CHAPTER 1: The Nature of Mathematics Daniel Bezalel A. Garcia Instructor I, Pangasinan State University - Urdaneta City Campus September, 2020 Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 1 Introduction Introduction Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 2 Introduction Introduction The coming of the ”Digital Age” contributes to our consumption and production of data. Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 3 Introduction Introduction The coming of the ”Digital Age” contributes to our consumption and production of data. In this fast-paced society, how often have you stopped and thought to appreciate the beauty of the things around you? Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 4 Introduction Introduction The coming of the ”Digital Age” contributes to our consumption and production of data. In this fast-paced society, how often have you stopped and thought to appreciate the beauty of the things around you? Have you every thought and pondered about the underlying principles that govern the universe? Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 5 Introduction Introduction The coming of the ”Digital Age” contributes to our consumption and production of data. In this fast-paced society, how often have you stopped and thought to appreciate the beauty of the things around you? Have you every thought and pondered about the underlying principles that govern the universe? Have you ever paused and pondered about the processes and mechanisms that make our lives easier and comfortable? Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 6 Introduction Introduction The coming of the ”Digital Age” contributes to our consumption and production of data. In this fast-paced society, how often have you stopped and thought to appreciate the beauty of the things around you? Have you every thought and pondered about the underlying principles that govern the universe? Have you ever paused and pondered about the processes and mechanisms that make our lives easier and comfortable? As rational creatures, we tend to identify and follow the patterns, whether consciously or subconsciously. Through these processes humans are able to survive up to this era. Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 7 Introduction Introduction The coming of the ”Digital Age” contributes to our consumption and production of data. In this fast-paced society, how often have you stopped and thought to appreciate the beauty of the things around you? Have you every thought and pondered about the underlying principles that govern the universe? Have you ever paused and pondered about the processes and mechanisms that make our lives easier and comfortable? As rational creatures, we tend to identify and follow the patterns, whether consciously or subconsciously. Through these processes humans are able to survive up to this era. In this chapter, we will look at the patterns and regularities in the world, and how mathematics played a huge part , both in nature and human endeavors. Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 8 Patterns and Numbers in Nature and the World What is Pattern? Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World What is Pattern? Pattern A pattern or patterns, are regular, repeated, or recurring forms or designs. Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World What is Pattern? Pattern A pattern or patterns, are regular, repeated, or recurring forms or designs. Studying these patterns enables us to identify relationships and to find logical connections. Through these we are able to form generalizations and make predictions. Example 1.1 What do you think will be the next emoji in the sequence above? Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 11 Patterns and Numbers in Nature and the World Other Examples: Logical Reasoning Test Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Other Examples: Logical Reasoning Test Example 1.2 Source: http://www.graduatewings.co.uk/how-to-improve-at-logical-reasoning Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Other Examples: Logical Reasoning Test Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 14 Patterns and Numbers in Nature and the World Other Examples: Logical Reasoning Test Example 1.3 Source: http://afppracticeexams.com.au/abstract-reasoning-exam-tips/ Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 15 Patterns and Numbers in Nature and the World Other Examples: What Number Comes Next? Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 16 Patterns and Numbers in Nature and the World Other Examples: What Number Comes Next? Example 1.3 What number comes next in 1, 3, 5, 7, 9, Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). ? Mathematics in the Modern World 17 Patterns and Numbers in Nature and the World Other Examples: What Number Comes Next? Example 1.3 What number comes next in 1, 3, 5, 7, 9, ? Answer: 11 Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Other Examples: What Number Comes Next? Example 1.3 What number comes next in 1, 3, 5, 7, 9, ? Answer: 11 Example 1.4 1, 8, 27, 64, 125, Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). ? Mathematics in the Modern World 19 Patterns and Numbers in Nature and the World Other Examples: What Number Comes Next? Example 1.3 What number comes next in 1, 3, 5, 7, 9, ? Answer: 11 Example 1.4 1, 8, 27, 64, 125, ? Answer: 216 Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Line Symmetry Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 21 Patterns and Numbers in Nature and the World Patterns in Nature: Line Symmetry Line/Bilateral Symmetry This type of symmetry indicates that you can draw an imaginary line across an object and the resulting parts are mirror images of each other. Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 22 Patterns and Numbers in Nature and the World Patterns in Nature: Line Symmetry Line/Bilateral Symmetry This type of symmetry indicates that you can draw an imaginary line across an object and the resulting parts are mirror images of each other. Example 1.5: Bilateral Symmetry Source:https://biologydictionary.net/bilateral-symmetry/ Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 23 Patterns and Numbers in Nature and the World Patterns in Nature: Rotational Symmetry Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 24 Patterns and Numbers in Nature and the World Patterns in Nature: Rotational Symmetry Rotational Symmetry Rotational symmetry (or radial symmetry) is when an object is rotated in a certain direction around a point while achieving the same appearance. Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 25 Patterns and Numbers in Nature and the World Patterns in Nature: Rotational Symmetry Rotational Symmetry Rotational symmetry (or radial symmetry) is when an object is rotated in a certain direction around a point while achieving the same appearance. The smallest angle that a figure can be rotated while still preserving original formation is called angle of rotation. Let ρ be the angle of rotation, Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 26 Patterns and Numbers in Nature and the World Patterns in Nature: Rotational Symmetry Rotational Symmetry Rotational symmetry (or radial symmetry) is when an object is rotated in a certain direction around a point while achieving the same appearance. The smallest angle that a figure can be rotated while still preserving original formation is called angle of rotation. Let ρ be the angle of rotation, ρ= 360◦ n where in n is the order (n-fold rotational symmetry). Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Spiderwort Example 1.6: Spiderwort Source: https://mathstat.slu.edu/escher/index.php/Rotational Symmetry Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Sea Star Example 1.7: Sea Star Source: https://www.pinterest.ph/pin/417145984205155939/ Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 29 Patterns and Numbers in Nature and the World Patterns in Nature: Snowflakes Example 1.8: Snowflakes Source: https://www.noaa.gov/stories/how-do-snowflakes-formscience-behind-snow Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 30 Patterns and Numbers in Nature and the World Patterns in Nature: Translational Symmetry Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 31 Patterns and Numbers in Nature and the World Patterns in Nature: Translational Symmetry Translational Symmetry Translational symmetry is when an object is relocated to another position while maintaining its general or exact orientation. Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 32 Patterns and Numbers in Nature and the World Patterns in Nature: Translational Symmetry Translational Symmetry Translational symmetry is when an object is relocated to another position while maintaining its general or exact orientation. Example 1.9: Honeycomb Source: https://www.pinterest.ph/pin/294282156874810690/ Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Honeycomb and Packing Problems Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 34 Patterns and Numbers in Nature and the World Patterns in Nature: Honeycomb and Packing Problems Packing Problem Packing problems involve finding the optimum method of filling up a given space such as circle or spherical container. Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Honeycomb and Packing Problems Packing Problem Packing problems involve finding the optimum method of filling up a given space such as circle or spherical container. Example 1.10 Suppose you have a circles of radius 1 cm, each of which will have an area of πcm2 . We are then going to fill a plane with these circles using: Square packing Hexagonal packing Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Honeycomb and Packing Problems Example 1.10 cont. Square Packing Source: https://web.nmsu.edu/ snsm/classes/chem116/notes/crystals.html For square packing, each square will have an area of Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 37 Patterns and Numbers in Nature and the World Patterns in Nature: Honeycomb and Packing Problems Example 1.10 cont. Square Packing Source: https://web.nmsu.edu/ snsm/classes/chem116/notes/crystals.html For square packing, each square will have an area of 4cm2 . Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 38 Patterns and Numbers in Nature and the World Patterns in Nature: Honeycomb and Packing Problems Example 1.10 cont. Square Packing Source: https://web.nmsu.edu/ snsm/classes/chem116/notes/crystals.html For square packing, each square will have an area of 4cm2 . The percentage of the square’s area covered by circles will be Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 39 Patterns and Numbers in Nature and the World Patterns in Nature: Honeycomb and Packing Problems Example 1.10 cont. Square Packing Source: https://web.nmsu.edu/ snsm/classes/chem116/notes/crystals.html For square packing, each square will have an area of 4cm2 . The percentage of the square’s area covered by circles will be area of circles area of square Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia × 100% = Clegg, & Epp (2018). Mathematics in the Modern World 40 Patterns and Numbers in Nature and the World Patterns in Nature: Honeycomb and Packing Problems Example 1.10 cont. Square Packing Source: https://web.nmsu.edu/ snsm/classes/chem116/notes/crystals.html For square packing, each square will have an area of 4cm2 . The percentage of the square’s area covered by circles will be area of circles area of square Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia × 100% = Clegg, & Epp πcm2 4cm2 (2018). × 100% Mathematics in the Modern World 41 Patterns and Numbers in Nature and the World Patterns in Nature: Honeycomb and Packing Problems Example 1.10 cont. Square Packing Source: https://web.nmsu.edu/ snsm/classes/chem116/notes/crystals.html For square packing, each square will have an area of 4cm2 . The percentage of the square’s area covered by circles will be area of circles area of square Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia × 100% = Clegg, & Epp πcm2 4cm2 (2018). × 100% ≈ 78.54% Mathematics in the Modern World 42 Patterns and Numbers in Nature and the World Patterns in Nature: Honeycomb and Packing Problems Example 1.10 cont. Hexagonal Packing Source: https://www.researchgate.net/figure/Circle-packing-patterns-Square... For hexagonal packing, we can think of each hexagon as composed of six equilateral triangles with side equal to 2 cm. Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Honeycomb and Packing Problems Example 1.10 cont. Hexagonal Packing Source: https://www.researchgate.net/figure/Circle-packing-patterns-Square... For hexagonal packing, we can think of each hexagon as composed of six equilateral triangles with side equal to 2 cm. The percentage of the hexagon’s area covered by circles will be Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Honeycomb and Packing Problems Example 1.10 cont. Hexagonal Packing Source: https://www.researchgate.net/figure/Circle-packing-patterns-Square... For hexagonal packing, we can think of each hexagon as composed of six equilateral triangles with side equal to 2 cm. The percentage of the hexagon’s area covered by circles will be area of circles area of hexagon Aufman, Lockwood, Nation, × 100% = Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Honeycomb and Packing Problems Example 1.10 cont. Hexagonal Packing Source: https://www.researchgate.net/figure/Circle-packing-patterns-Square... For hexagonal packing, we can think of each hexagon as composed of six equilateral triangles with side equal to 2 cm. The percentage of the hexagon’s area covered by circles will be area of circles area of hexagon Aufman, Lockwood, Nation, × 100% = Clegg, & Epp 2 3πcm √ 6 3cm2 (2018). × 100% Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Honeycomb and Packing Problems Example 1.10 cont. Hexagonal Packing Source: https://www.researchgate.net/figure/Circle-packing-patterns-Square... For hexagonal packing, we can think of each hexagon as composed of six equilateral triangles with side equal to 2 cm. The percentage of the hexagon’s area covered by circles will be area of circles area of hexagon Aufman, Lockwood, Nation, × 100% = Clegg, & Epp 2 3πcm √ 6 3cm2 (2018). × 100% ≈ 90.69% Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Stripes and Spots Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Stripes and Spots Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Stripes and Spots According to the theory of Alan Turing, chemical reactions and diffusion processes in cells determine the growth patterns. Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 50 Patterns and Numbers in Nature and the World Patterns in Nature: Stripes and Spots According to the theory of Alan Turing, chemical reactions and diffusion processes in cells determine the growth patterns. A new model by Harvard University researchers predicts that there are three variables that could affect the orientation of these stripes: Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Stripes and Spots According to the theory of Alan Turing, chemical reactions and diffusion processes in cells determine the growth patterns. A new model by Harvard University researchers predicts that there are three variables that could affect the orientation of these stripes: (1) substance that amplifies the density of stripe patterns; Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Stripes and Spots According to the theory of Alan Turing, chemical reactions and diffusion processes in cells determine the growth patterns. A new model by Harvard University researchers predicts that there are three variables that could affect the orientation of these stripes: (1) substance that amplifies the density of stripe patterns; (2) the substance that changes one of the parameters involved in stripe formation; and Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Stripes and Spots According to the theory of Alan Turing, chemical reactions and diffusion processes in cells determine the growth patterns. A new model by Harvard University researchers predicts that there are three variables that could affect the orientation of these stripes: (1) substance that amplifies the density of stripe patterns; (2) the substance that changes one of the parameters involved in stripe formation; and (3) physical change in the direction of the origin of the stripe. Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 54 Patterns and Numbers in Nature and the World Patterns in Nature: The Sunflower Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 55 Patterns and Numbers in Nature and the World Patterns in Nature: The Sunflower Source: https://asknature.org/strategy/fibonacci-sequence-optimizes-packing/ Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Nautilus Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Nautilus Source: https://www.pinterest.ph/pin/292311832039678146/ Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 58 Patterns and Numbers in Nature and the World Patterns in Nature: Spiral Galaxy Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Spiral Galaxy Source: https://blogs.unimelb.edu.au/sciencecommunication/2018/09/23/theuniverse-in-a-spiral/ Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 61 Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay The role of mathematics in the world population is through modeling the population growth or decay. Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 62 Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay The role of mathematics in the world population is through modeling the population growth or decay. Exponential Growth Formula for Population A = P ert where in Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 63 Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay The role of mathematics in the world population is through modeling the population growth or decay. Exponential Growth Formula for Population A = P ert where in A is the size of the population after it grows Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 64 Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay The role of mathematics in the world population is through modeling the population growth or decay. Exponential Growth Formula for Population A = P ert where in A is the size of the population after it grows P is the initial number of people Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 65 Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay The role of mathematics in the world population is through modeling the population growth or decay. Exponential Growth Formula for Population A = P ert where in A is the size of the population after it grows P is the initial number of people r is the rate of growth Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 66 Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay The role of mathematics in the world population is through modeling the population growth or decay. Exponential Growth Formula for Population A = P ert where in A is the size of the population after it grows P is the initial number of people r is the rate of growth t is the time Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 67 Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay The role of mathematics in the world population is through modeling the population growth or decay. Exponential Growth Formula for Population A = P ert where in A is the size of the population after it grows P is the initial number of people r is the rate of growth t is the time e is the Euler’s constant Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 68 Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay Example 1.11 Five years ago the population of Pangasinan is 2,787,326. Now, the population of Pangasinan is 2,956,726. Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay Example 1.11 Five years ago the population of Pangasinan is 2,787,326. Now, the population of Pangasinan is 2,956,726. 1 What is the rate of growth? Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 71 Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay Example 1.11 Five years ago the population of Pangasinan is 2,787,326. Now, the population of Pangasinan is 2,956,726. 1 What is the rate of growth? 2 What will be the population 10 years from the initial P ? Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 72 Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay Example 1.11 Five years ago the population of Pangasinan is 2,787,326. Now, the population of Pangasinan is 2,956,726. 1 What is the rate of growth? 2 What will be the population 10 years from the initial P ? 3 What will be the population 15 years from the initial P ? Solution: Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 73 Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay Example 1.11 Five years ago the population of Pangasinan is 2,787,326. Now, the population of Pangasinan is 2,956,726. 1 What is the rate of growth? 2 What will be the population 10 years from the initial P ? 3 What will be the population 15 years from the initial P ? Solution: For sub-item (1): Rearrange the equation, Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 74 Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay Example 1.11 Five years ago the population of Pangasinan is 2,787,326. Now, the population of Pangasinan is 2,956,726. 1 What is the rate of growth? 2 What will be the population 10 years from the initial P ? 3 What will be the population 15 years from the initial P ? Solution: For sub-item (1): Rearrange the equation, A = P ert ⇒ Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia A P = ert Clegg, & Epp (2018). Mathematics in the Modern World 75 Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay Example 1.11 Five years ago the population of Pangasinan is 2,787,326. Now, the population of Pangasinan is 2,956,726. 1 What is the rate of growth? 2 What will be the population 10 years from the initial P ? 3 What will be the population 15 years from the initial P ? Solution: For sub-item (1): Rearrange the equation, A = P ert ⇒ Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia A P = ert ⇒ ln ( PA ) = ln (ert ) Clegg, & Epp (2018). Mathematics in the Modern World 76 Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay Example 1.11 Five years ago the population of Pangasinan is 2,787,326. Now, the population of Pangasinan is 2,956,726. 1 What is the rate of growth? 2 What will be the population 10 years from the initial P ? 3 What will be the population 15 years from the initial P ? Solution: For sub-item (1): Rearrange the equation, A = P ert ⇒ A P = ert ⇒ ln ( PA ) = ln (ert ) ⇒ A ) ln ( P t =r Thus, A ln ( P ) t =r⇒ ) ln ( 2,956,726 2,787,326 5 Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, = r ⇒ r ≈ 0.01179997236 or r ≈ 1.18% & Epp (2018). Mathematics in the Modern World 77 Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 78 Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay Solution (cont.): For sub-item (2): Use the equation and plug in the values, Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 79 Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay Solution (cont.): For sub-item (2): Use the equation and plug in the values, A = P ert ⇒ A ≈ (2, 787, 326)e(0.0118)(10) ⇒ A ≈ 3, 136, 422 Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 80 Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay Solution (cont.): For sub-item (2): Use the equation and plug in the values, A = P ert ⇒ A ≈ (2, 787, 326)e(0.0118)(10) ⇒ A ≈ 3, 136, 422 For sub-item (3): Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 81 Patterns and Numbers in Nature and the World Patterns in Nature: Population Growth and Decay Solution (cont.): For sub-item (2): Use the equation and plug in the values, A = P ert ⇒ A ≈ (2, 787, 326)e(0.0118)(10) ⇒ A ≈ 3, 136, 422 For sub-item (3): A = P ert ⇒ A ≈ (2, 787, 326)e(0.0118)(15) ⇒ A ≈ 3, 327, 038 Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 82 The Fibonacci Sequence Sequence Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World The Fibonacci Sequence Sequence Sequence A sequence is an ordered list of numbers, called terms, that may have repeated values. The arrangement of terms is set by a definite rule. Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World The Fibonacci Sequence Sequence Sequence A sequence is an ordered list of numbers, called terms, that may have repeated values. The arrangement of terms is set by a definite rule. Example 1.12 Analyze the given sequence for its rule and identify the next term. Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World The Fibonacci Sequence Sequence Sequence A sequence is an ordered list of numbers, called terms, that may have repeated values. The arrangement of terms is set by a definite rule. Example 1.12 Analyze the given sequence for its rule and identify the next term. 1 1, 10, 100, 1000, 10000, 100000 Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World The Fibonacci Sequence Sequence Sequence A sequence is an ordered list of numbers, called terms, that may have repeated values. The arrangement of terms is set by a definite rule. Example 1.12 Analyze the given sequence for its rule and identify the next term. 1 1, 10, 100, 1000, 10000, 100000 2 2, 5, 9, 14, 20, 27, 35, 44 Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World The Fibonacci Sequence Sequence Sequence A sequence is an ordered list of numbers, called terms, that may have repeated values. The arrangement of terms is set by a definite rule. Example 1.12 Analyze the given sequence for its rule and identify the next term. 1 1, 10, 100, 1000, 10000, 100000 2 2, 5, 9, 14, 20, 27, 35, 44 3 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 88 The Fibonacci Sequence The Fibonacci Sequence Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World The Fibonacci Sequence The Fibonacci Sequence Named after Leonardo of Pisa or widely known as Fibonacci. Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World The Fibonacci Sequence The Fibonacci Sequence Named after Leonardo of Pisa or widely known as Fibonacci. While the sequence is widely known as Fibonacci sequence, the pattern is said to have been discovered much earlier in India. Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World The Fibonacci Sequence The Fibonacci Sequence Named after Leonardo of Pisa or widely known as Fibonacci. While the sequence is widely known as Fibonacci sequence, the pattern is said to have been discovered much earlier in India. The ratios of the successive Fibonacci numbers approach the number φ, also known as the Golden ratio Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World The Fibonacci Sequence The Fibonacci Sequence Named after Leonardo of Pisa or widely known as Fibonacci. While the sequence is widely known as Fibonacci sequence, the pattern is said to have been discovered much earlier in India. The ratios of the successive Fibonacci numbers approach the number φ, also known as the Golden ratio Aufman, 1 1 2 1 3 2 5 3 8 5 13 8 21 13 34 21 55 34 89 55 1 2 1.5 1.667 1.600 1.625 1.615 1.619 1.618 1.618 Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World The Fibonacci Sequence The Fibonacci Sequence Named after Leonardo of Pisa or widely known as Fibonacci. While the sequence is widely known as Fibonacci sequence, the pattern is said to have been discovered much earlier in India. The ratios of the successive Fibonacci numbers approach the number φ, also known as the Golden ratio 1 1 2 1 3 2 5 3 8 5 13 8 21 13 34 21 55 34 89 55 1 2 1.5 1.667 1.600 1.625 1.615 1.619 1.618 1.618 Thus, the Golden ratio is approximately equal to 1.618. Next, n try calculating Fφn Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World The Fibonacci Sequence The Fibonacci Sequence Named after Leonardo of Pisa or widely known as Fibonacci. While the sequence is widely known as Fibonacci sequence, the pattern is said to have been discovered much earlier in India. The ratios of the successive Fibonacci numbers approach the number φ, also known as the Golden ratio 1 1 2 1 3 2 5 3 8 5 13 8 21 13 34 21 55 34 89 55 1 2 1.5 1.667 1.600 1.625 1.615 1.619 1.618 1.618 Thus, the Golden ratio is approximately equal to 1.618. Next, n try calculating Fφn φn Fn Aufman, 1.6181 1 1.6182 1 1.6183 2 1.6184 3 1.6185 5 1.61810 55 1.61812 144 1.618 2.168 2.118 2.285 2.218 2.236 2.236 Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World The Fibonacci Sequence The Fibonacci Sequence Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 96 The Fibonacci Sequence The Fibonacci Sequence √ √ n n Hence, Fφn will approach the value 5. Rearranging Fφn = 5 to φn Fn ≈ √ gives us the idea that the nth Fibonacci number, Fn , is 5 the nearest whole number to Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp φn √ . 5 (2018). Mathematics in the Modern World 97 The Fibonacci Sequence The Fibonacci Sequence √ √ n n Hence, Fφn will approach the value 5. Rearranging Fφn = 5 to φn Fn ≈ √ gives us the idea that the nth Fibonacci number, Fn , is 5 the nearest whole number to φn √ . 5 The exact equation for the nth Fibonacci number is Formula for the nth Fibonacci number Fn = φn √ 5 ± 1√ φn 5 where in if n is even use − and else (odd), use +. Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 98 The Fibonacci Sequence The Fibonacci Sequence √ √ n n Hence, Fφn will approach the value 5. Rearranging Fφn = 5 to φn Fn ≈ √ gives us the idea that the nth Fibonacci number, Fn , is 5 the nearest whole number to φn √ . 5 The exact equation for the nth Fibonacci number is Formula for the nth Fibonacci number Fn = φn √ 5 ± 1√ φn 5 where in if n is even use − and else (odd), use +. Try! Find the Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 99 The Fibonacci Sequence The Fibonacci Sequence √ √ n n Hence, Fφn will approach the value 5. Rearranging Fφn = 5 to φn Fn ≈ √ gives us the idea that the nth Fibonacci number, Fn , is 5 the nearest whole number to φn √ . 5 The exact equation for the nth Fibonacci number is Formula for the nth Fibonacci number Fn = φn √ 5 ± 1√ φn 5 where in if n is even use − and else (odd), use +. Try! Find the 1 30th Fibonacci number Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 100 The Fibonacci Sequence The Fibonacci Sequence √ √ n n Hence, Fφn will approach the value 5. Rearranging Fφn = 5 to φn Fn ≈ √ gives us the idea that the nth Fibonacci number, Fn , is 5 the nearest whole number to φn √ . 5 The exact equation for the nth Fibonacci number is Formula for the nth Fibonacci number Fn = φn √ 5 ± 1√ φn 5 where in if n is even use − and else (odd), use +. Try! Find the 1 30th Fibonacci number 2 21st Fibonacci number Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 101 The Fibonacci Sequence The Fibonacci Sequence √ √ n n Hence, Fφn will approach the value 5. Rearranging Fφn = 5 to φn Fn ≈ √ gives us the idea that the nth Fibonacci number, Fn , is 5 the nearest whole number to φn √ . 5 The exact equation for the nth Fibonacci number is Formula for the nth Fibonacci number Fn = φn √ 5 ± 1√ φn 5 where in if n is even use − and else (odd), use +. Try! Find the 1 30th Fibonacci number 2 21st Fibonacci number th Fibonacci number 100Lockwood, Nation, Clegg, & 3 Aufman, Epp (2018). Mathematics in the Modern World The Fibonacci Sequence The Fibonacci Sequence Try to calculate the following (round off it to three decimal places): 1 1.618 Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 103 The Fibonacci Sequence The Fibonacci Sequence Try to calculate the following (round off it to three decimal places): 1 1.618 = 0.618 1.6182 Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 104 The Fibonacci Sequence The Fibonacci Sequence Try to calculate the following (round off it to three decimal places): 1 1.618 = 0.618 1.6182 = 2.618 Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 105 The Fibonacci Sequence The Fibonacci Sequence Try to calculate the following (round off it to three decimal places): 1 1.618 = 0.618 1.6182 = 2.618 = φ + 1 (*) 1.6183 Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 106 The Fibonacci Sequence The Fibonacci Sequence Try to calculate the following (round off it to three decimal places): 1 1.618 = 0.618 1.6182 = 2.618 = φ + 1 (*) 1.6183 = 4.236 Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 107 The Fibonacci Sequence The Fibonacci Sequence Try to calculate the following (round off it to three decimal places): 1 1.618 = 0.618 1.6182 = 2.618 = φ + 1 (*) 1.6183 = 4.236 = 2φ + 1 1.6184 Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 108 The Fibonacci Sequence The Fibonacci Sequence Try to calculate the following (round off it to three decimal places): 1 1.618 = 0.618 1.6182 = 2.618 = φ + 1 (*) 1.6183 = 4.236 = 2φ + 1 1.6184 = 6.854 Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 109 The Fibonacci Sequence The Fibonacci Sequence Try to calculate the following (round off it to three decimal places): 1 1.618 = 0.618 1.6182 = 2.618 = φ + 1 (*) 1.6183 = 4.236 = 2φ + 1 1.6184 = 6.854 = 3φ + 2 1.6185 Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 110 The Fibonacci Sequence The Fibonacci Sequence Try to calculate the following (round off it to three decimal places): 1 1.618 = 0.618 1.6182 = 2.618 = φ + 1 (*) 1.6183 = 4.236 = 2φ + 1 1.6184 = 6.854 = 3φ + 2 1.6185 = 11.089 Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 111 The Fibonacci Sequence The Fibonacci Sequence Try to calculate the following (round off it to three decimal places): 1 1.618 = 0.618 1.6182 = 2.618 = φ + 1 (*) 1.6183 = 4.236 = 2φ + 1 1.6184 = 6.854 = 3φ + 2 1.6185 = 11.089 = 5φ + 3 Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 112 The Fibonacci Sequence The Fibonacci Sequence Try to calculate the following (round off it to three decimal places): 1 1.618 = 0.618 1.6182 = 2.618 = φ + 1 (*) 1.6183 = 4.236 = 2φ + 1 1.6184 = 6.854 = 3φ + 2 1.6185 = 11.089 = 5φ + 3 Thus, this gives us the idea that φn = Fn φ + Fn−1 . Using (*) or known as the Golden relation, we can derive the exact value of the Golden ratio. Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 113 The Fibonacci Sequence The Fibonacci Sequence Golden Ratio The Golden ratio φ is unique and φ ∈ R+ satisfying the Golden relation. Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 114 The Fibonacci Sequence The Fibonacci Sequence Golden Ratio The Golden ratio φ is unique and φ ∈ R+ satisfying the Golden relation. Rearranging the Golden relation as φ2 − φ − 1 = 0 then use Quadratic formula, √ √ √ −(−1)+ (−1)2 −4(1)(−1) 1+ 5 −b+ b2 −4ac φ= = = = 1.618033989... ∈ Q∗ 2a 2 2(1) Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 115 The Fibonacci Sequence The Fibonacci Sequence Golden Ratio The Golden ratio φ is unique and φ ∈ R+ satisfying the Golden relation. Rearranging the Golden relation as φ2 − φ − 1 = 0 then use Quadratic formula, √ √ √ −(−1)+ (−1)2 −4(1)(−1) 1+ 5 −b+ b2 −4ac φ= = = = 1.618033989... ∈ Q∗ 2a 2 2(1) Geometrically, it can also be visualized as a rectangle perfectly formed by a square and another rectangle which can be repeated infinitely inside each section. Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 116 The Fibonacci Sequence The Golden Rectangle and Golden Spiral Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World The Fibonacci Sequence The Golden Rectangle and Golden Spiral Source: https://www.sciencedirect.com/science/article/pii/S1110016815000265 Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 118 The Fibonacci Sequence Applications Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World The Fibonacci Sequence Applications Logo Creation Source: https://www.invisionapp.com/inside-design/golden-ratio-designers/ Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 120 The Fibonacci Sequence Applications Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 121 The Fibonacci Sequence Applications Architecture Source: https://sites.google.com/site/funwithfibonacci/architecture/theparthenon Aufman, Lockwood, Nation, Daniel Bezalel A. Garcia Clegg, & Epp (2018). Mathematics in the Modern World 122 Mathematics for Our World Mathematics for Our World Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Mathematics for Our World Mathematics for Our World Mathematics for Organization (specifically, Statistics and Big data analysis) Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Mathematics for Our World Mathematics for Our World Mathematics for Organization (specifically, Statistics and Big data analysis) Mathematics for Prediction Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World Mathematics for Our World Mathematics for Our World Mathematics for Organization (specifically, Statistics and Big data analysis) Mathematics for Prediction Mathematics for Control Therefore, mathematics is indispensable Aufman, Lockwood, Nation, Clegg, & Epp (2018). Mathematics in the Modern World