 6 possible outcomes for this event (1, 2, 3, 4, 5, and 6), so the probability of rolling a 6 is 1/6. 6. Counting in Combinatorics: Combinatorics is a branch of mathematics that deals with counting, arrangements, and combinations of objects. It includes topics like permutations, combinations, and the binomial theorem, which all involve counting principles. 7. Counting in Set Theory: In set theory, counting is used to determine the size or cardinality of sets and to compare sets based on their elements. 8. Counting in Algebra and Calculus: Counting can also be related to algebraic and calculus concepts when dealing with discrete and continuous quantities. For instance, you might count the number of solutions to an equation, or integrate a function to find the total count of certain objects within a continuous domain. MATHEMATICS AS A STUDY OF PATTERNS INTRODUCTION In Mathematics, a pattern is a repeated arrangement of numbers, shapes, colours and so on. The Pattern can be related to any type of event or object. If the set of numbers are related to each other in a specific rule, then the rule or manner is called a pattern. Sometimes, patterns are also known as a sequence. Patterns are finite or infinite in numbers. Patterns are regular, repeated or recurring forms or designs. Examples: Layout of floor tiles Designs of buildings The way we tie our shoelaces Studying patterns helps us in identifying relationships and finding local connections to form generalizations and make predictions Rules for Patterns in Mathematics To construct a pattern, we have to know about some rules. To know about the rule for any pattern, we have to understand the nature of the sequence and the difference between the two successive terms.  Repeating – A type of pattern, in which the rule keeps repeating over and over. Growing – If the numbers are present in the increasing form, then the pattern is known as a growing pattern. Example 34, 40, 46, 52, ….. Shrinking – In the shrinking pattern, the numbers are in decreasing form. Example: 42, 40, 38, 36 ….. Types of Patterns: 1. Shape Patterns Logical Patterns Geometric Patterns 2. Letter Patterns 3. Number Patterns 1. Shape Pattern Shape patterns follow a certain sequence or order of shapes, i.e., they are repeated. The shapes can be simple shapes like circles, squares, rectangles, triangles, etc., or other objects such as arrows, flowers, moons, and stars. 32 1. Shape Pattern Logical Patterns- it involves abstract reasoning test that requires flexible thinking, creativity, judgment, ang logical problem solving. It requires examinees to recognize patterns similarities or differences between a given sequence of figures. Geometric Patterns- A motif, pattern, or design depicting abstract, nonrepresentational shapes such as lines, circles, ellipses, triangles, rectangles, and polygons. Patterns made from shapes are similar to patterns made from numbers because the pattern is determined by a rule. 33 EXAMPLE: 2. Letter Pattern A sequence that consists of letters or English alphabets is known as a letter pattern. A letter pattern establishes a common relationship between all the letters.  For example: A, C, E, G, I, K, M… In the above pattern, one letter has been removed after every alphabet. 3. Number Pattern- are patterns in which a list number that follows a certain sequence. Generally, the patterns establish the relationship between two numbers. It is also known as the sequences of series in numbers. The most common type of pattern in mathematics is the number pattern, where a list of numbers follows a certain sequence based on a rule. There are different types of number patterns: Arithmetic Pattern Geometric Pattern Fibonacci Pattern Triangular Number Pattern Square Number Pattern Cube Number Pattern Arithmetic Pattern Another name for arithmetic pattern is algebraic pattern. In such a pattern, the sequences are based on the addition or subtraction of the terms. If we know two or more terms in the sequence, then we can use addition or subtraction to find the arithmetic pattern. Example 1: In the pattern 65, 64, 63, 62, 61, we are subtracting the consecutive numbers by 1 or each number gets decreased by 1. Example 2: In the pattern given below: Each number is getting increased by 5. Geometric Pattern A sequence of numbers that are based on multiplication and division is known as a geometric pattern. If we are given two or more numbers in the sequence, we can easily find the unknown numbers in the pattern using multiplication and division operations. Example 1: Simran has $10 as her savings in the first month. In the second month, her savings doubled. So, now she has 10 x 2= $20. Again, in the third month her savings doubled. Now she has 20 x 2= $40. So the pattern here is $10, $20, $40. Example 2: In the pattern given below, each number is getting divided by 5. 3125, 625, 125, 25, 5  Fibonacci Pattern A sequence of numbers in which each number in the sequence is obtained by adding the two previous numbers together is known as the Fibonacci series or pattern. This sequence starts with 0 and 1. We add the two numbers to get the third number in the sequence. The sequence 0, 1, 1, 2, 3, 5, 8, 13 is the Fibonacci pattern. The pattern that is followed here is 0 + 1 = 1, 1 + 1 = 2 , 1 + 2 = 3 , 2 + 3 = 5, 3 + 5 = 8.  Triangular Number Pattern The representation of the numbers in the form of an equilateral triangle arranged in a series or sequence is known as a triangular number pattern. The numbers in the triangular pattern are in a sequence of 1, 3, 6, 10, 15, 21, 28, 36, 45 and so on. The numbers in the triangular pattern are represented by dots. Also, the pattern can be described as 0 + 1 = 1, 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10, 10 + 5 = 15 and so Square Number Pattern In the square number pattern, each number is a square of consecutive natural numbers. The pattern is written as: ,…= 1, 4, 9, 16, 25… Also, the pattern can be 1 + 3 = 4, 4 + 5 = 9, 9 + 7 = 16, 16 + 9 = 25 and so on. Basically, we add consecutive odd numbers in the sequence. Cube Number Pattern In a cube number pattern, each number is the cube of consecutive natural numbers. The pattern is written as: ,…= 1, 8, 27, 64, 125… The above number patterns are the ones that are commonly used. There are more number patterns. For example: odd number pattern, even number patterns, multiples pattern, etc. Solve and identify the following pattern problems: Example 3: Complete the pattern: AB, BC, CD, DE, ____, ____ Solution: The first term is the combination of first and second alphabets. The second term is the combination of second and third alphabets. The third term is the combination of third and fourth alphabets. The fourth term is the combination of fourth and fifth term. Similarly, the next two terms will be EF and FG.  Example 4: Correct answer: c Explanation: The letters present in 3rd, 6th, 9th...etc (multiples of 3) positions are repeated thrice Example 5: Additional problem excercises SOLVE AND IDENTIFY THE FOLLOWING NUMBER PATTER IF IT IS ARITHMETIC PATTERN, GEOMETRIC PATTERN, FIB PATTERN, TRIANGULAR NUMBER PATTERN,SQUARE NUM PATTERN, & CUBE NUMBER PATTERN. -7, -3, 1, 5, ?. 9A ANSWER:9 & arithmetic pattern 1, 3, 9, 27, 81, ?. 243 ANSWER: 243 & geometric pattern 10, 3, -4, -11, ?. -18 ANSWER: -18 & arithmetic pattern SOLVE AND IDENTIFY THE FOLLOWING NUMBER PATTER IF IT IS ARITHMETIC PATTERN, GEOMETRIC PATTERN, FIB PATTERN, TRIANGULAR NUMBER PATTERN,SQUARE NUM PATTERN, & CUBE NUMBER PATTERN. 3, 8, 6, 11, 9, 14, ?. 14A ANSWER: 12 & arithmetic pattern 1, 5, 16, 34, 59, ?. 91A ANSWER: 91 & arithmetic pattern 80, 40, 120, 60, 180, ?. ANSWER: 90 & geometric pattern 1, 3, 6, 10, ?. ANSWER: 15 & triangular number pattern SOLVE AND IDENTIFY THE FOLLOWING NUMBER PATTER IF IT IS ARITHMETIC PATTERN, GEOMETRIC PATTERN, FIB PATTERN, TRIANGULAR NUMBER PATTERN,SQUARE NUM PATTERN, & CUBE NUMBER PATTERN. 9,16, 25, 36, ?. ANSWER: 49 & square number pattern 27, 64, 125, ?. ANSWER: 216 & cube number pattern 5, 8, 13, 21, ?. ANSWER: 34 & fibonacci pattern WHICH SHAPES COMES NEXT IN THE SEQUENCE? 57 WHICH SHAPES COMES NEXT IN THE SEQUENCE? 58 WHICH SHAPES COMES NEXT IN THE SEQUENCE? 59 WHICH SHAPES COMES NEXT IN THE SEQUENCE? 60 WHICH SHAPES COMES NEXT IN THE SEQUENCE? 61 FIND THE NEXT LETTER PATTERN IN THE SEQUENCE? 62 FIND THE NEXT LETTER PATTERN IN THE SEQUENCE? 63 FIND THE NEXT LETTER PATTERN IN THE SEQUENCE? 64 TRANSLATE THE CODE LANGUAGE: 65 TRANSLATE THE CODE LANGUAGE: 66 WHICH NUMBER SHOULD REPLACE IN QUESTION MARK “?” 4 6 4 ? 1 4 5 8 a. 11 b. 12 7 11 2 1 c. 13 d. 14 12 5 0 2 67  Mathematics and nature are intricately connected, with numerous patterns and sequences found in the natural world that can be described mathematically. These patterns and sequences can be observed in various aspects of nature, from the shapes of plants and animals to the formations of galaxies and the behavior of natural phenomena.  Let's explore some examples of mathematics in nature: 1. Fibonacci Sequence and Golden Ratio: The Fibonacci sequence is a mathematical sequence in which each number is the sum of the two preceding numbers. It is named after Leonardo of Pisa, also known as Fibonacci, who introduced it to the Western world in his book Liber Abaci in 1202. The sequence commonly starts with 0 and 1, although some authors start it with 1 and 1 or 1 and 2. Starting from 0 and 1, the sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. 70  Fibonacci spiral patterns appear in many plants, such as pinecones, pineapples, and sunflowers. The patterns consist of spirals that curve around a surface in both the “sinister” form (clockwise) and the “dexter” form (counterclockwise). 71 Let's explore some examples of mathematics in nature: 1. Fibonacci Sequence and Golden Ratio:  Golden Ratio: The Golden Ratio, approximately 1.618, is often found in natural objects. It's closely related to the Fibonacci sequence and appears in the spiral patterns of shells, galaxies, hurricanes, and even in the proportions of animal bodies. 72 The Golden Ratio The Golden Ratio is an irrational number roughly equal to 1.618 and is symbolized by the Greek letter phi. In this ratio, the digits after the decimal keep going and never end, like 1.61803398874989484820…The purpose of the golden ratio is to create a strong visual through balance and proportion. Parthenon Notre Dame Cathedral Mona Lisa  74 FURBONACCI SEQUENCE PROVES THAT CAT ARE PURRFECT!!! Sample Problem: If you have a wooden board that is 0.75 meters wide, how long should you cut it such that the Golden Ratio is observed? Use 1.618 as the value of the Golden Ratio.  Let's explore some examples of mathematics in nature: 2. Fractals: Fractals are complex geometric shapes that can be split into parts, each of which is a reduced-scale copy of the whole. This property, known as self- similarity, is found in various natural objects such as snowflakes, mountain ranges, lightning bolts, and coastlines. The branching pattern of trees, rivers, and blood vessels also follows fractal geometry. 77 Let's explore some examples of mathematics in nature: 3. Symmetry: Symmetry is a fundamental aspect of many natural forms. For example, bilateral symmetry is common in animals, where the body is divided into two mirror-image halves. Radial symmetry is found in starfish, jellyfish, and many flowers. Crystals and snowflakes exhibit geometric symmetry at the molecular level. 78  A shape or an object has symmetry if it can be divided into two identical pieces. In a symmetrical shape, one-half is the mirror image of the other half. The imaginary axis or line along which the figure can be folded to obtain the symmetrical halves is called the line of symmetry.  In Mathematics, symmetry means that one shape is identical to the other shape when it is moved, rotated, or flipped. If an object does not have symmetry, we say that the object is asymmetrical. The concept of symmetry is commonly found in geometry. There are two kinds of Symmetries: Bilateral symmetry Radial symmetry Bilateral symmetry Bilateral or Reflective symmetry is when the mirrored elements are arranged around a center line. Radial symmetry Radial Symmetry is - is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation. A more common way of describing rotational symmetry is by order of rotation. Radial symmetry The smallest that a figure can be rotated while still preserving the original formation is called the angle of rotation. Angle of rotation = 360º n where n is the order of rotation. Radial symmetry Example Problems: Let's explore some examples of mathematics in nature: 4. Spirals: Spirals are common in nature and can be seen in shells, hurricanes, and galaxies. The logarithmic spiral, where the distance between the turns increases in a consistent ratio, is a mathematical pattern observed in many of these structures. 87  Let's explore some examples of mathematics in nature: 5. Tessellations: Tessellations are patterns made of shapes that fit together without any gaps or overlaps. These are observed in the honeycombs of bees, the scales of fish and reptiles, and the patterns on certain animals like giraffes and leopards. 88  Let's explore some examples of mathematics in nature: 6. Chaos Theory: Chaos theory describes how small changes in initial conditions can lead to vastly different outcomes, a concept famously known as the "butterfly effect." This theory applies to weather patterns, population dynamics, and the behavior of complex systems in nature. 89 INDIVIDUAL ACTIVITY! Provide a picture with yourself that shows some Patterns around you as your background. Instructions: 1. The picture must be 8”x7” in size 2. Print it on an A4 bond paper provided your Name, Course, Year and Section, and the date of submission.  Mathematics is deeply integrated into everyday life, often in ways we might not immediately recognize. Its applications range from simple tasks to complex decision-making processes. 92 Here are some real-life applications of mathematics in various fields: 1. Economics and Finance Mathematics is a core part of economics and finance. Many concepts such as algebra, profit and loss, discounts, statistics, and probability rely on mathematical principles 93 Here are some real-life applications of mathematics in various fields: 2. Engineering and Architecture Mathematics is not only confined to mathematics but also finds applications in engineering and architecture. It is used in fields like civil engineering, structural design, and urban planning to determine the position and orientation of objects and structures. 94 Here are some real-life applications of mathematics in various fields: 3. Physics and Kinematics Mathematics is also applied in physics and kinematics. It helps in analyzing the motion of objects and understanding concepts like velocity, acceleration, and trajectory 95 Here are some real-life applications of mathematics in various fields: 4. Computer Science Mathematics plays a significant role in computer science. It is the underlying code that turns ideas into reality in modern technologies. Mathematical principles ensure the efficiency, variety, and security of computers and computer networks. Concepts like network optimization, queuing theory, and encryption rely on mathematical principles 96 Here are some real-life applications of mathematics in various fields: 5.Medicine and Healthcare Mathematics is used in various aspects of medicine and healthcare. It helps in modeling biological processes, designing effective treatments, predicting the spread of diseases, and analyzing medical data. 97 Here are some real-life applications of mathematics in various fields: Other Applications Mathematics has countless other applications in everyday life. Some examples include weather prediction, music, video games, 3D modeling and animation, crime analysis, and data encryption. 98 END!!! 99