Mathematics in Nature Lesson Notes PDF

REVIEWER IN MMW Lesson 2: Numbers and Patterns Lesson 1: Mathematics in Nature Pattern It is an arrangement that helps observers Patterns in Nature and the Regularities in...

REVIEWER IN MMW Lesson 2: Numbers and Patterns Lesson 1: Mathematics in Nature Pattern It is an arrangement that helps observers Patterns in Nature and the Regularities in anticipate what they might see or what the World happens next. Patterns and counting are correlative. Counting happens when G. H. Hardy there is a pattern. A British Mathematician who Patterns can be sequential, characterized mathematics as the study of spatial, temporal, and even patterns. linguistic. All these phenomena create a Word Patterns repetition of names or events called deal with the metrical regularity. patterns of poems and the syntactic patterns. Regularities in the World (Collins 2018) States the fact that the same thing always Numbers Patterns happens under the same circumstances. deals with the prediction of the next number in a sequence. Patterns Discernable regularities in the world or a Who Was Fibonacci? man-made design. His real name is Leonardo of Pisa. He is a European Mathematician from Patterns in Nature 1175-1250. Visible regularities of form found in the He discovered the Fibonacci Sequence. world. Rabbits NATURAL PATTERNS It is used to investigate the Fibonacci sequence by knowing how fast rabbit Symmetry breeds. Proportion and Balance. Spiral Fibonacci Sequence Revolves around the point. Meander Numbers are found by adding the Series of curves and bends. numbers before them. Wave 1, 1, 2, 3, 5, 8, 13, 21, 34, … Oscillation or vibrations of a physical world. The Rule Fracture or Crack Formula: Xn = Xn-1 + Xn-2 Separation of an object from an object or material. Golden Ratio Stripes In mathematics, two quantities are in the Made of a series of bands or strips. Golden ratio if their ratio is the same as Fractal their sum to the larger of the two Never-ending patterns. quantities. “De Devina Proportioned” by Luca Paciolli. IMPORTANCE OF MATHETICS IN LIFE Phi Irrational number 1.61803398… Katie Kim (2015) Math is a subject that makes Golden Section in Architecture students either jump for joy or Parthenon in Greece rip their hair out. Great Pyramid of Egypt. Lesson 3: The Language of Mathematics and Sets SET THEORY Mathematical Language Set Theory It is the system used to communicate Branch of mathematics that studied sets. mathematical ideas. The study of sets became a fundamental Mathematics as language has theory in mathematics in the 1870s. symbols to express a formula or to represent a constant. Georg Cantor It has syntax to make the A German Mathematician Who expression well-formed to make Introduced Set Theory. the characters and symbols clear and valid that do not violate the The Language of Sets rules. Set Characteristics of Mathematical Language is a well-defined collection of objects; the objects are called the elements or 1. Precision - means able to make very fine members. distinction. To describe a set, we use braces {}, and 2. Concise - means able to say things briefly. use capital letters to represent it. 3. Powerful - means able to express complex The symbol ε denotes that an object is an thoughts with relative ease. element of a set. Expression versus Sentences Ways to Represent a Set An expression or a mathematical expression is a finite combination of 1. Roster method or Tabulation method symbols that is well-defined according to It is the method where the sets are rules that depend on the context. enumerated or listed, and each an expression or mathematical element is separated by a comma. expression is a finite For example: A = {a, e, i, o, u} 2. Rule method or Set Builder Notation well-defined combination of symbols It is a method that is used to describe according to rules that depend on the the elements or members of the set. context. For example: A= {x|x is a collection Closed Sentence of vowel letters} It is a sentence with a truth value of true (or false). Finite Set Open Sentence It is a set whose elements are limited or It is a sentence when it is unknown countable, and the last element can be whether it is true or false. identified. A = {x | x is a positive integer less than Mathematical Convention 10} Or A = {1, 2, 3, 4, 5, 6, 7, 8, 9} is a fact, name, notation, or usage which is generally agreed upon by mathematicians. Infinite Set Mathematicians abide by conventions to It is a set whose elements are unlimited allow other mathematicians to or uncountable, and the last element understand what they write without cannot be specified. constantly having to redefine basic terms. For example, let A = {x|x is a set of whole numbers} Or A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9,...} Ellipsis Intersection of Sets Three dots in enumerating the elements The intersection of A and B is the set of of the set. all elements 𝑥 in 𝒰 such that 𝑥 is in 𝐴 It indicates a continuing pattern. and 𝑥 is in 𝐵. It is denoted as 𝐴 ∩ 𝐵. Unit Set The intersection of A and B in the It is a set with only one element symbol is defined by A ∩ 𝐵 = 𝑥 | 𝑥 ∈ 𝐴 For example, A = {1}. 𝑎𝑛𝑑 𝑥 ∈ 𝐵. The word “and” is inclusive. Empty Set/Null set Example: Let A = {1, 2}.and B = {2, It is a unique set with no elements. 3}, then A ∩ B = {2}. For example, A = {} or ∅ Complements of a Sets The Cardinality Number of a Set The complement of A is the set of all It is the number of elements or elements 𝑥 in 𝒰 such that 𝑥 is not in 𝐴. members in the set. It is denoted as 𝐴′. It is denoted by 𝑛(𝐴). The complement of A in the symbol is For finite sets A, 𝑛(𝐴) is the number defined as 𝐴′ = 𝑥 ∈ 𝒰 | 𝑥 ∉ 𝐴. of elements of A. Let 𝒰 = 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓, 𝑔 and 𝐴 = 𝑎, 𝑏, 𝑐, For infinite sets A, write 𝑛(𝐴) = ∞. 𝑑, 𝑒 then 𝐴′ = 𝑓, 𝑔. Equal Sets Difference of Sets Two sets are equal if they have the same The difference between A and B is the set elements. of elements 𝑥 in 𝒰 such that 𝑥 is in 𝐴 and For example, let A = {1, 2, 3, 4, 5} and 𝑥 is not in 𝐵. B = {5, 4, 3, 2, 1}. It is denoted as 𝐴 − 𝐵. The difference between A and B in Universal Set symbol is defined as A − B = 𝑥 | 𝑥 ∈ 𝐴 It is a set that contains everything. 𝑎𝑛𝑑 𝑥 ∉ 𝐵 = 𝐴 ∩ 𝐵′ It is denoted by U. Let 𝐴 = 𝑎, 𝑏, 𝑐 and 𝐵 = 𝑐, 𝑑, 𝑒 then 𝐴 − A set U that includes all the elements 𝐵 = 𝑎, 𝑏. under consideration in a particular discussion. Subset When we define a set and we take pieces of that set, we can form what is called a subset. Operations of Sets Union of Sets The union of A and B is the set of all elements in 𝑥 in 𝒰 such that 𝑥 is in 𝐴 or 𝑥 is in 𝐵. It is denoted by 𝐴 ∪ 𝐵. The union of two sets A and B in symbol is defined by 𝐴 ∪ 𝐵 = 𝑥 | 𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐵. The word “or” is inclusive. Example: Let A = {1, 2} and B = {2, 3}, then A ∪ B = {1, 2, 3}.