Basic Mathematics
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TABLE OF CONTENTS BASIC ARITHMETIC .................................................................................................................. 3 Beginning Terminology............................................................................................................ 3 Kinds Of Numbers .................................................................................................................... 4 ADDITION OF WHOLE NUMBERS ......................................................................................... 7 Simple and Complex Addition ................................................................................................ 7 Numbers in Word Form ......................................................................................................... 10 SUBTRACTION OF WHOLE NUMBERS ............................................................................. 13 Simple Subtraction ................................................................................................................. 13 Complex Subtraction ............................................................................................................. 14 CHECKING ADDITION AND SUBTRACTION ..................................................................... 19 MULTIPLYING WHOLE NUMBERS ...................................................................................... 21 Simple Multiplication .............................................................................................................. 22 Complex Multiplication........................................................................................................... 22 DIVIDING WHOLE NUMBERS ............................................................................................... 28 Long Division........................................................................................................................... 29 Short Division .......................................................................................................................... 32 FRACTIONS ............................................................................................................................... 35 Kinds of Fractions................................................................................................................... 35 Reduction of Fractions to Lowest Terms ............................................................................ 37 Addition and Subtraction of Common Fractions ................................................................ 38 Multiplication of Common Fractions .................................................................................... 41 Division of Common Fractions ............................................................................................. 42 DECIMAL NUMBERS ............................................................................................................... 47 The Decimal System.............................................................................................................. 47 Adding Decimals..................................................................................................................... 49 Subtracting Decimals ............................................................................................................. 50 Multiplying Decimals .............................................................................................................. 52 Dividing Decimals ................................................................................................................... 53 Rounding Off Decimals ......................................................................................................... 54 PERCENTAGES ........................................................................................................................ 59 Percents ................................................................................................................................... 59 Percentages ............................................................................................................................ 61 FORMULAS ................................................................................................................................ 64 Geometric Formulas .............................................................................................................. 64 LINEAR MEASUREMENT ....................................................................................................... 66 ANGULAR MEASUREMENT .................................................................................................. 68 Circle ........................................................................................................................................ 68 Angles ...................................................................................................................................... 69 AREA MEASUREMENT ........................................................................................................... 73 Changing Units of Area Measurement................................................................................ 74 Determining the Area of a Rectangle .................................................................................. 77 Determining the Area of a Parallelogram ........................................................................... 78 Determining the Area of a Trapezoid .................................................................................. 79 Determining the Area of a Triangle 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Determining the Area of a Circle.......................................................................................... 81 Determining the Area of a Sector ........................................................................................ 82 Determining the Area of a Cylinder ..................................................................................... 82 VOLUME MEASUREMENT ..................................................................................................... 88 Converting Units of Volume Measure ................................................................................. 89 Computing the Volume of a Cube ....................................................................................... 90 Determining the Area of a Rectangular Solid .................................................................... 90 Determining the Volume of a Cylinder ................................................................................ 91 Application of Volume Measure to Liquid Measure .......................................................... 92 SQUARE ROOTS ...................................................................................................................... 96 Perfect Squares of Whole Numbers in Square Root ........................................................ 96 EXPONENTS (POWERS OF TEN) ...................................................................................... 100 Using Powers of Ten ........................................................................................................... 100 Multiplication and Division of Exponential Numbers ....................................................... 103 INTRODUCTION TO BASIC ALGEBRA ............................................................................. 106 Ratio ....................................................................................................................................... 106 Proportion .............................................................................................................................. 106 Signed Numbers ................................................................................................................... 107 The X and Y Axes ................................................................................................................ 108 Adding Signed Numbers ..................................................................................................... 110 Subtracting Signed Numbers ............................................................................................. 111 Multiplying Signed Numbers ............................................................................................... 112 Dividing Signed Numbers ................................................................................................... 112 Algebraic Equations ............................................................................................................. 112 Use of Parentheses, Brackets, Braces, and Bars ................................................................. 113 Letters Used as Symbols .................................................................................................... 114 Use of Sub-Numbers ........................................................................................................... 114 Use of Coefficients, Exponents, Bases, Powers, and Terms........................................ 114 RECIPROCAL OPERATIONS............................................................................................... 114 Writing Formulas .................................................................................................................. 115 Transposing........................................................................................................................... 115 Solving Equations with Radicals ........................................................................................ 116 INTRODUCTION TO TRIGONOMETRY ............................................................................. 121 Hypotenuse of a Right Triangle ......................................................................................... 121 Trigonometric Functions of an Angle ................................................................................ 123 The Tangent Function ......................................................................................................... 124 The Sine and Cosine Function ........................................................................................... 125 THE METRIC SYSTEM........................................................................................................... 131 Metrication ............................................................................................................................. 131 Affect of Change to Industry ............................................................................................... 131 Basic Principles of the Metric System ............................................................................... 131 Metric Abbreviations ............................................................................................................ 133 The Metric Scale................................................................................................................... 134 Practice Exercises in Metric Measurement ...................................................................... 136 Conversion Factors .............................................................................................................. 138 Conversion Table of Length ............................................................................................... 138
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Conversion Table for Area .................................................................................................. 138 Conversion Table for Pressure .......................................................................................... 139 USING THE ELECTRONIC CALCULATOR ....................................................................... 143 The Addition Key .................................................................................................................. 143 The Subtraction Key ............................................................................................................ 145 The Multiplication Key ......................................................................................................... 145 The Division Key................................................................................................................... 146 The Memory Key .................................................................................................................. 146 The Square and Square Root Keys .................................................................................. 148 Mixed Calculations ............................................................................................................... 148 The Sine, Cosine, and Tangent Keys ............................................................................... 149 The Reciprocal Key.............................................................................................................. 150
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LIST OF EXHIBITS Exhibit 1 Description of Common Fractional Forms ..................................................................... 4 Exhibit 2 Adding with Pictures ....................................................................................................... 8 Exhibit 3 Multiplication Table ...................................................................................................... 21 Exhibit 4 15/5=3 ........................................................................................................................... 28 Exhibit 5 15/3=5 ........................................................................................................................... 29 Exhibit 6 Subtraction of a Least Common Denominator.............................................................. 40 Exhibit 7 Multiplication of Common Fractions ............................................................................ 42 Exhibit 8 Division of Common Fractions ..................................................................................... 43 Exhibit 9 Percent ........................................................................................................................... 59 Exhibit 10 Fractional Parts of an Inch (Enlarged) ........................................................................ 66 Exhibit 11 Enlarged View of Decimal Fractions .......................................................................... 67 Exhibit 12 The Circle, Center, and Circumference....................................................................... 68 Exhibit 13 Showing Radius (R) and Diameter (D) ....................................................................... 69 Exhibit 14 Angle Formation within a Circle................................................................................. 69 Exhibit 15 Illustrated Examples of Various Forms....................................................................... 69 Exhibit 16 The Semicircular Protractor ........................................................................................ 70 Exhibit 17 Areas of Linear Measures in Different Units .............................................................. 73 Exhibit 18 Diagrams Showing the Areas of a Square Inch, Square Foot, and Square Yard ........ 74 Exhibit 19 Area of a Rectangle ..................................................................................................... 77 Exhibit 20 Area of a Parallelogram .............................................................................................. 78 Exhibit 21 Area of a Trapezoid ..................................................................................................... 79 Exhibit 22 Area of a Trapezoid ..................................................................................................... 80 Exhibit 23 Area of a Triangle ....................................................................................................... 80 Exhibit 24 Illustration of a Sector ................................................................................................. 82 Exhibit 25 Lateral Surface of a Cylinder ...................................................................................... 83 Exhibit 26 Total Area of a Cylinder ............................................................................................. 83 Exhibit 27 Volume Measurements................................................................................................ 88 Exhibit 28 Determining the Area of a Rectangular Solid ............................................................. 91 Exhibit 29 Determining the Volume of a Cylinder ....................................................................... 91 Exhibit 30 Positive Numbers ...................................................................................................... 107 Exhibit 31 Adding and Subtracting Like and Unlike Numbers .................................................. 108 Exhibit 32 Pairs of Numbers ....................................................................................................... 108 Exhibit 33 X and Y Axes ............................................................................................................ 109 Exhibit 34 The Coordinate System ............................................................................................. 109 Exhibit 35 The Reciprocal .......................................................................................................... 115 Exhibit 36 Right Triangle ........................................................................................................... 121 Exhibit 37 The Function ............................................................................................................. 123 Exhibit 38 The Tangent Function ............................................................................................... 124 Exhibit 39 Roof Support ............................................................................................................. 125 Exhibit 40 The Sine and Cosine Function .................................................................................. 126 Exhibit 41 Dimensioned Drawing .............................................................................................. 134 Exhibit 42 Dimensioned Drawing with Note for Standard Unit Application ............................. 134 Exhibit 43 Application of Metric Scale ...................................................................................... 135 Exhibit 44 Application of Metric Scale ...................................................................................... 135
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Exhibit 45 Application of Metric Scale ...................................................................................... 135 Exhibit 46 The POWER ON and CLEAR MEMORY Keys ...................................................... 143 Exhibit 47 CLEAR ENTRY Key................................................................................................ 144 Exhibit 48 PLUS and EQUAL Keys .......................................................................................... 144 Exhibit 49 MINUS, MULTIPLICATION, and DIVISION Keys .............................................. 145 Exhibit 50 STORE and RECALL Keys...................................................................................... 146 Exhibit 51 SQUARE and SQUARE ROOT Keys ...................................................................... 148 Exhibit 52 SINE, COSINE, and TANGENT Keys..................................................................... 149 Exhibit 53 The RECIPROCAL Key ........................................................................................... 150
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PREFACE This Basic Mathematics manual contains fundamental information about mathematical terms, formulas, and functions. Read and study the information and examples provided. The Knowledge Check Activities that accompany each section of the manual are to be completed by the learner at the end of each section. Compare your answers with the answers provided. Completion of these Knowledge Check Activities and the evaluation by your instructor will help you know when you have obtained sufficient basic skills in basic mathematics. Parts of this manual are self-paced so that the learner may obtain the knowledge and skills presented at his or her initiative and individual rate of speed. However, an instructor or supervisor should lead the training program, provide guidance for learning, and provide additional instruction as needed. The exhibits in this manual show details that are necessary and helpful for clarification of the points covered. These illustrations may not be reproduced to scale.
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BASIC MATHEMATICS Learner Objectives Upon completion of this unit, the student will have a general concept of numbers and digits, kinds of numbers, mathematical signs, and symbols. Mathematics is the foundation upon which modem day life depends. Diesel transportation, high-speed computers, jet planes, submarines, telephones, and televisions are just a few of the limitless number of products and mechanisms that depend upon mathematics for development and production. Arithmetic is the simplest form of mathematics and is used everyday to solve most of the common problems encountered in work, play, and living. Basic arithmetic includes addition, subtraction, multiplication, and division. The four different operations of arithmetic are easy to recognize because of the use of signs to indicate the type of operation being performed (see below). 1. Plus sign (+) 2. Minus sign (-) 3. Multiplication sign (x) 4. Division sign (รท) The equal sign (=) is used to show equal or even values. It does not indicate the answer to an arithmetic question. For example, two plus two equals four, or stated another way is 2 + 2 = 4. The values on each side of the equal sign are equal.
Beginning Terminology Numbers A number is a symbol or word commonly used to express value or quantity. However, any number expressed belongs to a system of numbers. The Arabic number system is the one used most often in the United States. This system includes ten numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These numerals may be combined to express any desired number. Digits In the Arabic number system, the location or position of a numeral (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) in the written whole number expresses its place value. The word digit is a name given to the place or position of each numeral in a whole number. The first numeral in the extreme right place (digit) of a whole number is in the ones column. The numeral in the next position (or digit) to the left is in the tens column; the third digit, hundreds; the fourth digit thousands; etc. (see the following chart).
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Millions HundredTenThousands Hundreds Tens Ones thousands thousands 8 7 6 5 4 3 2 A whole number like 432 is a simple way of saying 400 + 30 + 2. The number 432 has three digits. Like words formed by combining letters, whole numbers are formed by combining digits.
Kinds of Numbers Whole Numbers Whole numbers refer to complete units where there is NO fractional part (or rational number). Numbers such as 30 and 50 and quantities like 140 machine screws, 10 spools, and 43 outlets are examples of types of whole numbers. Whole numbers are also used to describe measurements like 75 feet or 12 inches and other material values such as $810. All of these examples represent whole numbers because the values do not contain a fraction (rational number). Whole numbers may be written in the form of words. For example: Three hundred fifty-seven or four thousand six hundred ninety-eight. Rational Numbers A rational number is a part of a whole unit or quantity. Common rational numbers include ¼, ½, and ¾. For example, if a square, triangle, or circle is divided into two equal parts, one of these parts is a rational number of the whole square, whole triangle, or whole circle(see Exhibit 1). EXHIBIT 1 Description of Common Fractional Forms
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Decimal Numbers A decimal number is a type of rational number that can be written to show parts of a whole number. The difference in decimal numbers and whole numbers is the use of a decimal point placed directly to the right of the ones column. The following chart shows the position of the period and decimal digits. Thousands .005
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Hundreds .05
Tens .5
Ones 5
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ADDITION OF WHOLE NUMBERS Learner Objectives Upon completion of this unit, the student will: 1. Solve addition problems of whole numbers. 2. Solve addition problems with numbers written in word form. Simple addition of whole numbers is the adding of numbers that result in single digit answers in each column. Examples of simple addition are:
3 + 6 9
322 + 132 454
11132 12136 + 74211 97479
Notice that in each example the added columns result in a single digit no larger than 9. Addition with regrouping consists of adding digits which result in column answers higher than 9 or double digits. This double-digit column answer must be regrouped. The number in the ones place is a part of the answer, and the number in the tens place is added to the next column.
Simple Addition and Addition with Regrouping A number line can be used to show how numbers are added together. A number line is a picture that shows numerals in order of value. Example:
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The number 2 can be added with the number 3 on the number line. Example:
Adding whole numbers can also be explained by adding pictures that show the added amount. An example is shown in Exhibit 2. EXHIBIT 2 Adding with Pictures
The same numbers can be added still another way by the column method. Example: 2 +3 5 This method uses no equal sign. Whole numbers are usually added this way. More examples:
5 +5 10
414 + 141 555
Addition is nothing more than a procedure of adding all the numbers in each column in a problem. The answer to the addition problem is called the sum. Whole numbers can be easily added using a few basic rules. Review the following example, and remember each step of the problem.
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Example: Add 2765 + 972 + 857 + 1724 Step 1
Arrange each number in column form with all ones in the ones column, all tens in the tens column, hundreds in the hundreds column, etc.
Step 2
Add the numerals in the ones column, from top to bottom (5 + 2 + 7 + 4 = 18). Place 8 in the ones column and regroup the 1 to the tens column.
Step 3
Add the numerals in the tens column and remember to add the 1 carried over from the ones column (6 + 7 + 5 + 2 + 1 = 21). Place 1 in the answer section of the tens column, and regroup the 2 to the hundreds column.
Step 4
Perform the same addition steps for the hundreds and thousands columns.
Digit Placement
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Numbers in Word Form It is sometimes necessary to rewrite numbers from their word form because it is obviously difficult to add numbers accurately that are in word form. Addition of these numbers is accomplished through translation from word to digit form and adding columns. Examples of this process are shown below. Example: Find the sum of fifty-two, thirty-one, and sixty-seven. Solution: Translate words to numbers in column form. Fifty-two Thirty-one Sixty-seven
52 31 + 67 150
Example: Find the sum of forty-five, fifteen, and ten. Solution: Translate words to numbers in column form. Forty-five Fifteen Ten
45 15 +10 70
Expanded Notation Expanded notation is when a number is written out to show all of the place values and is helpful because it is a sum of the numbers of each digit and place value. Examples: 42 = 40 + 2 287 = 200 + 80 + 7 5263 = 5000 + 200 + 60 + 3
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KNOWLEDGE CHECK Addition Add the following whole numbers. 1. a.
222
b.
+ 222
2. a.
813 + 267
3. a.
813 222 + 318
318
c.
+ 421
b.
b.
611
d.
+ 116
924 + 429
c.
618 + 861
1021 611 + 421
c.
1026 96 + 861
1021
+ 1210
d.
d.
411 + 946
1261 1621 + 6211
Expanded Notation Write the following numbers in expanded notation. 45= 264=
Bonu Problems Add the following numbers (Hint: Write the numbers out as whole numbers first). Add: Four thousand two hundred forty-five plus five hundred sixty-two plus nine thousand eight hundred eleven.
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ANSWERS TO KNOWLEDGE CHECK 1. a.
444
2. a.
1080
3. a.
1353
b.
739
727
d.
2231
b. 1353
c. 1479
d.
1357
b.
c.
d.
9093
2053
c.
1983
Expanded Notation 45= 40 + 5 264= 200 + 60 + 4 Bonus Problem 4245 562 + 9811 14618
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