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MATHEMATICS IN THE MODERN WORLD
Mathematics is the study of the relationships among numbers, quantities, and shapes. It includes arithmetic, algebra, trigonometry, geometry, statistics and calculus.
- Mathematics nurtures human characteristics like the power of creativity, reasoning, critical thinking, spatial thinking and others. It provides the opportunity to solve both simple and complex problems in many real-world contexts using a variety of strategies.
- is a universal way to make sense of the world and to communicate understanding of concepts and rules using mathematical signs, symbols, proofs, language and conventions.
- Mathematics helps organize patterns and regularities in the world. Most patterns found in nature were later on associated with numerals.
- Mathematics helps predict the behavior of nature and phenomena in the world. It is instrumental to control natural phenomena for the betterment of the human race.
- Mathematics is regarded as a science of patterns and it helps students to utilize, recognize and generalize patterns that exist in numbers, in shapes and in the world around them. A. Patterns and Numbers in Nature and the World
- Patterns in nature are visible regularities found in the natural world. The patterns can sometimes
- Natural Patterns such as spirals, symmetries, mosaic, stripes and spots
- Plato, Pythagoras, Empedocles and other early Greek philosophers studied patterns to explain order in nature. Examples: (19th Century)
- Joseph Plateau – examined soap films, concept of minimal surface.
- Ernst Haeckel – painted hundreds of marine organisms to emphasize their symmetry
- D’ ArcyThompson pioneered the study of growth patterns , showing that simple equations could explain spiral growth. (20th Century)
- Alan Turing – predicted mechanisms of morphogenesis which gives rise to patterns of spots and stripes
- Aristid Lindenmayer and Benoît Mandelbrot showed how the mathematics of fractals could create plant growth patterns.
- W. Gary Smith adopted 8 patterns in his landscape work: scattered, fractured, mosaic, naturalistic drift, serpentine, spiral, radial and dendritic. B. The Fibonacci Sequence Leonardo Pisano Bigollo lived between 1170 and 1250 in Italy. His nickname, “Fibonacci” means “son of Bonacci”. He is famous for the Fibonacci Sequence. November 23 is named “Fibonacci Day.” 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181… The terms of the Fibonacci sequence are obtained by simply adding two consecutive numbers : 1+1= 1+2= 2+3= 3+5= 5+8=13… The Golden Ratio The ratio of any two successive Fibonacci Numbers is very close to what is referred to as the Golden Ratio and represented by phi (φ) and is approximately equal to 1.618034 ... such as 21/13 = 1.615... 34/21 = 1.619... 55/34 = 1.617... 89/55 = 1.618... 144/89 = 1.617... 233/144 = 1.618... The Golden Spiral The golden spiral is a logarithmic spiral whose growth factor is phi , the golden ratio. The golden spiral gets wider (or further from its origin) by a factor of phi for every quarter turn it makes.
C. Patterns and Regularities in the World as organized by Mathematics Patterns, relationships and functions constitute a unifying theme of mathematics. Many beautiful phenomena observed in nature can be described in mathematical terms. The world is made up of orders (such as the regular cycles of day and night, recurrence of seasons and alternate sunrise and sunset) and symmetry such as the fractal patterns from which similarity, predictability and regularity in nature and the world consequently exist. Examples of Spectacular Patterns Symmetries Mathematics helps organize patterns and regularities in the world. The great secret uncovered by mathematics: Nature’s patterns are not just there to be admired ; they are vital clues to the rules that govern natural processes. What is it for? D. Phenomena in the World as Predicted by Mathematics
- To describe symmetry-breaking processes.
- To help us unravel the puzzles of nature, a useful way to think about nature.
- To exercise the human mind in abstracting the results of observation to find similarities and differences between phenomena. To summarize, formalize, interpolate and extrapolate from recorded observations.
TYPES OF PATTERNS
- Spirals - series of circles, curve pattern
- Symmetries (Bilateral or Mirror/Radial or Rotational /Three fold /Four fold /Five fold /Six fold) - the mirror of the other one, identical
- Mosaics (Tessellations) - regular/irregular stones
- Stripes - pattern of strips
- Spots - like cats - Dendritic - pattern of branching, root - Scattered - raindrops, clouds - Fractured - cracks - Naturalistic Drift - wave, desert - Serpentine / Meander - snake - Foam - bubbles, bear We live in the Universe full of Patterns
- Ian Stewart MATHEMATICS IS EVERYWHERE! E. Nature and Occurrences in the World as Controlled by Mathematics Logic and creativity are essential aspects of Mathematics. For a particular person, the essence
Vocabulary vs Sentences Every language has its vocabulary (the words), and its rules for combining these words into complete thoughts (the sentences). Mathematics is no exception. As a first step in discussing the mathematical language, we will make a very broad classification between the ‘nouns’ of mathematics (used to name mathematical objects of interest) and the ‘sentences’ of mathematics (which state complete mathematical thoughts). B. Mathematical Expressions and Sentences English: Nouns vs Sentences Mathematics: Expressions vs Sentences The mathematical analogue of a ‘noun’ will be called an expression. Thus, an expression is a name given to a mathematical object of interest. Whereas in English we need to talk about people, places, and things, we’ll see that mathematics has many different ‘objects of interest’. The mathematical analogue of a ‘sentence’ will also be called a sentence. A mathematical sentence, just as an English sentence, must state a complete thought. The table below summarizes the analogy. Since people frequently need to work with numbers, these are the most common type of mathematical expression. And, numbers have lots of different names. For example, the expressions 7 5+2 7(1) 14 ÷ 2 (8 – 2) + 1 1+1+1+1+1+1+ The simple idea—that numbers have lots of different names—is extremely important in mathematics. English has the same concept: synonyms are words that have the same (or nearly the same) meaning. However, this ‘same object, different name’ idea plays a much more fundamental role in mathematics than in English.
- The number ‘six’ has lots of different names. Give names satisfying the following properties. There may be more than one correct answer. a) the ‘standard’ name 6 b) a name using a plus sign, + 5+ c) a name using a minus sign, - 8 – 2 d) a name using a division sign, ÷ 12÷ MATHEMATICAL EXPRESSION
- A mathematical expression is a mathematical analogue of an English noun.
- It DOES NOT state a complete thought.
- The most common expressions are numbers, variables, sets, functions Examples: 6 5x y- MATHEMATICAL SENTENCE
- A mathematical sentence is a mathematical analogue of an English sentence.
- It states a complete thought Examples: 1 + 6 = 7 , 8x = 16 , x – 3 > 2
- It always used relation symbols such as = , ≠ , < , > , ≤ or ≥.
- Mathematical sentences have verbs.
- In 1 + 6 = 7 , the verb is equal “ = “. A mathematical sentence can
- Always TRUE
- Always FALSE
- Sometimes True / Sometimes False Ideas regarding sentences 1. Sentences have verbs. Just as English sentences have verbs, so do mathematical sentences. In the mathematical sentence “3 + 4 = 7 “, the verb is “ = “. If you read the sentence as ‘three plus four is equal to seven’, then it’s easy to ‘hear’ the verb. Indeed, the equal sign “= “is one of the most popular mathematical verbs. 2. Truth of Sentences Sentences can be true or false. The notion of truth (i.e., the property of being true or false) is of fundamental importance in the mathematical language.
Conventions in the Mathematical Language Languages have conventions. In English, for example, it is conventional to capitalize proper names (like ‘Juan’ and ‘Mandaluyong’). This convention makes it easy for a reader to distinguish between a common noun (like ‘teacher’, a Christmas song) and a proper noun (like ‘Ms. Reyes’). Mathematics also has its conventions, which help readers distinguish between different types of mathematical expressions. There are two things to consider to understand the meaning of mathematical symbols
- Context that refers to a particular topic being studied and it is important to understand the context to understand mathematical symbols. Example: different meaning of “is”:
- 2 is a square root of 4.
- 2 is less than 5.
- 2 is a prime number.
- Convention is a technique used by mathematicians, engineers, scientists in which each particular symbol has particular meaning. Examples: position of numbers and symbols; subscripts and superscripts; Greek and Latin Letters **D. Four Basic Concepts
- SETS** A set is a well-defined collection of objects called elements or members of the set. Class, collection and family are words that are synonymous to sets. Capital letters are usually used to denote a set and lowercase letters are used to denote the elements of the set. Example: A = {a, e, i, o u} The statement that an element a belongs to a set s is written as a ∈ S Example: a ∈ A, x ∉ A We also write a, b ∈ S when both a and b belong to S. Suppose every element of a set A also belongs to a set B, that is, suppose a ∈ A implies a ∈ B, then A is called a subset of B, or A is said to be contained in B, which is written as A ⊆ B. Example: B = {letters of the English alphabet} Then A ⊂ B. Two sets are equal if they both have exactly the same elements, or equivalently, if each is contained in the other. That is A = B if A ⊆ B or B ⊆ A. Example: If A = {1, 2, 3}and B = {3, 2, 1},then A = B. The negations of a ∈ A, A ⊆ B, and A = B are written as a ∉ A, A ⊈ B and A ≠ B, respectively. It is common practice in mathematics to put a vertical line or slanted line through a symbol to indicate the opposite or negative meaning of the symbol. The statement A ⊆ B does not exclude the possibility that = B. In fact, for any set A, we have since, trivially, every element in A belongs to B. However, if A ⊆ C and A ≠ B, then we say that A is a proper subset of B (sometimes written as A ⊂ B). Suppose every element of a set A belongs to a set B, and every element of B belongs to a set C. Then clearly every element of A belongs to C. In other words, if A ⊆ B and B ⊆ C , then A ⊆ C. 2. SPECIFYING SETS ● roster/tabular method – obtained by simply listing or enumerating the elements of a set, enclosed by braces and separated by a comma example: A = {2, 4, 6 ,8} ● rule/descriptive method – obtained by describing the set with a general rule and usually represented by a set builder notation. example: B = {x: x is an even integer, x > 0} 3. KINDS OF SETS 1. empty/null/void set is a set without any element denoted by { } or ∅ and is regarded as a subset of every other set. 2. Finite set is a set with countable number of elements 3. Infinite set is a set with an uncountable number of elements and often characterized by ellipses. 4. Universal set is some large fixed set also known as the universe of discourse denoted by U. 5. Disjoint sets are sets with no elements in common otherwise, they are called joint sets. 6. Equivalent sets are two or more sets with the same cardinal number.
Classes of Sets Given a set S, we may wish to talk about some of its subsets. Thus, we would be considering a “set of sets”. Whenever such a situation arises, we use the terms class of sets or a collection of sets. Or a” subcollection” that has meanings analogous to subset. Example Let S = {1, 2, 3, 4} Find 1) A, the class of subsets of S which contains exactly 3 elements of S Answer: A = {(1,2,3), (1,2,4), (1,3,4), (2,3,4)} 2) B, the class of subsets of S which contains the numeral 2 and two other elements of S Answer: B = {{2,1,3), (2,1,4), (2,3,4)} Power Sets For any given set S, we may consider the class of all subsets of S. This class is called the power set of S, denoted by P(S). If S is finite, then so is P(S). In fact, the number of elements in P(S) is 2 raised to the power of S; that is n(P(S)) = 2n(S) or 2S Example:
- Suppose S = {1, 2, 3}. Find the power set of S. Solution: n(P(S)) = 2n(S)= 23 = 8 P(S) = {(1), (2), (3), (1, 2), (1, 3), (2, 3), (1, 2, 3), ({ })}
- Suppose S = {h, o, p, e}. Find the power set of S. Solution: n(P(S)) = 2n(S)= 24 = 16 P(S) = {(h), (o), (p), (e), (h, o), (h, p), (h, e), (o, p), (o, e), (p, e), (h, o, p), (h, o, e), (h, p, e), (o, p, e), (h, o, p, e), ({ })} 2. Relations A relation is a rule that relates values from a set of values (called the domain or x-values) to a second set of values (called the range or y-values). The elements of the domain can be imagined as input to a machine that applies a rule to these inputs to generate one or more outputs. A relation is also a set of ordered pairs (x, y). Examples:
- (-2, -3), (-1, -2), (0, -1), (1, 0)
- (8, 1), (8, 2), (8, 4), (8, 8)
- (-1, 0), (0, -1), (1, 0), (2, 3)
- (6, 2), (6, 3), (4, 2), (4, 1) Kinds of Relations
- One-to-one
- One-to-many
- Many-to-one
- Many-to-many Relations can be represented through ordered pairs, tables of values, mapping diagrams and graphs. Relation Symbols Relation symbols behave like adjectives that refer to a property rather than an object. Statement of relationship Example: 2 is less than 5 Example: “Equals” and “is an element of” are two other examples of relations 3. FUNCTIONS A function is a relation where each element in the domain is related to only one value in the range by some rule. It is a set of ordered pairs (x, y) such that no two ordered pairs have the same x–value but different y–values. Thus, only one- to-one and one-to-many relations exist as functions. DOMAINS (x value) SHOULD BE UNIQUE 1. f = {(0, 1), (1, 2), (2, 3), (3, 4)} answer: function because all the domains are unique 2. g = {(-1, 1), (0, 0), (1, 1), (2, 4)} answer: function because all the domains are unique 3. h = {(1, 3), (1, 4), (2, 5), (2, 6)} answer: not a function because the ordered pairs (1, 3) and (1, 4) have the same domain Evaluating Functions To evaluate a function is to replace the variable in the function, such as x, with a value from the function’s domain and calculate the result and is denoted by f(x) for some value of x in the domain of f. Operations on Functions To perform operations on functions, the rules for adding, subtracting, multiplying and dividing algebraic expressions as well as fractions are utilized. Adding, subtracting and multiplying two or more functions together will result in another function while dividing functions will result in another function if the denominator is not a zero function. Definition
- A composite function is a function within a function. The process of obtaining a composite function is called composition function.
4. Binary Operations Binary literally means consisting of two parts. Mathematically speaking, binary numbers belong to a system of numbers called the binary system that uses the base 2. Thus, binary numbers are numbers made up of only 0’s and 1’s that has many applications in the digital world especially in programming. A single binary digit is known as a bit so that the binary number 11001 has 5 bits. A binary number is written with a subscript 2 to distinguish it from a decimal number – the one we are familiar with that uses 10 as a base. Binary Numbers to Decimal To convert a binary number to its decimal equivalent, use the powers of 2 and its multiplicative inverse for binary numbers containing a decimal part. For the decimal part, use the negative powers of two converted to their fractional and decimal equivalents such as:
The most common connectives and their symbols are: and/but ∧; or ∨; if ...,then →. Example: Your dress is beautiful and I like its color. S ∧ C G. Formality Formality is a relational concept. An expression can be more or less formal relative to another expression, involving an ordering of expressions. But it is said that no expression is absolutely formal or absolutely informal and all linguistic expressions lie somewhere in the middle of absolute formality and absolute informality. An expression is said to be completely formal when it is context independent, definite and precise. It represents a clear distinction that is undeviating regardless of context. Generalizations:
- Mathematics is a language in itself. Hence, it is useful in communicating important ideas.
- Mathematics as a language is clear and objective. 3.Language conventions are necessary in mathematics for it to be understood by all. PROBLEM SOLVING AND REASONING Problem Solving is the ability to make choices , interpret , formulate , model and investigate problem situations , and communicate solutions effectively. Reasoning is a sophisticated capacity for logical thought and actions , such as analyzing, proving, evaluating, explaining, inferring, justifying, and generalizing. There are two major types of reasoning , inductive and deductive is a process by which someone creates a conclusion as well as how they believe their conclusion to be true. ● Deductive reasoning Deductive reasoning is the process of reaching a conclusion by applying general assumptions, procedures, or principles. It starts with a few general ideas , called premises, and applies them to particular situations. Deductive arguments are meant to prove a conclusion. A conclusion is right if it can be proven by recognized rules, laws, theories, and other widely accepted truths. The most basic form of deductive reasoning in practice is the syllogism, where two premises that share some idea support a conclusion. It may be easier to think of syllogisms as the following theorem: If A=B and C=A, then B=C. DEDUCTIVE REASONING- GENERAL > SPECIFIC ● INDUCTIVE REASONING Inductive Reasoning is the process of reaching a general conclusion by examining specific examples. The conclusion formed by using inductive reasoning is often called a conjecture, which may or may not be correct. It uses specific observations before reaching a conclusion. It starts with a few particular premises then creates a pattern that can give way to a broad idea that is likely true. Inductive reasoning is based on finding a conclusion that is most likely to fit the premises. It is usually used when making predictions, creating generalizations, and analyzing cause and effect. Inductive arguments are meant to predict a conclusion and try to show that the conclusion is probable based on the given premises. An inductive argument is considered either as weak or strong based on whether the conclusion is likely to explain the premises. INDUCTIVE REASONING- SPECIFIC > GENERAL Determine Types of Reasoning
- During the past 10 years, a tree has produced plums every other year. Last year the tree did not produce plums. So, this year the tree will produce plums. - DEDUCTIVE
- All students need an internet connection these days. Juan is a student. Therefore, Juan needs internet connection. - DEDUCTIVE
- All Dan Brown novels are worth reading. Origin is a Dan Brown novel. Therefore, Origin is worth reading. - DEDUCTIVE
- Maria is not allowed to go outside her home according to the IATF law. People under 21 years old are not allowed to go out these days. Therefore, Maria is under 21 years old. - DEDUCTIVE
B. Intuition, Proof & Certainty Intuition is the ability to understand something instinctively, without the need for conscious reasoning. Mathematical Proof is an argument, which convinces other people that something is true. Proof is an inferential argument for a mathematical statement. Certainty is a total continuity and validity of inquiries to the highest degree of precision. It is a conclusion that is beyond doubt. C. Polya’s Four Steps One of the foremost recent mathematicians to make a study of problem solving was George Polya (1887–1985). He was born in Hungary and moved to the United States in 1940. After his brief stay at Brown University, he moved to Stanford University in 1942 and taught there until his retirement. While at Stanford, he published 10 books and a number of articles for mathematics journals. Of the books Polya published, How to Solve It (1945) is one of his best known. In the book, he outlined a strategy for solving problems from virtually any discipline. Polya’s Four-Step Problem-Solving Strategy
1. Understand the Problem. You must have a clear understanding of the problem. To help you understand the problem, consider the following questions: ● Can you restate the problem in your own words? ● Can you determine what is known about these types of problems? ● Is there missing information that, if known, would allow you to solve the problem? ● Is there extraneous information that is not needed to solve the problem? 2. Devise a Plan. Successful problem solvers use a variety of techniques when they attempt to solve a problem. Some frequently used procedures are the following. ● Make a list of the known information and information that is needed. ● Draw a diagram. ● Make a table or a chart. ● Work backwards. ● Try to solve a similar but simpler problem. ● Look for a pattern. ● Write an equation. If necessary, define what each variable represents. ● Perform an experiment. ● Guess at a solution and then check your result
- Carry Out the Plan. After devising a plan, you must carry it out. ● Work carefully. ● Keep an accurate and neat record of all your attempts ● Realize that some of your initial plans will not work and that you may have to devise another plan or modify your existing plan. 4. Review the Solution. Once you have found a solution, check the solution. ● Ensure that the solution is consistent with the facts of the problem. ● Interpret the solution in the context of the problem. ● Ask yourself whether there are generalizations of the solution that could apply to other problems. D. Problem Solving Strategies ● Guess and check ● Act it out ● Draw ● List/Tabulate E. Mathematical Problems Involving Patterns Terms of a sequence A sequence is an ordered list of numbers. The numbers in a sequence separated by commas are the terms of the sequence.
