Math 111 Reviewer PDF

Math 111 art, architecture, nature, and even human anatomy. Module 1: The Nature of  Golden Rectangle and Spiral: Mathematics Common geometric figures derived...

Math 111 art, architecture, nature, and even human anatomy. Module 1: The Nature of  Golden Rectangle and Spiral: Mathematics Common geometric figures derived Unit 1: Introduction to Mathematics from Fibonacci numbers. Mathematics is the study of patterns, numbers, and relationships that exist Examples of Fibonacci Sequence in in nature and our daily world. It is Nature: applied to various phenomena and is a  Sunflowers: Seed spirals follow universal language for communication Fibonacci numbers. and understanding across different  Pineapples: The nubs form spirals fields. that follow Fibonacci numbers Key Concepts: (e.g., 5, 8, and 13).  Mathematics and Nature: The  Pine Cones: Spiral arrangements exploration of patterns in nature of scales also follow Fibonacci highlights the integral role of patterns. mathematics in understanding the world. Applications of the Golden Ratio:  The Origins of Mathematics: Its  Nature: development is linked to the  Appears in growth patterns of discovery of patterns, sequences, plants, seashells, animal and counting systems. proportions, and DNA structure. Patterns and Numbers in Nature and  Art and Architecture: the World  Used by famous artists like Patterns in Nature: Leonardo da Vinci and  Patterns: Regularities of form Michelangelo. seen in the environment (e.g.,  Found in structures like the snowflakes, sand dunes, and Parthenon and Pyramids. waves).  Mathematical Patterns: Nature’s patterns can be modeled mathematically and give insight into natural processes. Fibonacci Sequence:  Discovered by Leonardo of Pisa (Fibonacci), this sequence (1, 1, 2, 3, 5, 8, etc.) explains growth  Fibonacci (Leonardo of Pisa): patterns and shapes in nature Introduced the Fibonacci (like flower petals and pine cones). sequence (1, 1, 2, 3, 5, 8,...), Golden Ratio: found in nature, art, and  Golden Ratio (φ): The relationship mathematics. between Fibonacci numbers,  Leonardo da Vinci: approximately 1.618. It appears in Applied the golden ratio in  works like the Vitruvian Man and The Last Supper for ideal proportions.  Michelangelo: Used geometry and proportion in sculptures like David to  Johannes Kepler: achieve balance and beauty. Discovered the golden ratio in planetary orbits, showing mathematics’ role in cosmic harmony.  Phidias: An ancient Greek sculptor who is believed to have applied the golden ratio in his works, including the design of the Parthenon.  Plato: Described the Platonic solids  Vitruvius: and discussed their Ancient Roman architect connection to the golden ratio whose ideas of symmetry and in his work Timaeus. proportion influenced later figures like Leonardo.  Pythagoras: Developed the Pythagorean theorem and explored mathematical relationships in nature and music.  Euclid:  Luca Pacioli: Known as the "Father of Italian mathematician who Geometry," he systematized explored the golden ratio in geometry and explored the his book "De Divina golden ratio in his work. Proportione" with illustrations by Leonardo da Vinci.  Renaissance Artists (Raphael, inspired by Da Vinci's Botticelli, Rembrandt): principles. Applied the golden ratio in their compositions. Raffaello Sanzio da Urbino/Raphael (1483-1520) in The School of Athens, Alessandro di Mariano di Vanni Filipepi/Botticelli  George Pierre Seurat (1445-1510) in The Birth of (1859–1891) Venus, and Rembrandt A French post-Impressionist Harmenszoon Van Rijn painter, applied the golden (1606-1669)/Rembrandt in ratio in works like Bathers at his self-portraits. Asnières, The Bridge at  Raphael Courbevoie, and A Sunday on La Grande Jatte to define horizons, focal points, and balance in his compositions. Patterns and Regularities in Nature Types of Patterns:  Symmetry: Balanced proportions in natural objects. Bilateral Symmetry: One side mirrors the other (e.g.,  Rebrandt animals).  Botticeli Radial Symmetry: Circular symmetry around a central point (e.g., flowers).  Salvador Domingo Felipe Jacinto Dali/ Salvador Dalí (1904-1989): Used the golden ratio in his artwork, notably in The Sacrament of the Last Supper,   Engineering and IT: Development of technologies, cryptography, and computer science.  Social Sciences and Political Science: Game theory, voting patterns, and economic models. Practical Applications: Mathematics is applied in daily life  Fractals: Complex shapes where activities like market pricing, each part mirrors the whole (e.g., transportation, and architecture. snowflakes, coastlines). Module 2: Mathematical Language and Symbols Characteristics of Mathematical Language Galileo's Insight:  Galileo Galilei stated, “Mathematics is the language in which God has written the universe.” This highlights the universal nature of mathematics.  Spirals: Seen in galaxies, typhoons, shells, etc. 1. Communication System  Definition: A system used by mathematicians to express ideas, combining natural language with technical terms and symbols.  Symbolic Notation: Symbols are used to represent formulas and constants (e.g., π). Behavior of Nature: 2. Use of Symbols Patterns like honeycombs, zebra  Representation: Symbols express stripes, and spider webs showcase operations and concepts. mathematical rules governing their  Syntax: Follows specific rules to formation. ensure clarity.  Conventions: Symbols have Applications of Mathematics in the agreed-upon meanings, e.g.:+ (add), World − (subtract), × (multiply),÷ Fields Using Mathematics: (divide).  Forensic Science: Image analysis 3. Precision, Conciseness, and Power and projectile motion.  Precision: Each symbol has a  Medicine: Drug development and clear meaning. protein modeling.  Conciseness: Complex ideas can  be expressed succinctly (e.g., 8 + 2. Equivalent Sets: Two sets with 2=10) the same number of elements.  Power: Mathematics simplifies  Example: complex concepts and enhances Set 1: {Θ,¥,£} problem-solving. Set 2: {Φ,Ψ,Λ} Both have 3 elements. Expression vs. Sentence 3. Universal Set: Contains all sets  Expressions under investigation, denoted U. Definition: A finite 4. Cardinality: The number of combination of symbols that elements in a set, denoted n(A). does not convey a complete  Example: n(E)=5 for thought. E={a,e,i,o,u}. Examples: 5, 2 + 3, (6−2)+1. 5. Subsets: Set A is a subset of B if  Sentences every element of A is also in B.  Definition: Complete thoughts  Example: If A={1,2} and that can be evaluated as true B={1,2,3,4}, then A⊆ B. or false. 6. Union and Intersection:  Examples: 2+3=5 (True),  Union: A∪ B is the set of elements 2+3=6 (False). in A or B.  Example: If A={1,2} and Conventions in Mathematical B={2,3}, then A∪ B={1,2,3}. Language  Intersection: A∩ B is the set of Order of Operations elements common to both A and B. (BODMAS/PEMDAS) Example: If A={1,2} and  BODMAS: Brackets, Order, B={2,3}, then A∩ B={2}. Division, Multiplication, Addition, Subtraction B. Relations  PEMDAS: Parentheses,  Definition: A relation pairs each Exponents, element in one set with elements Multiplication/Division (left to in another, forming ordered pairs. right), Addition/Subtraction (left  Example: to right) Consider regular holidays in the Philippines, where A. Language of Sets A={New Year, Holy Week} and 1. Set Representation: B={January 1, April 15}. The  Roster Method: Lists elements relation could explicitly R={(New Year,January 1),(Hol y Week,April 15)}. C. Functions  Definition: A relation where each  Rule Method: Describes elements element in the domain with a property. corresponds to exactly one element in the range.  Example:  A function mapping side lengths of squares to their perimeters.  Table: D. Language of Binary Operations 1. Closure Property: An operation is closed if it produces results within the same set.  Example: The set of integers is closed under addition since adding two integers always yields another integer. 2. Associative Property: The way in which numbers are grouped does not change the sum or product.  Example: (a+b)+c=a+(b+c). 3. Identity Property: There exists an element in a set that, when used in an operation, leaves other elements unchanged.  Example: The identity element for addition is 0 because a+0=a. 4. Inverse Property: For every element in a set, there is a corresponding element that reverses the operation.  Example: The inverse of a under addition is −a because a+(−a)=0.