Notes ModMath PDF

Lesson 1. 2. Fractals o These are never-ending patterns that are The Nature of Mathematics self-similar across different scales. The...

Lesson 1. 2. Fractals o These are never-ending patterns that are The Nature of Mathematics self-similar across different scales. The image just reappears over and over again no matter how many times the object is Mathematics magnified. - study of pattern and structure o They are created by repeating a simple - A useful way to think about nature and our process over and over in an ongoing world feedback loop. - A tool to quantify, organize and control our o The branching pattern of trees world, predict phenomena and make life o leaves of ferns and umbellifers easier 3. Spirals Mathematical pattern o Coiled pattern revolving about a center - Patterns are things that are repetitive, which point can be found in nature as color, shape, action, o Common in plants and animals or some other sequences that are almost 4. Chaos, flow, meanders everywhere. o A dynamical system is chaotic if it is - Mathematics expresses patterns. sensitive to initial condition (the so-called - These sequences that repeat, follows a rule or “butterfly effect” rules. A rule is a way to calculate or solve a o Flow are patterns of whirling vortices problem. created by the unsteady separation of flow - An organization of shapes and symbols of a fluid over obstructing objects distributed in regular interval o Meanders are sinuous bends in rivers or - Regular, repeated, or recurring forms of other channels, which form as a fluid flows designs around bends 5. Waves Types of Pattern in Nature o Disturbances that carry energy as they move 1. Symmetry o Mechanical waves, dunes or crescent o According to the American Heritage dunes Dictionary, symmetry is an exact 6. Bubbles, foam correspondence of form and constituent o A soap bubble forms a sphere, a surface configuration on opposite sides of a with minimal area (minimal surface) — dividing line or plane or about a center or an the smallest possible surface area for the axis. It indicates that you can draw an volume enclosed imaginary line across an object and the o A foam is a mass of bubbles; foams of resulting parts are mirror images of each different materials occur in nature other. 7. Tessellations a) Reflective Symmetry - It is also called o patterns formed by repeating tiles all mirror symmetry or line symmetry. It is over a flat surface made with a line going through an 8. Cracks object which divides it into two o linear openings that form in materials to pieces which are mirror images of relieve stress each other. This is often termed as 9. Spots, Stripes bilateral symmetry as it divides the o patterns have an evolutionary object into two (“bi “ means two) explanation mirror images. b) Rotational or Radial Symmetry - if Significance of patterns the figure rotates at its center, the figure should be the same as the Prediction original Decision-making c) Five point symmetry – starfish, sea Reasoning urchins and sea lilies d) 6-fold symmetry - snowflakes  Reminder of the interconnectedness of all Note: things and the delicate equilibrium that Arithmetic Sequence - “common difference” sustains life Geometric Sequence – “common ratio” Lesson 2. Quadratic Sequence Fibonacci Sequence - The difference between the terms are arranged in arithmetic sequence - Example: Sequence 1 3 6 10 15 - list of numbers or elements arranged in a +2 +3 +4 +5 certain pattern - Each element will be called “terms” o Triangle Number Sequence – the - First term = 𝑎1 pattern of these sequence form - second term = 𝑎2 equilateral triangles - nth term = 𝑎𝑛 o Cube Number Sequence – the pattern - It can be finite or infinite of these sequence form cubes - It can be ascending or descending order o Square Number Sequence – the - Each term can be evaluated through direct pattern of these sequence form squares formula, relations from the preceding term or Fibonacci sequence table of values - named after the Italian mathematician Arithmetic Sequence Leonardo of Pisa, who is commonly known - Each term have a common difference as Fibonacci. between their preceding terms (plus) - The next term of the sequence is the sum - Examples: of the two preceding terms - 1, 2, 3, 4, 5,…. The sequence has a F0 0 common difference of 1 F1 1 - 2, 4, 6, 8, 10,.. The sequence has a F2 1 common difference of 2 F3 2 - Equation for the nth term F4 3 𝐴𝑛 =𝐴1+ (𝑛 −1) 𝑑 F5 5 F6 8 - where 𝐴1 = first term - 𝑑 is the common difference Golden Ratio Geometric Sequence - The limit of the ratio of two consecutive terms of the Fibonacci sequence - Each term have a common ratio between Approximately  = 1.618034 their preceding terms (multiply) Seen usually in nature - Examples: Mathematical ratio describing - 1, 2, 4, 8, 16,…. The sequence has a aesthetically pleasing common ratio of 2 For Fibonnaci, Sequence for the nth term - 1, 4, 16, 64,.. The sequence has a common ratio of 4 𝜑𝑁 −(1−𝜑)𝑁 - Equation for the nth term 𝐹𝑁 = ------------------- 𝐴𝑛 =𝐴1𝑟𝑛−1 √5 - where 𝐴1 = first term - 𝑟 is the common ratio For the 5th term = 5 1.6180345 −(1−1.618034)5 𝐹5 = ---------------------------------- ≈ 5 √ 5 Fibonacci Series Spiral Symbols and its interpretations - Formed by drawing a spiral guided by stacking squares with dimensions from the Fibonnaci Sequence - Known for its aesthetic and symmetrical appearance Mathematical Language and Symbols The language of mathematics - makes it easy to express the kinds of thoughts that mathematicians likes to express - Characteristics of the language of mathematics: Variables ✓Precise - able to make very fine - are symbols used to represent some values distinctions - Usually in the form of an alphabet ✓Concise - able to say things briefly Example: 𝐴𝑥 + b ✓Powerful - able to express complex Set thoughts with relative ease - A collection of numbers or values written as - Mathematical symbols are the components a set roster or set builder used in forming these Mathematical language o Set roster A = {1,2,3} o Set builder 𝐴 ={𝑥 𝜖 ℝ|𝑎