GE4 Mathematics in the Modern World Module 1 PDF

Prelim Part 2 GE4: Mathematics in the Modern World Module 1:The Nature of Mathematics Learning Outcomes: The students will be able to: 1. Discuss the language, symbols, and conventions in mathematics; 2. Explain the nature of mathematics as a language; 3....

Prelim Part 2 GE4: Mathematics in the Modern World Module 1:The Nature of Mathematics Learning Outcomes: The students will be able to: 1. Discuss the language, symbols, and conventions in mathematics; 2. Explain the nature of mathematics as a language; 3. Perform operations on mathematical expressions correctly; 4. Acknowledge that mathematics is a useful language. Prepared by: Daniel O. Roxas Instructor  GE4: Mathematics in the Modern World Class Policies and Course Requirements 1. Students are expected to submit their problem sets and exercises on time. No time extension will be given unless permitted by the professor/instructor. Any output submitted beyond the scheduled date shall receive a point deduction. 2. Any form of cheating is not allowed. Students that are caught/proved cheating will automatically receive a failing grade. 3. Students are expected to display proper decorum and behaviour especially when communicating with the professor/instructor. This is also applicable when communicating online or thru social media. 4. Use black or blue pens when answering problem sets and exercises. GRADING SYSTEM Course Requirements Percentile Assignments/Exercises 20% Quizzes 20% Problem Sets 20% Term Examinations 40% 100% Property of and for the exclusive use of ASCOT. Reproduction, distribution, uploading or posting online of any part of this document, without prior permission of ASCOT, is strictly prohibited. Page 1  GE4: Mathematics in the Modern World UNIT 2: Mathematical Language and Symbols Section 1: Characteristics of Mathematical Language The language of mathematics makes it easy to express the kinds of thoughts that mathematicians like to express. It is:  Precise (able to make very fine distinctions);  Concise (able to say things briefly);  Powerful (able to express complex thoughts with relative ease). The language of mathematics can be learned, but requires the efforts needed to learn any foreign language. The goal is to get extensive practice with mathematical language ideas, to enhance your ability to correctly read, write, speak, and understand mathematics. 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). The classification of mathematical ‘nouns’ versus ‘sentences’ does not typically appear in math books. However, there is tremendous benefit to be derived from this classification of the basic building blocks of mathematics. Without such an understanding, people are more likely to fall prey to common syntax errors – for example, inappropriately setting things equal to zero, or stringing things together with equal signs, as if ‘=’ means ‘I’m going to the next step.’ Section 2: Expressions vs. Sentences English: Nouns vs. Sentences In English, nouns are used to name things we want to talk about (like people, places, and things); whereas sentences are used to state complete thoughts. A typical English sentence has at least one noun, and at least one verb. For example, consider the sentence Carol loves mathematics. Here, ‘Carol’ and ‘mathematics’ are nouns, ‘loves’ is a verb. 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 much different ‘object 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 on the next page summarizes the analogy. Property of and for the exclusive use of ASCOT. Reproduction, distribution, uploading or posting online of any part of this document, without prior permission of ASCOT, is strictly prohibited. Page 2  GE4: Mathematics in the Modern World ENGLISH MATHEMATICS Name given to an object of NOUN (person, place, thing) EXPRESSION interest: Examples: Carol, Aurora, book Examples: 5, 2 + 3, A complete thought: SENTENCE SENTENCE Examples: Examples: The capital of Aurora is Baler. 3+4=7 The capital of Aurora is 3+4=8 Dipaculao. Numbers have lots of different names 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 5 2+3 10÷2 (6 – 2) + 1 1+1+1+1+1 all look different, but are all just different names for the same number. This 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, the ‘same object, different name’ idea plays a much more fundamental role in mathematics than in English, as you will see throughout the book. EXAMPLE 1: 1. Give several synonyms for the English word ‘beautiful’. 2. The number “three” has lots of different names. Give names satisfying the following properties. There may be more than one correct answer. a. A name using a plus sign, + b. A name using a minus sign, – c. A name using a division sign, ÷ Solution/Answer: 1. Pretty, Handsome, Gorgeous 2. Different names for number ‘three’: a. 1 + 2, 3 + 0 b. 5 – 2, 200 – 197 c. 6 ÷ 2, 18 ÷ 6 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. Property of and for the exclusive use of ASCOT. Reproduction, distribution, uploading or posting online of any part of this document, without prior permission of ASCOT, is strictly prohibited. Page 3  GE4: Mathematics in the Modern World 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. EXAMPLE 2: Truth Values Determine the truth value of each mathematical sentence: a. 7 – 3 = 4 b. 7 – 3 = 5 c. x + 10 = 11 Answer/Solution: a. 7 – 3 = 4 Always true b. 7 – 3 = 5 Always false; because 7 – 3 = 4 c. x + 10 = 11 Sometimes true/Sometimes false; because when x = 1, then the sentence is true, otherwise (i.e. if x = 2, then we have 2 + 10 = 11 is false), it is false. Convention in Languages Languages have conventions. In English, for example, it is conventional to capitalize proper names (like ‘Carol’ and ‘Baler’). This convention makes it easy for a reader to distinguish between a common noun (like ‘carol’, a Christmas song) and a proper noun (like ‘Carol’). Mathematics also has its conventions, which helps readers distinguish between different types of mathematical expressions. ASSIGNMENT 2: Determine whether the item is an English noun/Mathematical expression or English sentence/Mathematical sentence. If the item is a sentence, encircle the verb. (30 pts) Examples: Carol loves mathematics English sentence 3+4=7 Mathematical sentence 1. Manila Bay ____________ 2. 5 + 2 ____________ 3. The sand is white. ____________ 4. 1 + 1 = 2 ____________ 5. 1 + 1 > 3 ____________ 6. x ____________ 7. x ≠ 2 ____________ 8. y = mx + b ____________ 9. This sentence is false. ____________ 10. Virus ____________ 11. 3 + 5 – 6 ____________ 12. a < g ____________ 13. a + b ____________ 14. 5 + 2(6 – 5) ____________ 15. Aurora State College of Technology ____________ Property of and for the exclusive use of ASCOT. Reproduction, distribution, uploading or posting online of any part of this document, without prior permission of ASCOT, is strictly prohibited. Page 4  GE4: Mathematics in the Modern World Definitions in Mathematics With several examples behind us, it is now time to make things more precise. In order to communicate effectively, people must agree on the meanings of certain words and phrases. When there is ambiguity, confusion can result. Consider the following conversation in a car at a noisy intersection: Carol: “Turn left!” Bob: “I didn’t hear you. Left?” Carol: “Right!” Question: Which way will Bob turn? It depends on how Bob interprets the word ‘right’. If he interprets ‘right’ as the opposite of ‘left’, then he will turn right. If he interprets ‘right’ as the opposite of ‘left’, then he will turn right. If he interprets ‘right’ as ‘correct’, then he will turn left. Although there are certainly instances in mathematics where context is used to determine correct meaning, there is much less ambiguity allowed in mathematics than in English. The primary way that ambiguity is avoided is by the use of definitions. By defining words and phrases, it is assured that everyone agrees on their meaning. Here’s our first definition: Expressions An expression is the mathematical analogue of an English noun; it is a correct arrangement of mathematical symbols used to represent a mathematical object of interest. An expression does NOT state a complete thought; in particular, it does not make sense to ask if an expression is true or false. In most mathematics books, the word ‘expression’ is never defined, but is used as a convenient catch-all to talk about anything (including sentences) to which the author wants to draw attention. In our discussion, expression and sentences are totally different entities. They don’t overlap. If something is an expression, then it’s not a sentence. If something is a sentence, then it’s not an expression. Be careful about this. The most common problem type involving an expression is: SIMPLIFY: To simplify an expression means to get a different name for the expression, that in some way is simpler. The notion of ‘simpler’, however, can have different meanings:  FEWER SYMBOLS: Often, ‘simpler’ means using fewer symbols. For example, ‘3 + 1 + 5’ and ‘9’ are both names for the same number, but ‘9’ uses fewer symbols.  FEWER OPERATIONS: Sometimes, ‘simpler’ means using fewer operations (an ‘operation’ is something like addition or multiplication). For example, ‘3 + 3 + 3 + 3 + 3’ and ‘5 × 3’ are both names for the same number, but the latter uses fewer operations. There are four additions used in ‘3 + 3 + 3 + 3 + 3’, but only one multiplication used in ‘5 × 3’.  BETTER SUITED FOR CURRENT USE: In some cases, ‘simpler’ means better suited for the current use. For example, we’ll see in a future section that the name is a great name for the number ‘1’ if we need to convert units of inches to units of feet.  PREFFERED STYLE OR FORMAT: Finally, ‘simpler’ often means in a preferred style or format. For example, (two-fourths) and (one-half) are both names for the same number, but people usually prefer Property of and for the exclusive use of ASCOT. Reproduction, distribution, uploading or posting online of any part of this document, without prior permission of ASCOT, is strictly prohibited. Page 5  GE4: Mathematics in the Modern World the names for the same number, but people usually prefer the name ; it is said to be in ‘reduced form’ or ‘simplest form’. Mathematical Sentences A mathematical sentence is the analogue of an English sentence; it is a correct arrangement of mathematical symbols that states a complete thought. It makes sense to ask about the TRUTH of a sentence: Is it true? Is it false? Is it sometimes true/sometimes false? A question commonly encountered , when presenting the sentence example ‘1 + 2 = 3’, is the following: If ‘=’ is the verb, then what is the ‘+’? Here’s the answer. The symbol ‘+’ is a connective; a connective is used to ‘connect’ objects of a given type to get a ‘compound’ object of the same type. Here, the numbers 1 and 2 are ‘connected’ to give the new number 1 + 2. A familiar English connective for nouns is the word ‘and’: ‘cat’ is a noun, ‘dog’ is a noun, ‘cat and dog’ is a ‘compound’ noun. There are two primary ways to decide whether something is a sentence, or not:  Read it aloud, and ask yourself the question: Does it state a complete thought? If the answer is ‘yes’, then it’s a sentence. Notice that expressions do not state a complete thought. Consider, for example, the number ‘1 + 2’. Say it aloud: ‘one plus two’. Have you stated a complete thought? NO! But, if you say ‘1 + 2 = 4’, then you have stated a complete (false) thought.  Alternately, you can ask yourself the question: Does it make sense to ask about the TRUTH of this object? Consider again the number ‘1 + 2’. Is ‘1 + 2’ true? Is ‘1 + 2’ false? These questions don’t make sense, because it doesn’t make sense to ask about the truth of an expression. ASSIGNMENT 3: Determine whether the given item is an expression or a sentence. If the item is a sentence, classify its truth value: Always true (AT); Always false (AF); or sometimes true/sometimes false (ST/SF). (20 pts) Examples: 1 + 2 Expression 1+1=2 Sentence, AT x+1=2 Sentence, ST/SF 1. 1 + 2 + 3 + 4 + 5 = 15 _____________ 2. 5 + 3(27 ÷ 3) _____________ 3. 9x + 5 = 14 _____________ 4. 25 – 5(13 – 9 + 1) = 1 _____________ 5. a + b = c _____________ 6. 6 – 9 = -3 _____________ 7. x + 1 _____________ 8. 100 _____________ 9. 10 > 3 _____________ 10. 2x < 4x _____________ Property of and for the exclusive use of ASCOT. Reproduction, distribution, uploading or posting online of any part of this document, without prior permission of ASCOT, is strictly prohibited. Page 6  GE4: Mathematics in the Modern World REFERENCES  Fisher, C. B. (n.d.) The Language of Mathematics. One Mathematical Cat, Please! Retrieved from: http://www.onemathematicalcat.org  Jamison, R.E. (2000). Learning the Language of Mathematics Property of and for the exclusive use of ASCOT. Reproduction, distribution, uploading or posting online of any part of this document, without prior permission of ASCOT, is strictly prohibited. Page 7