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1 MODULE 1 – THE NATURE OF MATHEMATICS OVERVIEW: This module provides a holistic understanding of mathematics and its relation in our modern world. We usually tend to identify and follow patterns, whether consciously or not. It feels natural to recognize patterns, and it’s like that our brain is programed to recognize them. Humans in early civilizations recognize the repeating interval of night and day, the cycle of seasons (winter, spring, summer, and fall), the falling and rising of tides, and many others. Similarly, flowers follow certain patterns such as arrangement of leaves and stems in a plant, the shape of snowflake, the flower’s petals, and even the shape of the snail’s shell. MODULE OBJECTIVES: After successfully completing the module, you should be able to: 1. Identify the patterns in nature and regularities in the world. 2. Explain the importance of mathematics in our life. 3. Express appreciation for mathematics as a human endeavor. COURSE MATERIALS: 1.1 Mathematics What is mathematics? Is it about arithmetic? The study of numbers? A body of formulas and rules for solving and equations? A useless obstacle course in school? Many people consider mathematics as a boring and formal science, but any good work in mathematics always has in it. Mathematics has beauty, simplicity, structure, imagination, and crazy ideas! Mathematics is a language. It enables us to communicate thoughts and meanings to each other. It is a powerful language, helping us represent and communicate ideas with precision. It is the language of science and technology. The world is built in the ideas of mathematics. Mathematics is therefore a tool, with applications in many aspects of our lives. Mathematics is a way of seeing, a way of making sense of the world. In other words, mathematics is not: only about numbers and arithmetic, a useless obstacle course in school, and study of formulas and techniques in computing. We encounter math everyday: Figure 1.1 For example, modern cars and machines run on calculus (Figure 1.1).
2 1.2 Fibonacci Numbers If you count the number of petals in most flowers, you will notice that they are either one petal, two petals, three petals, five, or eight. This sequence of numbers form the set {1, 1, 2, 3, 5, 8, 13,...} whose pattern was discovered by Fibonacci, a great European mathematician of the Middle Ages. His full name in Italian is Leonardo of Pisa (Figure 1.2), because he was born in Pisa, Italy around 1775. Fibonacci is the shortened word for the Latin term “filius Bonacci” which stands for “son of Bonaccio”. His father’s name was Guglielmo Bonaccio. (Figure 1.2) Numbers in nature are usually observed in Fibonacci. Surprisingly, these petal counts represent the numbers in Fibonacci sequence. (However, not all numbers of petals of a flower follow the patterns discovered by Fibonacci). (Figure 1.3) The principle behind the Fibonacci numbers is as follows: Let xn be the nth integer in the Fibonacci sequence, the next (n+1)th term xn – 1 is determined by adding nth and the (n – 1)th integers. Example: Let x1 = 1 be the first term, and x2 = 1 be the second term, the third term x3 is found by x3 = x1 + x2 = 1 + 1 = 2 The fourth term x4 is 2 + 1 = 3, the sum of the third and the second term. To find the new nth Fibonacci number, simply add the two numbers immediately preceding this nth number.
3 These numbers arranged in increasing order can be written as the sequence {1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...}. (Figure 1.4) There are 34 spiral rows that go in the clock-wise direction and 55 spiral rows that go in the counter clock-wise direction of the sunflower . 34 and 55 are both Fibonacci numbers. 1.3 Mathematics Around Us We encounter mathematics everyday, everywhere! Below are some examples of mathematics that can be found in nature. (Figure 1.5) Dead bees in hexagon (Figure 1.5) shows a hexagonal figure (a polygon of six sides) which is common in nature. (Figure 1.6) A leaf (Figure 1.6) uses its own mathematics to create the pathways that deliver nutrients to all its parts in the most efficient way possible.