Mathematics - Module

Mathematics Module 1. REAL NUMBERS 1.1 Introduction 1.2 Concepts 1.3 Properties of Real Numbers 2. INTEGERS 2.1 Introduction 2.2 Absolute Value 2.3 Operation on Integers 3. ALGEBRA 3.1 Introduction 3.2 Polynomials 3.3 Evaluating Algebraic Expression 3.3 Addition and Subtraction 3.3 Laws of Exponent 3.4 Multiplication and Division 4. LINEAR EQUATION 4.1 Linear Equation in One Variable 4.2 Properties of Equality 4.3 Translating Sentences into Equations 4.4 Problem Solving 5. REAL NUMBERS A real number is any value that represents a quantity along a number line. Real numbers are classified as rational or irrational. Under these classifications are natural numbers, whole numbers, integers, and fractions. Natural numbers are the numbers used for counting. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Whole numbers are formed by adding 0 to the set of natural numbers. {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Integers are formed by adding the negatives of the natural numbers to the whole numbers. {…-4, -3, -2, -1, 0, 1, 2, 3, 4, …} Rational numbers is any number that can be expressed as the quotient or fraction of two integers where b is not zero. The decimal representation of a rational number either terminates or repeats. {…-4.25, -4, -3, , -1, 0, , 1, 2, 3, 3.75, 4, …} Irrational numbers are numbers that cannot be written as a simple fraction. The decimal representation of an irrational number is neither terminating nor repeating. ( = –1.414214, = 1.73205, , etc…) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Test Yourself I. Classify the given numbers by placing a check mark () in the appropriate row. Set –15 0 9.5 0.112233… Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers II. Complete the statement using always, sometimes, or never. 1) A real number is _______________ a rational number. 2) An irrational number is _______________ a real number. 3) A negative integer is _______________ an irrational number. 4) A natural number is _______________ a whole number. 5) A fraction is ______________ an integer. 6) The square root of four is _______________ an integer. 7) An integer is _______________ a natural number. 8) Pi () is _______________ a rational number. 9) Zero is _______________ a natural number. 10) One-half is _______________ a rational number. ====================================================================== PROPERTIES OF REAL NUMBERS =========================================================== CLOSURE PROPERTY 1. Closure Property of Addition The sum of any two real numbers is a real number. a + b is a real number 2. Closure Property of Multiplication The product of any two real numbers is a real number. a ◦ b is a real number COMMUTATIVE PROPERTY 1. Commutative Property of Addition Two real numbers can be added in any order. a + b = b + a 2. Commutative Property of Multiplication Two real numbers can be multiplied in any order. a ◦ b = b ◦ a ASSOCIATIVE PROPERTY 1. Associative Property of Addition If three real numbers are added, it makes no difference which two are added first. (a + b) + c = a + (b + c) 2. Associative Property of Multiplication If three real numbers are multiplied, it makes no difference which two are multiplied first. (a ◦ b) ◦ c = a ◦ (b ◦ c) DISTRIBUTIVE PROPERTY 1. Distributive Property of Multiplication over Addition Multiplication distributes over addition. a ◦ (b + c) = a ◦ b + a ◦ c 2. Distributive Property of Multiplication over Subtraction Multiplication distributes over subtraction. a ◦ (b – c) = a ◦ b – a ◦ c IDENTITY PROPERTY 1. Identity Property of Addition Any number added to the identity element will remain unchanged. The identity element for addition is 0. a + 0 = 0 + a = a 2. Identity Property of Multiplication Any number multiplied to the identity element will remain unchanged. The identity element for multiplication is 1. a ◦ 1 = 1 ◦ a = a INVERSE PROPERTY 1. Inverse Property of Addition The sum of a number and its additive inverse (opposite) is the identity element 0. The additive inverses are (a) and (–a). a + (–a) = (–a) + a = 0 2. Inverse Property of Multiplication The product of a number and its multiplicative inverse (reciprocal) is the identity element 1. The multiplicative inverses are a and . a ◦ = ◦ a = 1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Test Yourself I. Identify the real number property that justifies each statement. 1) 4(x – 3) = 4x – 2 _____________________________ 2) (0.25)(4) = 1 _____________________________ 3) x + 0 = x _____________________________ 4) (4 + 6) + 9 = 4 + (6 + 9) _____________________________ 5) 1234(1) = 1234 _____________________________ 6) 5(x2y3) = (5x2)y3 _____________________________ 7) 9 ◦ x = x ◦ 9 _____________________________ 8) 12 ◦ = 1 _____________________________ 9) 3(2x + 1) = 6x + 3 _____________________________ 10) (–127) + (127) = 0 _____________________________ II. Complete each equation using the indicated property. 1) d + k = ____________, commutative 2) 18 – 6x = ____________, distributive 3) 12(xy) = ____________, associative 4) x + = ____________, distributive 5) (x + 3) + 7 = ____________, associative 6) 8(_______) = 1, inverse 7) 5 ◦ (_______) = 5, identity 8) –3(1) = ____________, identity 9) 0.19 + (–0.19) = __________, inverse 10) c ◦ b = ____________, commutative INTEGERS Integers consist of all positive whole numbers, all negative whole numbers, and zero. origin positive integers negative integers The number associated with a point on a number line is the coordinate of the point. The point on the number line assigned to 0 is called the origin. Numbers that are of the same distance from zero (the origin), but on opposite sides of zero are called opposites. The opposite of 4 is (–4). ABSOLUTE VALUE Absolute value is used to describe distance between two points on a number line. The absolute value of an integer is equal to its distance from 0. Distance is always a positive value. The absolute value of x is written as |x|. Example: Find the value of each expression. 1) |–8| = 8 2) |14| = 14 3) |0| = 0 ====================================================================== ADDING AND SUBTRACTING INTEGERS =========================================================== ADDITION OF INTEGERS 1. Adding LIKE SIGNS (+) plus (+) and (–) plus (–) To add integers with same sign, add the absolute values of the numbers and use the sign common to both integers. Example 1: Add (5) and (14) (5) + (14) = 19 Example 2: Add (–5) and (–14) (–5) + (–14) = (–19) 2. Adding UNLIKE SIGNS (+) plus (–) and (–) plus (+) To add integers with different signs, subtract the absolute values of the numbers and use the sign of the integer with greater value. Example 1: Add (–7) and (17) (–7) + (17) = 10 Example 2: Add (7) and (–17) (7) + (–17) = (–10) SUBTRACTION OF INTEGERS 1. Subtracting LIKE SIGNS (+) minus (+) and (–) minus (–) 1.1) (+) minus (+) To subtract integers with same sign, change the operation to addition and change the sign of the integer next to minus sign. Apply the rule for adding integers. Use the sign of integer with greater value. Example 1: Subtract (5) and (14) (5) – (14) → (5) + (–14) = (–9) Example 2: Subtract (14) and (5) (14) – (5) → (14) + (–5) = (9) 1.2) (–) minus (–) To subtract integers with same sign, change the operation to addition and change the sign of the integer next to minus sign. Apply the rule for adding integers. Use the sign of integer with greater value. Example 1: Subtract (–7) and (–18) (–7) – (–18) → (–7) + (18) = (11) Example 2: Subtract (–18) and (–7) (–18) – (–7) → (–18) + (7) = (–11) 2. Subtracting UNLIKE SIGNS (+) minus (–) and (–) minus (+) 2.1) (+) minus (–) To subtract integers with same sign, change the operation to addition and change the sign of the integer next to minus sign. Apply the rule for adding integers. Use the sign common to both integers. Example 1: Subtract (7) and (–12) (7) – (–12) → (7) + (12) = (19) Example 2: Subtract (12) and (–7) (12) – (–7) → (12) + (7) = (19) 2.2) (–) minus (+) To subtract integers with same sign, change the operation to addition and change the sign of the integer next to minus sign. Apply the rule for adding integers. Use the sign common to both integers. Example 1: Subtract (–4) and (16) (–4) – (16) → (–4) + (–16) = (–20) Example 2: Subtract (–16) and (4) (–16) – (4) → (–16) + (–4) = (–20) Test Yourself I. Evaluate the following expressions. 1) 234 + (–163) 11) –21 – (–38) + 45 2) 82 – (–157) 12) (–87) + 125 3) 4 – 15 + (–9) 13) 56 – 124 4) (–140) + (–82) 14) 117+ (–22) 5) –24 – 37 – (–41) 15) –91 – 58 6) –48 – (–72) 16) 12 – (–15) + 18 7) 65 + 19 17) (–95) + (–48) 8) 83 – 172 18) –164 – (–55) 9) (–46) + 36 19) 73 – (–123) 10) –35 – 74 20) –73 + 16 – (–9) II. Find the missing number in each subtraction sentence. 1) ⃝ – 25 = – 120 6) (–46) + ⃝ = 77 2) –124 + ⃝ = 15 7) ⃝ – 168 = –78 3) ⃝ – 72 = –145 8) 82 + ⃝ = –112 4) 216 + ⃝ = 91 9) ⃝ – (–140) = –49 5) ⃝ – (–19) =12 10) –19 + ⃝ = 53 ====================================================================== MULTIPLYING AND DIVIDING INTEGERS =========================================================== MULTIPLICATION OF INTEGERS 1. Multiplying LIKE SIGNS (+) times (+) and (–) times (–) To multiply integers with same sign, multiply the absolute values of the numbers and use the positive sign. Example 1: Add (12) and (7) (12) ◦ (7) = 84 Example 2: Add (–7) and (–12) (–7) ◦ (–12) = 84 2. Multiplying UNLIKE SIGNS (+) times (–) and (–) times (+) To multiply integers with same sign, multiply the absolute values of the numbers and use the negative sign. Example 1: Add (–8) and (14) (–8) ◦ (14) = –112 Example 2: Add (8) and (–14) (8) ◦ (–14) = –112 DIVISION OF INTEGERS 1. Dividing LIKE SIGNS (+) divided by (+) and (–) divided by (–) To divide integers with same sign, divide the absolute values of the numbers and use the positive sign. Example 1: Add (32) and (8) (32) ÷ (8) = 4 Example 2: Add (–7) and (–12) (–8) ÷ (–32) = 2. Dividing UNLIKE SIGNS (+) divided by (–) and (–) divided by (+) To divide integers with same sign, divide the absolute values of the numbers and use the negative sign. Example 1: Add (–7) and (35) (–7) ÷ (35) = – Example 2: Add (35) and (–7) (35) ÷ (–7) = –5 Test Yourself I. Evaluate the following expressions. 1) 294 ÷ (–14) 11) –27 – (– ) ÷ 45 2) 12 (–15) 12) (–82) ÷ 124 3) 15 ÷ (–240) 13) 56 ◦ (–3) 4) –140 ÷ (–82) 14) 110 ÷ (–22) 5) (–24)(30)(–) 15) –18 ◦ (–12) 6) –48(–2) 16) (–12)(6) ÷ 18 7) 125 ÷ 75 17) (–98) ÷ (–40) 8) 8 ◦ 42 18) –14 (–5) 9) (–42) ÷ 36 19) 120(– ) 10) –35(4) 20) –63 ÷ 7(–9) II. Write the missing factor in the ⃝ to complete each sentence. 1) (–32)(–12) = 24 ◦ ⃝ 6) ⃝ + –23 = –7 2) ⃝ + (–5) = –21 7) (–7)◦ ⃝ ◦(–5) = 70 3) –|144| + (16) = ⃝ 8) |–9|◦ ⃝ = (–12)(18) 4) (–324) = (–6) ◦ ⃝ 9) 0 + 50 = ⃝ 5) ⃝ + |–12| = –9 10) (–448) ◦ ⃝ = (–14) ALGEBRA Algebra is a branch of mathematics that involves expressions with variables. It generalizes the facts in arithmetic. The combination of numbers and variables with ordinary operations of arithmetic is called an expression or an algebraic expression. The language of algebra consists of numerals (0, 1, 2, 3 …), variables (x, y, z, etc.), and symbols or signs. The variable represents any number from a given replacement set. The replacement set is the set of values of the variable. A constant is a symbol which has exactly one number in its replacement set. Examples: Definitions Algebraic term – is a single number, a letter, or a product of several numbers or letters. Numerical Coefficient – is the number in an algebraic term. Literal Coefficient – is a letter used to represent a number. Base – a number or letter which an exponent refers. Exponent – a number or letter that indicates how many times a base will be multiplied by itself. Similar Terms – terms that have similar literal factors and in which each literal factor has the same exponent in all of the terms. Constant – a number, letter, or symbol whose value is fixed. Test Yourself I. In each algebraic expression, identify the numerical and literal coefficients, and write a similar term. Algebraic Expression Numerical Coefficient Literal Coefficient Similar Term Algebraic Translation of Verbal Phrases The table below shows the how verbal phrases are translated into an algebraic expression. Verbal Phrases Keyword Expression the sum of x and 7 plus the difference of 10 and s minus 4 more than t plus 8 less than p minus 12 greater than m plus k increased by 9 plus d decreased by 5 minus 2 plus b plus r minus 6 minus w added to 11 plus 3 subtracted from h minus the product of y and 14 times the quotient of c and 7 divide twice a number times five times a number times 17 multiplied by g times m divided by n divide half of v divide the ratio of 12 to s divide Is; is equal to; equals equals the square of x square the square root of y square root More examples: 1) The product of three and sum of x and y is twenty four. 2) Twelve less than the quotient of h and 3 is half of 7. 3) The sum of the square of x and 2 is 27. 4) Thrice the sum of x, y, and z is 12. 5) The ratio of the sum of x and 4 and the difference of y and 7. Test Yourself I. Translate each phrase into an algebraic expression. Use any letter to represent the unknown unless otherwise specified. 1) Ten greater than a number 2) Fourteen decreased by a number 3) Twelve less than the product of eight and h 4) The difference of r and twenty multiplied by two 5) Eleven times the quotient of x and y 6) The product of x and 4increased by the product x and 3 7) The difference of the square of x and 3 is 22. 8) The sum of four times a number and three is thirty five. 9) Eight less than the quotient of x and 3 is four. 10) The difference of twice a number and 5 is the sum of thrice a number and four. 12) The sum of the ratio of x and 10 and one-half is one-fourth 13) Three times the difference of 2 and x plus seven times a number is twelve more than twice the number. 14) The ratio of the difference of x and 8 and the sum of x and 12 is 24. 15) The sum of twelve and thrice a number is negative twenty eight. II. Translate each expression into a verbal phrase. 1) 2) 3) 4) 5) 11