Math Notes PDF

M a t h N O T E S By Sophia Adara L. Toor | Bs Psychology | 1F1 Topic: Mathematics in Our World: Discovering Patterns and Numbers in Nature What Is Mathematics? “Mathematics is the study of pattern—of patterns and structure in numbers, and patterns and str...

M a t h N O T E S By Sophia Adara L. Toor | Bs Psychology | 1F1 Topic: Mathematics in Our World: Discovering Patterns and Numbers in Nature What Is Mathematics? “Mathematics is the study of pattern—of patterns and structure in numbers, and patterns and structure in geometry.” - Keith Devlin 2, 3, 5, 8, 12, 17, 23,... 30 Patterns is always associated in math Any regularity that can be explained mathematically is a pattern. Math is rooted in patterns With patterns we can learn to predict Value of pi 3.1416 ⅓ =.33… A pattern is something that is recurring which could be repetitive and duplicative Shapes and Patterns in Mathematics Circle Set of all points in a plane that are equidistant from a fixed point.  Arc Part of the circumference of a circle. Triangle A closed two-dimensional shape with 3 sides, 3 vertices and 3 angles. Hexagon A closed 2-dimensional shape with 6 sides, 6 vertices and 6 angles. Sphere A 3-dimensional figure with every point on its surface equidistant from its center. Star A type of non-convex polygon Pyramid A polyhedron whose base is a polygon and whose lateral faces are triangles with a common vertex. Symmetry A shape is symmetrical if one-half of it is the mirror image of the other half. Tessellations Patterns made up of one or more shapes with no overlaps and no gaps Fractals Complex geometric shapes that are self-similar on all scales Voronoi A type of tessellation pattern in which a number of points scattered on a plane subdivides in exactly n cells enclosing a portion of the plane that is closest to each point. Spirals a curve which emanates from a point, moving farther away as it revolves around the point. Quiz #1 1. Which shape is most commonly associated with beehives? a. Square b. Hexagon c. Triangle d. Circle 2. Which of the following is an example of a Voronoi pattern in nature? a. Tree branches b. Animal Fur Patterns c. Soap Bubbles d. Spider Webs 3. Which of the following is not typically associated with the concept of Golden Ratio? a. Human Body Proportions b. Nautilus Shell c. Pyramids of Giza d. Equilateral Triangle 4. Which of the following best describes a tessellation in nature? a. The arrangement of petals on a flower b. The honeycomb structure in a beehive c. The spiral in a seashell d. The fractal patterns in a fern leaf 5. Which of the following is an example of symmetry in nature? a. A seashell’s spiral b. The branching of trees c. Butterfly wings  d. A honeycomb’s hexagonal cells Matching Type: Patterns in Nature A B ANSWERS 1. Fibonacci Sequence A. Butterfly Wings 2. Hexagon B. Snail Shell 3. Sphere C. Honeycomb 4. Spiral D. Soap Bubbles 5. Tessellations E. Sunflower Seed Arrangement 6. Voronoi Diagram F. Fern Leaf 7. Golden Ratio G. Turtle Shell 8. Fractal H. Parthenon Structure 9. Symmetry I. Sea Star 10. Triangle J. Pyramid of Giza K. Water droplet Relevance Of Mathematical Patterns To Science, To Life And To My Everyday Life Relevance to Science Natural Phenomena: Mathematical patterns help in explaining natural phenomena. Understanding these patterns allows scientists to predict and explain the structure and behavior of natural systems. Scientific Laws: Many scientific laws and theories are based on mathematical relationships and patterns. For instance, Newton’s laws of motion, Maxwell’s equations in electromagnetism, and the wave functions in quantum mechanics all rely on mathematical formulations. Data Analysis: In scientific research, identifying patterns in data is essential for making discoveries. Statistical methods and algorithms help scientists uncover correlations, trends, and anomalies, leading to new insights and advancements Relevance To Art Aesthetics: Patterns play a significant role in the aesthetics of art. Artists use mathematical concepts such as symmetry, proportion, and geometry to create visually appealing compositions. The Golden Ratio, for instance, has been used in art and architecture to achieve balance and beauty. Design and Architecture: Architects and designers often employ mathematical patterns to create structures that are both functional and aesthetically pleasing. The use of fractals, tessellations, and other geometric patterns can be seen in buildings, textiles, and decorative arts. Music: Music theory is heavily based on mathematical patterns. The arrangement of notes, scales, and rhythms involves mathematical relationships. Relevance to Everyday Life Problem-Solving: Recognizing patterns helps in problem-solving and decision-making. Whether it’s managing finances, planning schedules, or troubleshooting technical issues, understanding patterns can lead to more efficient and effective solutions. Technology: Many technological advancements are based on mathematical principles and patterns. Algorithms, which are at the heart of computer science, rely on pattern recognition to process information, solve problems, and perform tasks such as image recognition, data encryption, and artificial intelligence. Nature and the Environment: Understanding patterns in nature can lead to better environmental management and conservation efforts.  Golden Ratio and Fibonacci Sequence Leonardo Di Pisa (1170 - 1250) - Founder of the Fibonacci Sequence - Fibonacci Rabbit Problem FIBONACCI RABIT PROBLEM: “How many pairs of rabbits can be produced in a year if each pair of rabbits produces a new pair of rabbits every month, and rabbits take one month to mature?” The number of rabbit pairs each month follows the Fibonacci sequence: 1, 1, 2, 3, 5, 8, and so on. Each number in the sequence is the sum of the two preceding numbers. Overview of Fibonacci's Rabbit Problem  Initial Setup: The problem begins with one pair of newborn rabbits placed in a pen. The assumptions are that each pair produces one male and one female offspring monthly, can reproduce starting at one month old, and never die. Monthly Growth Calculation Month 1: Start with 1 pair (newborns). Month 2: Still 1 pair (too young to reproduce). Month 3: 2 pairs (the original pair reproduces). Month 4: 3 pairs (the original pair and their first offspring). Month 5: 5 pairs (three pairs can now reproduce). Continuing this pattern, the sequence follows the Fibonacci numbers: 1, 1, 2, 3, 5, 8, etc. Fibonacci Sequence Each number in the sequence is the sum of the two preceding numbers: 1+1=2 1+1=2 1+2=3 1+2=3 2+3=5 2+3=5 The Fibonacci Ratio, referred to as the Golden Ratio, is the limit of the ratio of successive Fibonacci numbers as the sequence progresses. Approximately equal to 1.6180344 or the Greek Symbol, Phi ɸ To solve it, divide each Fibonacci number by the preceding one, and as you move further along the sequence, these ratios converge to the Golden Ratio. Just divide each number to the one before it. Divide the bigger no. to the smaller no Golden Angle The Golden Angle is a specific angle that arises from the Golden Ratio, approximately 137.5 degrees. The Golden Angle happens when you break up a circle.  In sunflowers, the arrangement of seeds follows this angle to achieve the most efficient packing and optimal space usage. This pattern allows for the most seeds to fit in a given area and ensures that each seed has a uniform distribution around the center of the flower Binet Formula n= nth term (the term you are looking for) ɸ (Phi)- Golden Ratio (1.6180344) Fn= (1.6180344^n - (1-1.6180344)^n / square root of 5 Fn= (ɸ^n-(1-ɸ)^n/ square root of 5 The Golden Ratio: Concept and Applications The Golden Ratio, often denoted by the Greek letter Phi (φ), is an irrational number approximately equal to 1.618. This ratio has been celebrated throughout history for its aesthetic appeal and is frequently encountered in art, architecture, and nature. Applications in Art The Golden Ratio has been extensively utilized by artists to create compositions that are visually pleasing. Here are some notable applications: Famous Works: Artists like Leonardo da Vinci and Michelangelo employed the Golden Ratio in their masterpieces. For instance, in The Last Supper, Da Vinci structured key dimensions according to this ratio, creating a sense of balance and focus. Techniques: Artists often use techniques such as the Phi Grid and Golden Spiral to guide their compositions. The Phi Grid divides a canvas into sections that align with the Golden Ratio, while the Golden Spiral helps in placing elements harmoniously within a work. Visual Appeal: The use of the Golden Ratio helps artists position focal points effectively, enhancing viewer engagement. The placement of subjects along  intersecting lines derived from this ratio can create a natural flow within the artwork. Examples of Artworks Utilizing the Golden Ratio Artist Artwork Application of Golden Ratio Leonardo da Vinci Mona Lisa Focal points aligned with φ Michelangelo The Creation of Adam Proportions based on φ Georges Seurat Various Paintings Composition guided by spirals Presence in Nature The Golden Ratio is also prevalent in nature, often seen in various biological structures: Floral Patterns: The arrangement of leaves around a stem or the pattern of seeds in sunflowers often follows this ratio, optimizing light exposure and space. Animal Structures: The shells of nautilus and certain animal proportions exhibit growth patterns that align with φ, demonstrating efficiency and beauty in design. Contribution to Perception of Beauty The association of the Golden Ratio with beauty stems from its frequent occurrence in both art and nature. This ratio creates a sense of balance and harmony that resonates with human perception. Psychological studies suggest that people tend to find objects or compositions that adhere to this ratio more aesthetically pleasing compared to those that do not Introduction to languages and symbol in mathematics Mathematics is similar—it uses its own language and symbols to convey ideas and solve problems. What is a language? A systematic means of communicating by the use of sounds or conventional symbols. A structured system of communication that uses symbols, sounds, or signs to convey meaning. A set (finite or infinite) of sentences, each finite in length and constructed out of a finite set of elements. Components of Language A vocabulary of symbols or words A grammar consisting of rules of how these symbols may be used A syntax or propositional structure, which places the symbols in linear structures. A Discourse or narrative' consisting of strings of syntactic propositions. A community of people who use and understand these symbols. Importance of Languages and Symbols in Mathematics For comprehending mathematical idea To solve complex equations Demonstrates understanding and enables effective teaching. It's the bridge to share and discuss math concepts Fundamental to understanding concepts Enhances overall cognitive development and academic success. Mathematical Language Expressions Sentences Analogy of the Expression in the English Language and in Mathematics Language Expressions English: Does not express a complete thought Ex: Ricardo, Makati City, book Mathematics: A mathematical expression is a combination of numbers, variables, operators (such as +,-, x, +), and sometimes functions that represent a value or a quantity. Examples: 12.9, 9+2, 1/3, 3x+53x+5 2(a2+b2)2(a2+b2) 16-4 Sentence English: A complete thought Ex. The capital of the Philippines in Manila Mathematics: is a complete thought that can be either true or false. It often includes an equality or Inequality sign and makes a claim about the relationship between two expressions. Examples: 11+3=14 -3+7=4 3x+2 = 20 (true depending on the value of x) x-760 Key Differences (Math Expressions and Sentences) Nature: An expression [does not] assert a truth value; it simply represents a value. In contrast, a sentence asserts a relationship that can be evaluated as true or false. Components: Expressions can be composed of various mathematical elements but do not include relational operators like equals (=), greater than (>), or less than (< 5. Conjunction — Logical Connections 5.1. If-then, and/or 6. Adverb — the verb equal to equally 6.1. = 7. Preposition — Symbols that represents relationships 7.1. >, 0 } Some people use ":" instead of "|", so they write { x : x > 0 } How do you read this? A = {x | x ∈ N, 5 < x < 10} "set A is the set of all ‘x’ such that ‘x’ is a natural number between 5 and 10. B = { x | x is a two-digit odd number from 9 to 21} B contains all the odd numbers from 9 to 21 OR B contains all the odd numbers from 9 to 21 What are the different types of set? 4. Empty or Null Set Is a set that has no element. 5. Equivalent Sets Are sets that have the same number of elements. Meaning they have the same cardinality. Note. Cardinality(n) or cardinal number of a set is the number of elements of a set. A subset is a set that contains some or all elements of another set. Key Points: If A is a subset of B, every element in A is also in B. This is denoted as A⊆B A⊆B. The empty set (denoted as ∅) and the set itself are always subsets of any set. A set can have multiple subsets, including proper subsets, which contain at least one element but are not identical to the original set. Example: If we have a set B={1,2,3} B={1,2,3}, then: A={1,2} A={1,2} is a subset of B B (proper subset). C={1,2,3} C={1,2,3} is also a subset of B B (not a proper subset). D=∅ D=∅ is a subset of B. In summary, subsets are fundamental concepts in set theory that help us understand the relationships between different sets. 6. Equal Sets Are sets that have exactly the same elements. 7. Power Set It is the set composed of all the subsets of a given set. Example A = { 3, 5, 7 }, the power set of A denoted byP(A) is P(A) ={∅,{3}, {5}, {7}, {3,5}, {3,7},{5,7}, (3,5,7} 8. Joint Sets These are sets with common elements 9. DIsjoint Sets These are sets with NO common elements 10. Universal Set The set containing all possible elements for a particular discussion, denoted as U. The notations/symbols? ∈ - “an element of” ∉ - “not an element of” ⊆ - Equal Set used to indicate that equal sets are a subset of one another Example: A = { l, o, v, e } B = { e, l, v, o } A ⊆ B and B ⊆ A ⊄ “Not a subset of” used to indicate that equal sets are a subset of one another Symbols N represents natural numbers or all positive integers. W represents whole numbers. Z indicates integers. Q represents rational numbers or any number that can be expressed as a fraction of integers. R represents real numbers or any number that isn't imaginary. Operations Venn Diagram An illustration that uses overlapping circles to show the logical relationship between two or more sets of items Union (A ∪ B) The set of all elements that belong to set A, set B, or both. Example: A = {1, 2, 3}, B = {3, 4, 5}, A ∪ B = {1, 2, 3, 4, 5}. Intersection (A ∩ B) The set of elements that belong to both set A and set B Example: A = {1, 2, 3}, B = {3, 4, 5}, A = {1, 2, 3}, B = {3, 4, 5} Difference (A - B) The set of elements that belong to set A but NOT to set B. Example: A = {1, 2, 3}, B = {3, 4, 5} A - B = {1, 2}. Complement (A') The set of all elements in the universal set that are not in set A. Example: If U = {1, 2, 3, 4, 5, 6}, A = {1, 2, 3} then A' = {4, 5, 6} Cartesian Product A x B the Cartesian product A × B between two sets A and B is the set of all possible ordered pairs with the first element from A and the second element from B. Example: A = {brown, green, yellow} B = {red, blue, purple}, (brown, red), (brown, blue), (brown, purple), (green, red), (green, blue), (green, purple), (yellow, red), (yellow, blue), (yellow, purple) Cartesian Product A x B Example: A = {brown, green, yellow} B = {red, blue, purple}, (brown, red), (brown, blue), (brown, purple), (green, red), (green, blue), (green, purple), (yellow, red), (yellow, blue), (yellow, purple)