Mathematics in the Modern World PDF

Mathematics in the Modern World Ms. Jennifer I. Veloria  INTRODUCTION WHAT ARE YOUR EXPECTATIONS IN THIS SUBJECT? WHAT ARE YOUR EXPECTATIONS IN YOUR PROFESSOR? RATE YOUR COMPETENCY IN MATHEMATICS SUBJECT, 10 IS THE HIGHEST.  EXPECTATIONS 1. TIMELY COMPLETION OF OUTPUTS. 2. COURTESY, POLITEN...

Mathematics in the Modern World Ms. Jennifer I. Veloria  INTRODUCTION WHAT ARE YOUR EXPECTATIONS IN THIS SUBJECT? WHAT ARE YOUR EXPECTATIONS IN YOUR PROFESSOR? RATE YOUR COMPETENCY IN MATHEMATICS SUBJECT, 10 IS THE HIGHEST.  EXPECTATIONS 1. TIMELY COMPLETION OF OUTPUTS. 2. COURTESY, POLITENESS AND RESPECTFULNESS. 3. SOME INVOLEMENT IN THE CLASS. 4. ACADEMIC HONESTY. 5. COMMUNICATION 6. LEVEL OF PROFESSIONAL DISTANCE.  Course Description: This course deals with the nature of mathematics, appreciation of its practical, intellectual and aesthetic dimensions, and application of mathematical tools in daily life. The course begins with an introduction to the nature of mathematics as an exploration of patterns (in nature and the environment) and as an application of inductive and deductive reasoning. By exploring these topics, students are encouraged to go beyond the typical understanding of mathematics as merely a set of formulas but as a source of aesthetics in patterns of nature, for example, and a rich language in itself ( and of science) governed by logic and reasoning.  The course then proceeds to survey ways in which mathematics provides a tool for understanding and dealing with various aspects of present day living such as managing personal finances, making social choices, appreciating geometric designs, understanding codes used in data transmission and security and dividing limited resources fairly. These aspects will provide opportunities for actually doing mathematics in a broad range of exercises that bring out the various dimensions of mathematics as a way of knowing, and test the students’ understanding and capacity. Prelim Topics 1. Fundamental Operations with Numbers ∙ Computation with Algebraic Expressions ∙ Four System 3. Properties of Numbers ∙ System of Real Numbers ∙ Sets of Numbers ∙ Graphical Representatives of Real Numbers ∙ Properties ∙ Properties of addition and Multiplication of ∙ Additional Properties Real Numbers 4. Special Products ∙ Rule of Signs 5. Factoring ∙ Exponents and Powers ∙ Factorization Procedure ∙ Operations with Fractions ∙ Greatest Common Factor 2. Fundamental Operations with Algebraic ∙ Least Common Multiple Expressions 6. Fractions ∙ Algebraic Expressions ∙ Rational Algebraic Fractions ∙ Terms ∙ Operations with Algebraic Functions ∙ Degree ∙ Complex Fractions ∙ Grouping  FOUR OPERATIONS Four operations are fundamental in algebra, as in arithmetic. These are addition, subtraction, multiplication and division When two numbers a and b are added, there sum is indicated by a + b. Thus 3 + 2 = 5, When a number b is subtracted from a number a, the difference is indicated by a – b. Thus 6 – 2 = 4. Subtraction maybe defined in terms of addition. That is, we may define a- b to represent that number x such that x added to b yields a, or x + b = a. For example, 8 – 3 is that number x which when added to 3 yields 8, i.e., x + 3 = 8; thus 8 – 3 = 5. The product of two numbers a and b is a number c such that a x b = c. The operation of multiplication may be indicated by cross, a dot or parentheses. Thus 5 x 3 = 5. 3 = 5(3) = (5) (3) = 15, where the factors are 5 and 3 and the product is 15. When letters are used, as in algebra, the notation p x q is usually avoided since x may be confused with a letter representing a number.  When a number a is divided by a number b, the quotient obtained is written 𝑎 a ÷ b or or a/b 𝑏 Where a is called the dividend and b the divisor. The expression a/b is also called a fraction, having numerator a and denominator b. Division by zero is not defined. Division may be defined in terms of multiplication. That is, we may consider a/b as that number x which upon multiplication by b yields a, or bx = a. for example, 6/3 is that number x such that 3 multiplied by x yields 6, or 3x = 6; thus 6/3 = 2.  SYSTEM OF REAL NUMBERS The system of real numbers as we know it today is a result of gradual progress, as the following indicates. (1) Natural numbers 1, 2, 3, 4,... ( three dots means “ and so on”) used in counting are also known as the positive integers. If two such numbers are added or multiplied, the result is always a natural number. (2) Positive rational numbers or positive fractions are the quotients of two positive integers, such as 2/3, 8/5, 121/17. The positive rational numbers include the set of natural numbers. Thus the rational number 3/1 is the natural number 3. (3) Positive irrational numbers are numbers which we not rational, such as √2, 𝜋. (4) Zero, written 0, arose in order to enlarge the numbers system so as to permit such operations as 6 -6 or 10 – 10. Zero has the property that any number multiplied by zero is zero. Zero divided by any numbers ≠ 0 (i.e., not equal to zero) is zero. (5) Negative integers, negative rational numbers and negative irrational numbers such as -3, - 2/3, and – √2, arose in order to enlarge the number system so as to permit such operations as 2 – 8 , 𝜋 − 3 𝜋 or 2 − 2 √2.  When no sign is placed before a number, a plus sign is understood. Thus 5 is +, √2 is +√2. Zero is considered a rational numbers without sign. The real number system consists of the collection of positive and negative rational and irrational numbers and zero. Note. The word “real” is used in contradiction to still other numbers involving √−1, which will be taken up later and which are known as imaginary, although they are very useful in mathematics and the sciences. Unless otherwise specified we shall deal with real numbers. GRAPHICAL REPRESENTATION OF REAL NUMBERS  Absolute Value By the absolute value or numerical value of a number is meant the distance of the number from the origin on a number line. Absolute value is indicated by two vertical lines surrounding the number. Thus | −6|= 6, |+4| = 4, | − 3/4 = 3/4. PROPERTIES OF ADDITION AND MULTIPLICATION OF REAL NUMBERS (1) Commutative property for addition. The order addition of two numbers does not affect the result. Thus a + b = b + a, 5+ 3 = 3 + 5 = 8 (2) Associative property for addition. Terms of sum may be grouped in any manner without affecting the result. a + b + c = a + (b + c) = (a + b) + c, 3 + 4 + 1 = 3 + (4 + 1) = (3 + 4) + 1 = 8 (3) Commutative property for multiplication - The order of factors of a product does not affect the result. 𝑎 ∙ 𝑏 = 𝑏 ∙ 𝑎, 2 ∙ 5 = 5 ∙ 2 = 10 (4) Associative property for multiplication. The factors of the product may be grouped in any manner without affecting the result. 𝑎𝑏𝑐 = 𝑎(𝑏𝑐) = (𝑎𝑏)𝑐, 3 ∙ 4 ∙ 6 = 3(4 ∙ 6) = (3 ∙ 4)6 = 72 (5) Distributive property for multiplication over addition. The product of number a by the sum of two numbers (b + c) is equal to the sum of the products ab and ac. 𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐, 4(3 + 2) = 4 ∙ 3 + 4 ∙ 2 = 20 Extensions of these laws may be made. Thus we may add the numbers a, b, c, d, e by grouping in any order as (𝑎 + 𝑏) + 𝑐 + (𝑑 + 𝑒), 𝑎 + (𝑏 + 𝑐) + (𝑑 + 𝑒), etc. Similarity, in multiplication we may write (𝑎𝑏)𝑐(𝑑𝑒) 𝑜𝑟 𝑎(𝑏𝑐)(𝑑𝑒), the result being independent of order or grouping.  RULES OF SIGNS (1) To add two numbers with like signs, add their absolute values and prefix the common sign. Thus 3 + 4 = 7, (−3) + (−4) = −7. (2) To add two numbers with unlike signs, find the difference between their absolute values and prefix the sign of the numbers with greater absolute value. EXAMPLE 1: 17 + (−8) = 9, (−6) + 4 = −2, (−18) + 15 = −3 (3) To subtract one number b from another number a, change the operation to addition and replace b by its opposite, −𝑏. EXAMPLE 2: 12 − (7) = 12 + (−7) = 5, (−9) − (4) = −9 + (−4) = −13 2 − (−8) = 2 + 8 = 1  4) To multiply (or divide) two numbers having like signs, multiply (or divide) their absolute values and prefix a plus sign (or no sign). −6 EXAMPLE 3: (5)(3) = 15, (−5)(−3) = 15, −3 = 2 (5) To multiply (or divide) two numbers having unlike signs, multiply (or divide) their absolute values and prefix a minus sign. −12 EXAMPLE 4: (−3)(6) = −18, (3)(−6) = −18, 4 = −3 EXPONENT AND POWERS When a number a is multiplied by itself n times, the product a. a. a … a (n times) is indicated by the symbols a n which is referred to as “ the nth power of a” or “ a to the nth power” or “ a to the nth.” EXAMPLE 5: 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 = 2 5 = 32, (−5) 3 = (−5)(−5)(−5) = −125 2 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 = 2𝑥 3 𝑎 ∙ 𝑎 ∙ 𝑎 ∙ 𝑏 ∙ 𝑏 = 𝑎 3 𝑏2 (𝑎 − 𝑏)(𝑎 − 𝑏)(𝑎 − 𝑏) = (𝑎 − 𝑏) 3 In 𝑎 𝑛 , the number a is called the base and the positive integer n is the exponent  OPERATIONS WITH FRACTIONS Operation with fractions may be performed according to the following rules. (1) The value of fraction remains the same if its numerator and denominator are both multiplied or divided by the same number provided the number is not zero. 𝟑 𝟑.𝟐 𝟔 𝟏𝟓 𝟏𝟓÷𝟑 𝟓 EXAMPLE 6: 𝟒 = 𝟒.𝟐 = 𝟖 , 𝟏𝟖 = 𝟏𝟖÷𝟑 = 𝟔 (2) Changing the sign of the numerator or denominator of a fraction changes the sign of the fraction. EXAMPLE 7: −𝟑 𝟒 = − 𝟑 𝟒 = 𝟑 −𝟒 (3) Adding two fractions with a common denominator yields a fraction whose numerator is the sum of the numerators of the given fractions and whose denominator is the common denominator is the common denominator. 𝟑 𝟒 𝟑+𝟒 𝟕 EXAMPLE 8: 𝟓 + 𝟓 = 𝟓 = 𝟓 (4) The sum of difference of two fraction having different denominators may be found by writing the fractions with a common denominator. 𝟑 𝟒 𝟏𝟖 𝟐𝟎 𝟑𝟖 𝟖 𝟒 EXAMPLE 9: 𝟓 + 𝟔 = 𝟑𝟎 + 𝟑𝟎 = 𝟑𝟎 𝒐𝒓 𝟏 𝟑𝟎 𝒐𝒓 𝟏 𝟏𝟓 (6) The reciprocal of a fraction is a fraction whose numerator is denominator of the given fraction and whose denominator is the numerator of the given fraction. Thus the reciprocal of 3 (i.e., 3/1) similarity the reciprocals of 5/8 and –4/3 are 8/5 and 3/−4 𝑜𝑟 − 3/4, respectively. (7) To divide two fractions, multiply the first by the reciprocal of the second. FUNDAMENTAL OPERATIONS WITH NUMBERS EXAMPLE 11: 𝑎/𝑏 ÷ 𝑐/𝑑 = 𝑎/𝑏 ∙ 𝑑/𝑐 = 𝑎𝑑/𝑏𝑐 2/3 ÷ 4/5 = 2/3 ∙ 5/4 = 10/12 = 5/6 This result may be established as follows: 𝑎 𝑏 ÷ 𝑐 𝑑 = 𝑎/𝑏 𝑐/𝑑 = 𝑎/𝑏∙𝑏𝑑 𝑐/𝑑∙𝑏𝑑 = 𝑎𝑑 𝑏c  Examples: 𝟖 𝟑 𝟔 +𝟔 = 𝟖 𝟑 𝟏𝟎 𝟕 +𝟕 + 𝟕 = 𝟑 𝟑 𝟑 𝟑 𝟒 + 𝟑 + 𝟐 + 𝟓 = 𝟏 𝟓 𝟗 𝟐 𝟕 + 𝟔 + 𝟑 − 𝟔 = Examples: 𝟐 𝟖 5. 4 + + 𝟏𝟎 = 𝟓 𝟗 𝟐 𝟒 6. 4 − = 𝟑 𝟕 𝟒 𝟓 𝟒 7. − + = 𝟗 𝟔 𝟑 Examples: 𝟐 𝟖 8. 4 ∙ ∙ 𝟏𝟎 = 𝟓 𝟗 𝟐 𝟒 9. 4 ∙ = 𝟑 𝟕 𝟒 𝟓 𝟒 10. ∙ ∙ = 𝟗 𝟔 𝟑 Examples: 𝟐 𝟖 11. 4 ÷ = 𝟓 𝟗 𝟐 𝟏𝟔 12. ÷ = 𝟑 𝟗 𝟐𝟎 𝟏𝟓 13. ÷ = 𝟔 𝟐𝟒 𝟐𝟓 1𝟒 14. ÷ = 𝟑𝟎 𝟐𝟖  MODULE 2: FUNDAMENTAL OPERATIONS WITH ALGEBRAIC EXPRESSIONS ALGEBRAIC EXPRESSIONS TERMS DEGREE GROUPING COMPUTATION WITH ALGEBRAIC EXPRESSIONS  ALGEBRAIC EXPRESSIONS An algebraic expression is a combination of ordinary numbers and letters which represent numbers. A term consists of products and quotients of ordinary numbers and letters which represent numbers. A monomial is an algebraic expression consisting of only one term. Because of this definition, monomials are sometimes simply called terms. A binomial is an algebraic expression consisting of two terms A trinomial is an algebraic expression consisting of three terms. A multinomial is an algebraic expression consisting of more than one term. TERMS One factor of term is said to be the coefficient of the rest of the term. Thus in the term 5x3y2 , 5x3 is the coefficient of y2 , 5y2 is the coefficient of x3, and 5 is the coefficient of x3y2. If a term consist of the product of an ordinary number and one or more letters, we call the number the numerical coefficient (or simply the coefficient) of the term. Thus in – 5x3y2 , -5 is the numerical coefficient or simply the coefficient. Like terms, or similar terms, are terms which differ only in numerical coefficients. For example 7xy and – 2xy are like terms; 3x2y4 and – 1/2 x2y4 are like terms; however, - 2a2b3 and – 3a2b7 are unlike terms. Two or more like terms in an algebraic expression may be combined into one term. Thus 7x2y – 4x2y + 2x2y may be combined and written 5x2y. A term is integral and rational in certain literal (letters which represent numbers) if the term consist of (a)positive integer powers of the variables multiplied by a factor not containing any variable, or (b)no variables at all For example, the terms 6x2y3 , - 5y4 , 7, - 4x and √3x3y6 are integral and rational in the variables present. However, 3√x is not rational in x, 4/x is not integral in x. A polynomial is a monomial or multinomial in which every term is integral and rational. For example, 3x2y3 - 5x4y +2, 2x4 – 7x3 + 3x2 – 5x +2, 4xy + z, and 3x2 are polynomials. However, 3x2 – 4/x and 4√y + 3 are not polynomials DEGREE The degree of monomial is the sum of all the exponents in the variables in the term. Thus the degree of 4x3y2z is 3 + 2 + 1 = 6. The degree of constant, such as 6, 0, - √3, or π is zero. The degree of polynomial is the same as that of the term having highest degree and nonzero coefficient. Thus 7x3y2 – 4xz5 + 2x3y has terms of degree 5, 6 and 4 respectively; hence the degree of the polynomial is 6.  GROUPING A symbol of grouping such as parentheses ( ), brackets [ ], or braces { } is often use to show that the terms contained in them are considered as single quality. For example, the sum of two algebraic expressions 5x2 – 3x + y and 2x – 3y may be written (5x2 – 3x + y) + (2x – 3y). The difference of these may be written (5x2 – 3x + y) – (2x – 3y), and their product (5x2 – 3x + y) (2x- 3y).  Removal of symbols of groupings is governed by the following laws. (1) If a + sign precedes a symbol of grouping, this symbol of grouping may be removed without affecting the terms contained. Thus (3x + 7y) + (4xy – 3x3 ) = 3x + 7y + 4xy – 3x3 (2) If a – sign precedes a symbol of grouping, this symbol of grouping may be removed if each sign of the terms is changed. Thus (3x + 7y) – (4xy – 3x3 ) = 3x + 7y – 4xy + 3x3 (3) If more than one symbol of grouping is present, the inner ones are to be removed first. Thus 2x – {4x3 – (3x2 – 5y)} = 2x – {4x3 – 3x2 + 5y} = 2x – 4x3 + 3x2 – 5y COMPUTATION WITH ALGEBRAIC EXPRESSIONS Addition of algebraic expressions is achieved by combining like terms. In order to accomplish this addition, the expressions may be arranged in rows with like terms in the same column; these columns are then added. EXAMPLE 1: Add 7x + 3y3 – 4xy, 3x – 2y3 + 7xy, and 2xy – 5x – 6y3. Subtraction of two algebraic expressions is achieved by changing the sign of every term in the expression which is being subtracted (sometimes called the subtrahend) and adding this result to other expression (called the minuend). EXAMPLE 2: Subtract 2x2 – 3xy + 5y2 from 10x2 – 2xy – 3y2 Multiplication of algebraic expressions is achieved by multiplying the terms in the factors of the expressions. To multiply two or more monomials: Use the laws of exponents, the rules of signs, and the commutative and associative properties of multiplication. EXAMPLE 3: Multiply – 3x2y3z, 2x4y and – 4xy4z2 EXAMPLE 4: Multiply 3xy – 4x3 + 2xy2 by 5x2y4 EXAMPLE 5: Multiply – 3x + 9 + x2 by 3–x