Mathematics in Modern World, Lecture notes of Mathematics

Download Mathematics in Modern World and more Lecture notes Mathematics in PDF only on Docsity! CHAPTER 1 | Module 1 THE NATURE OF MATHEMATICS Mathematics explains phenomenon, behavior, structure of nature through patterns, constants which are represented by numbers and equations; predicted. Fibonacci numbers  The numbers in Fibonacci sequence, and characterized by every numbers after the first two is the sum of the two preceding numbers. F(n) = f(n-1) + f(n-2) Examples: 0, 1, 1, 2, 3, 5, 8, 13, 21, 33, 55, 88, 143,………….. Fibonacci Series  A series of numbers in which each number (Fibonacci numbers) is the sum of the two preceding numbers. This sequence of numbers is not just found in flowers. These are found in just about everywhere in nature. Fibonacci in the pineapples There are a lot of interesting patterns in nature. Not just the spiral patterns but are as well geometric such as the following: The patterns in Romanesco broccoli, the different patterns of snowflakes, the basalt columns in North Iceland and the honeycomb are just a few patterns that nature have crafted that gives us amazement. But the truth is, mathematics can reason out how it came to be. Golden Ratio  The ratio of two consecutive Fibonacci numbers as n becomes large, it approaches the golden ratio; that is, This can be verified by measuring some parts of the human body. Some examples are the length of the arm , height, the distance of the fingertips to the elbow. "The ratio of a person's height, to the height of his or her navel is roughly the golden ratio" - Markowsky (1992) Another name for the Golden Ratio is the divine proportion. This must be so because human beauty is based on divine proportion. Who is Fibonacci? Leonardo Pisano (Leonardo of Pisa)  Fibonacci is short for the the Latin of "filius Bonacci" which means "son of Bonacci"  He observed numbers in nature; most popular is the Fibonacci numbers. FIBONACCI NUMBERS  The Fibonacci numbers are nature's numbering system  They appear in nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, pinecone, or the scales of pineapple.  Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees. FIBONACCI SPIRAL  Spirals arise from a property of growth called self- similarity or scaling - the tendency to grow in size but to maintain the same shape.  Not all organisms grow in this self-similar manner. We have seen that adult people, for example, are not just scaled up babies: babies have larger heads, shorter legs, and a longer torso relative to their size.  But if we look for example at the shell of the chambered nautilus, we see a different growth pattern. As the nautilus outgrows each chamber, it builds new chambers for itself, always the same shape - if you imagine a very long-lived nautilus, its shell will spiral around and around, growing ever larger but always looking exactly the same at every scale. CHAPTER 2 | Module 1 MATHEMATICAL LANGUAGE Language facilitates communication and meaning making. It allows people to express themselves and maintain their identity. Read the following statements, and tell whether you agree or disagree with the given statements. - Mathematics is awesome. - Magaling ako sa matematika. - Les maths sont amusantes. - 我愛數學 - ∀𝑥∈R,𝑥2 ≥0 - (m + n)3 = m3 + mn3 + n3 - Παρακαλώ - Hartelijk bedankt There are unfamiliar characters in the written Mandarin language, which makes learning Mandarin more difficult. But it has its alphabet, the same way the English or Greek languages have their own alphabet. Mathematics is also a language, it has its own symbol system, as other languages have their own alphabet. Characteristics of Mathematical Language 1. Precise o make very fine distinction or definitions among set of mathematical symbols 2. Concise o mathematicians can express otherwise long expositions or sentences briefly using the language of Mathematics. 3. Powerful o one can express complex thoughts with relative ease. For example, consider the sentence “The sum of any two real numbers is also a real number.” In mathematical notation, this declarative sentence can be written as: ∀𝑎 , 𝑏 ∈ R, 𝑎 + 𝑏 ∈ R. Mathematics is a symbolic Language Some of the symbols we will use in the module are the following: ∑ “the sum of” ∃ “there exists” ∀ “for every / for any” ∈ “element of / member of” ∈ ̸ “not an element of / not a member of” ⊆ “subset of” ⇒ “if ..., then” ⇔ “if and only if” R “set of real numbers” N “set of natural numbers” Z “set of integers” Q “set of rational numbers” Mathematical Language  can describe a subset of the real world using only these symbols. o In Physics - Free falling bodies, speed, and acceleration; o In Biology - modeling diseases.  It describes abstract structures. o e.g., Pure Mathematics - Abstract Algebra, Linear Algebra, Real Analysis, and Complex Analysis. Therefore, Mathematics is the language of the sciences, business, economics, music, architecture, arts, and politics. There is an intimate connection between the language of Mathematics and the English language. In English Language 1. First ask what this something is. 2. Relate it with what you already know and classify it. 3. Think about the rules that govern such things. In Mathematical Language 1. You usually look for a definition of the mathematical word or concept. 2. You apply this based on the meanings attributed to it by identifying valid applications or operations of the concept.