Mathematics in Our World Notes PDF

Chapter 1: Mathematics in Our World spots form 6. Aristid Lindenmayer & Benoit Mandelbrot: showed how Mathematics fractals c...

Chapter 1: Mathematics in Our World spots form 6. Aristid Lindenmayer & Benoit Mandelbrot: showed how Mathematics fractals can model plant growth – study of relationships among numbers, quantities, and shapes – includes arithmetic, algebra, trigonometry, geometry, B. The Fibonacci Sequence statistics, calculus – helps organize patterns and regularities in the world – a sequence where each number is the sum of the two – being a science of patterns, it helps us recognize and before it generalize patterns in numbers, shapes, and the world Example: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,... A. Patterns and Numbers in Nature and the World Sequence Patterns – an ordered list of numbers based on a specific rule – each number is called a term – regular, repeated, or recurring forms or designs – seen in nature and can be modeled mathematically – examples: Who is Fibonacci? Symmetries – Leonardo of Pisa – introduced the Fibonacci Sequence in the book Liber Spirals Abaci Waves Foams Golden Ratio (Φ ≈ 1.618) Tessellations – ratios of Fibonacci numbers get closer to Φ – seen in art, nature, and architecture – examples: Mona Lisa, Notre Dame, Parthenon Symmetry – when an object can be divided into mirror-image parts Binet’s Formula for Fibonacci Types of Symmetry: Fₙ = [(1 + √5)/2]ⁿ – [(1 – √5)/2]ⁿ / √5 1. Line or Bilateral Symmetry 2. Rotational Symmetry C. Mathematics for Our World – For Organization Spiral Helps in arranging and understanding regularities – a curve starting from a point, moving farther away as it – For Prediction spins around the point Helps predict behavior in nature and real-life situations – For Control Helps humans control events and improve life through Wave innovations – a moving disturbance of one or more quantities (like sound, water, or light) Sequences 1. Arithmetic Sequence Foams – same number is added or subtracted every time – materials made by trapping gas in a liquid or solid – Example: 3, 6, 9, 12,... Tessellation (Tiling) 2. Geometric Sequence – covering a surface with shapes without gaps or overlaps – each term is multiplied or divided by the same number – Example: 2, 4, 8, 16,... History of Pattern Study 1. Plato, Pythagoras, Empedocles: early Greek thinkers 3. Quadratic Sequence who studied order in nature – second differences between terms are constant 2. Joseph Plateau: studied soap films and minimal – Example: 1, 4, 9, 16, 25,... surfaces 3. Ernst Haeckel: painted symmetrical marine life 4. D’Arcy Thompson: used math to explain growth patterns 5. Alan Turing: predicted how patterns like stripes and 4. Harmonic Sequence – formed by taking reciprocals of an arithmetic sequence 2. Conditional Statement – Example: 1, 1/2, 1/3, 1/4,... – says if one thing is true, then something else is also true 3. Existential Statement 5. Mixed Sequence – says that there is at least one element for which the – formed using more than one rule (like addition and property is true multiplication) – Example: 2, 4, 7, 11, 16,... Chapter 3. The Language of Sets 6. Square Number Sequence – formed by squaring numbers Definition of a Set – Example: 1² = 1, 2² = 4, 3² = 9, 4² = 16,... A set is a well-defined collection of distinct objects. 7. Cube Number Sequence The objects in a set are called elements. – formed by cubing numbers The symbol ∈ means "is an element of," and ∉ – Example: 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64,... means "is not an element of." Chapter 2: Mathematical Language and Symbols Ways of Describing a Set Mathematical Language 1. Roster or Tabular Method – listing the elements separated by commas and enclosed in braces {}. – system used to communicate mathematical ideas – precise, concise, powerful, clear, and objective 2. Rule or Descriptive Method – describing the common characteristics of the elements using set-builder notation. Variable – a quantity that can change in a mathematical problem Kinds of Sets or experiment 1. Empty / Null / Void Set – a set with no elements, denoted by ∅ or {}. Mathematical Expressions and Sentences 2. Finite Set – a set with countable elements. Mathematical Expression 3. Infinite Set – a set with uncountable elements. – made up of terms separated by plus (+) or minus (–) signs 4. Universal Set (U) – the set that contains all possible elements under consideration. Literal Coefficient: the variable part of a term Numerical Coefficient: the constant number multiplied with the variable Relationships of Sets 1. Equal Sets – sets that contain exactly the same elements. Types of Expressions 2. Equivalent Sets – sets that have the same Monomial – one term number of elements. Binomial – two terms 3. Joint Sets – sets with at least one common element. Trinomial – three terms 4. Disjoint Sets – sets with no elements in common. Mathematical Sentence – compares two expressions using a comparison sign (like =, >,