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Chapter One Whole Numbers and Introduction to Algebra Written by Sue C. Little North Harris College Offered with thanks by Mr. Roberts Clement Middle School, RUSD Place Value and Names for Numbers Ten-millions Millions Hundred-thousands Ten-thousands Thousands Hundreds Tens Ones Hundred-millions Billions Ten-billions Hundred-billions The position of each digit in a number determines its place value. 3 5 6 8 9 4 0 2 Ten-millions Millions Hundred-thousands Ten-thousands Thousands Hundreds Tens Ones Hundred-millions Billions Ten-billions Hundred-billions A whole number such as 35,689,402 is written in standard form. The columns separate the digits into groups of threes. Each group of three digits is a period. Billions Millions Thousands Ones 3 5 6 8 9 4 0 2 Ten-millions Millions Hundred-thousands Ten-thousands Thousands Hundreds Tens Ones Hundred-millions Billions Ten-billions Hundred-billions To write a whole number in words, write the number in each period followed by the name of the period. 3 5 6 8 9 4 0 2 thirty-five million, six hundred eighty-nine thousand, four hundred two Helpful Hint The name of the ones period is not used when reading and writing whole numbers. Also, the word “and” is not used when reading and writing whole numbers. It is used when reading and writing mixed numbers and some decimal values as shown later. Standard Form 4,786 = Expanded Form 4000 + 700 + 80 + 6 The place value of a digit can be used to write a number in expanded form. The expanded form of a number shows each digit of the number with its place value. Adding Whole Numbers and Perimeter 3 + 4 = 7 addend addend sum Addition Property of 0 The sum of 0 and any number is that number. 8+0=8 Commutative Property of Addition Changing the order of two addends does not change their sum. 4+2=2+4 Associative Property of Addition Changing the grouping of addends does not change their sum. 3 + (4 + 2) = (3 + 4) + 2 Descriptions of problems solved through addition may include any of these key words or phrases: Key Words Examples Symbols added to 3 added to 9 3+9 plus 5 plus 22 5 + 22 more than 7 more than 8 7+8 total total of 6 and 5 6+5 increased by 16 increased by 7 16 + 7 sum sum of 50 and 11 50 + 11 A polygon is a flat figure formed by line segments connected at their ends. Geometric figures such as triangles, squares, and rectangles are called polygons. triangle square rectangle Finding the Perimeter of a Polygon The perimeter of a polygon is the distance around the polygon. Subtracting Whole Numbers Subtraction is finding the difference of two numbers. Subtraction Properties of 0 The difference of any number and that same number is 0. 9-9=0 The difference of any number and 0 is the same number. 7-0=7 Descriptions of problems solved by subtraction may include any of these key words or phrases: Key Words Examples Symbols subtract subtract 3 from 9 9-3 difference difference of 8 and 2 8-2 less 12 less 8 12 - 8 take away 14 take away 9 14 - 9 decreased by subtracted from 16 decreased by 7 5 subtracted from 9 16 - 7 9-5 Rounding and Estimating Rounding a whole number means approximating it. 20 23 30 23 rounded to the nearest ten is 20. 40 48 50 48 rounded to the nearest ten is 50. 10 15 20 15 rounded to the nearest ten is 20. Rounding Whole Numbers to a Given Place Value Step 1. Locate the digit to the right of the given place value. Step 2. If this digit is 5 or greater, add 1 to the digit in the given place value and replace each digit to its right by 0. Step 3. If this digit is less than 5, replace it and each digit to its right by 0. Making estimates is often the quickest way to solve real-life problems when their solutions do not need to be exact. Multiplying Whole Numbers and Area Multiplication is repeated addition with a different notation. 4 + 4 + 4 + 4 + 4 = 5 x 4 = 20 5 fours factor product Multiplication Property of 0 The product of 0 and any number is 0. 90=0 06=0 Multiplication Property of 1 The product of 1 and any number is that same number. 91=9 16=6 Commutative Property of Multiplication Changing the order of two factors does not change their product. 63=36 Associative Property of Multiplication Changing the grouping of factors does not change their product. 5 ( 2 3) = (5 2) 3 Distributive Property Multiplication distributes over addition. 5(3 + 4) = 5 3 + 5 4 There are several words or phrases that indicate the operation of multiplication. Some of these are as follows: Key Words Examples Symbols multiply multiply 4 by 3 43 product times product of 2 and 5 2 5 7 times 6 76 1 1 square inch Area 1 5 inches 3 inches Area of a rectangle = length width = (5 inches)(3 inches) = 15 square inches Dividing Whole Numbers Division is the process of separating a quantity into equal parts. quotient 20 5 4 dividend 6 3 18 14 2 7 divisor Division Properties of 1 The quotient of any number and that same number is 1. 1 6 1 5 5 771 6 The quotient of any number and 1 is that same number. 5 6 6 15 1 7 1 7 Division Properties of 0 The quotient of 0 and any number (except 0) is 0. 0 0 0 5 0 0 7 0 6 The quotient of any number and 0 is not a number. We say that 6 0 5 7 0 are undefined (u). 0 Here are some key words and phrases that indicate the operation of division. Key Words Examples Symbols divide divide 15 by 3 15 3 quotient quotient of 12 and 6 12 6 divided by 8 divided by 4 48 divided or shared equally $20 divided equally among 5 people 20 5 How do you find an average? A student’s prealgebra grades at the end of the semester are: 90, 85, 95, 70, 80, 100, 98, 82, 90, 90. How do you find his average? Find the sum of the scores and then divide the sum by the number of scores. Exponents and Order of Operations An exponent is a shorthand notation for repeated multiplication. 3•3•3•3•3 3 is a factor 5 times Using an exponent, this product can be written as base 3 5 exponent base 3 5 exponent Read as “three to the fifth power” or “the fifth power of three.” This is called exponential notation. The exponent, 5, indicates how many times the base, 3, is a factor. 3•3•3•3•3 3 is a factor 5 times 1 4 4= is read as “four to the first power.” 44= 4 is read as “four to the second power or four squared.” 3 444= 4 is read as “four to the third power or four cubed.” 4 4 4 4 = 44 is read as “four to the fourth power.” 2 To evaluate exponential notation, we write the expression as a product and then find the value of the product. 3 = 3 • 3 • 3 • 3 • 3 = 243 5 Order of Operations 1. Do all operations within grouping symbols such as parentheses or brackets. 2. Evaluate any expressions with exponents. 3. Multiply or divide in order from left to right. 4. Add or subtract in order from left to right. Introduction to Variables and Algebraic Expressions A combination of operations on letters (variables) and numbers is called an algebraic expression. Algebraic Expressions 5+x 6y 3y–4+x 4x means 4 x and xy means x y Keywords and phrases suggesting addition, subtraction, multiplication, or division. Addition Subtraction Multiplication Division sum difference product quotient plus minus times divided by added to subtracted from multiply into more than less than twice per increased by decreased by of for every total less double slower than faster than