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Math 300 Basic College Mathematics Chapter 1 The Whole Numbers Math 300 M-G 4e Chapter 1; Rev: Mar 2011 Page 1 of 22 1.2 Place Value and Names for Numbers Place Value A digit is a number 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9 that names a place-value location. For large numbers, digits are separated by commas into groups of three called periods. Each period has a name: ones, thousands, millions, billions, trillions, and so on. See the place value chart below. Place-Value Chart for Whole Numbers HundredTenThousands Thousands Thousands Hundreds 100,000 10,000 1,000 100 Tens Ones 10 1 Beginning with the digit to your right, the place value goes as follows: ones… tens… hundreds,… thousands… ten-thousands… hundredthousands,… millions… ten-millions… hundred-millions,… billions… ten-billions... hundred-billions,… trillions… and so on. You must learn the place value chart for whole numbers. It will help you write the place value of a digit as well as writing a number in words. Example: Determine the place value of the digit 5 in the whole number. 1. 45, 987 Write the whole number in words. 2. 45,987 Math 300 M-G 4e Chapter 1; Rev: Mar 2011 Page 2 of 22 1.3 Adding Whole Numbers and Perimeter Sum – The total when adding 2 or more numbers together. Addends – The numbers that are added together. Additive identity –The sum of zero and any number is that number. Example: 7 0 7 Associative law of addition –Changing the grouping of addends does not change their sum. Example: 3 5 7 3 12 15 and 3 5 7 8 7 15 Commutative law of addition –Changing the order of two addends does not change their sum. Example: 2 3 5 and 3 2 5 Perimeter – The distance around an object. To find the perimeter of an object, add all sides. Example: Add. 1. 1521 + 348 (Hint: Begin adding the digits in a column from right to left.) 2. 10,120 + 12,989 + 5738 (Hint: Set this problem up vertically by lining up the numbers from right to left) Math 300 M-G 4e Chapter 1; Rev: Mar 2011 Page 3 of 22 1.4 Subtracting Whole Numbers Subtraction is finding the difference of two numbers. The first number is called the minuend and the second is called the subtrahend. Subtraction Properties of Zero (1) The difference of any number and that same number is equal to zero. Example: 11 11 0 (2) The difference of any number and zero is equal to that same number. Example: 45 0 45 Examples: Subtract. 1. 526 – 323 (Hint: Begin by subtracting in a column from right to left.) Subtracting with Borrowing When subtracting vertically, if a digit in the second number (bottom #) is larger than the corresponding digit in the first number (top #), borrowing is needed. Sometimes we may have to borrow from more than one place. Let’s look at the examples below to help us practice borrowing. Be sure to take notes and ask plenty of questions. 2. 7007 – 6349 3. 6000 – 3149 Math 300 M-G 4e Chapter 1; Rev: Mar 2011 4. 9035 – 7489 Page 4 of 22 1.4 Subtracting Whole Numbers (cont) Solving Problems by Subtracting Key Words or Phrases subtract difference less less than take away decreased by subtracted from Examples subtract 5 from 8 the difference of 10 and 2 17 less 3 2 less than 20 14 take away 9 7 decreased by 5 9 subtracted from 12 Symbols 8–5 10 – 2 17 – 3 20 – 2 14 – 9 7–5 12 – 9 Solve. 5. Subtract 9 from 21. 6. When Lou and Judy Zawislak began a trip, the odometer read 55,492. When the trip was over, the odometer read 59,320. How many miles did they drive on their trip? Math 300 M-G 4e Chapter 1; Rev: Mar 2011 Page 5 of 22 1.5 Rounding and Estimating Steps for rounding a number 1. Locate the digit in the given place value. 2. Look at the digit to the right of the given place value. 3. When looking at the digit to the right of the given place value, determine one of the following: a. If this digit is 5 or higher, add 1 to the digit in the given place value and replace each digit to its right with zero. All digits to the left will remain the same. or b. If this digit is less than 5, keep the digit in the given place value the same and replace each digit to its right with zero. All digits to the left will remain the same. 1. Round 467 to the nearest ten. 2. Round 4645 to the nearest hundred. 3. Round 7500 to the nearest thousand. When estimating, you will need to round first, then perform the operation: 4. Estimate the difference by first rounding to the nearest hundred, the performing the operation: 6852 - 1748 _____ Math 300 M-G 4e Chapter 1; Rev: Mar 2011 Page 6 of 22 1.6 Multiplying Whole Numbers and Area The numbers that we multiply are called factors. The result of a multiplication problem is called a product. Example: 3 x 5 = 15 {In this case, 3 and 5 are the factors and 15 is the product.} The product of zero and any whole number is equal to zero. The product of 1 and any whole number is equal to that whole number. This also called the multiplicative identity. Commutative law of multiplication – You can multiply two numbers in any order as long as you are multiplying throughout the problem. Rule: a b b a (The answer will be the same) Associative law of multiplication – It does not matter how you group the numbers, the answer will be the same as long as you are multiplying throughout the problem. Rule: a b c a b c (The answer will be the same) The area of rectangle can be found by using the formula below: Area = length width or A l w Multiply. 1. 6078 Math 300 M-G 4e Chapter 1; Rev: Mar 2011 Page 7 of 22 1.6 Multiplying Whole Numbers and Area (cont) Multiply. 2. 432 375 Estimate the product by first rounding to the nearest hundred, then multiplying: 3. 355 x 299 4. What is the area of this region? 129 yd 65 yd Solving Problems by Multiplying Key Words or Phrases multiply product times Example multiply 5 by 7 the product of 3 and 2 10 times 13 Symbols 57 32 10 13 Solve. 5. One ounce of hulled sunflower seeds contains 14 grams of fat. How many grams of fat are in 6 ounces of hulled sunflower seeds? Math 300 M-G 4e Chapter 1; Rev: Mar 2011 Page 8 of 22 1.7 Dividing Whole Numbers To divide, you should divide the dividend by the divisor to get the quotient. The dividend is the number to be divided. The divisor is the by which the divided is divided. The quotient is the answer. Example: 20 5 4 20 4 5 4 5 20 In the above problems 20 is the dividend, 5 is the divisor, and 4 is the quotient. Other rules to make note of when dividing: Division by 1 – Any number divided by 1 is equal to that number. Examples: 7 7 1 Dividing a number by itself – Any number (nonzero) divided by itself is equal to 1. Examples: 55 1 55 Dividends of 0 – Zero divided by any number (nonzero) is equal to zero. Examples: 0 0 11 Excluding division by 0 – Division by 0 is undefined or not defined. Examples: 3 undefined 0 Math 300 M-G 4e Chapter 1; Rev: Mar 2011 Page 9 of 22 1.7 Dividing Whole Numbers (cont) With larger numbers, you will need to use a process called long division. When using long division you may run into a case where you are left with a remainder. A remainder is the number that is left over from the dividend. The remainder is to be less than the divisor. The symbol for a remainder is R. Divide. 1. 56 1 2. 699 3 3. 6 4846 4. 27 9724 Math 300 M-G 4e Chapter 1; Rev: Mar 2011 Page 10 of 22 1.7 Dividing Whole Numbers (cont) Key Words or Phrases divide Examples divide 10 by 5 quotient the quotient of 64 and 4 9 divided by 3 divided by divided or shared equally among $100 divided equally among five people Symbols 10 5 64 64 4 or 4 10 5 or 9 3 100 100 5 or 5 9 3 or Solve. 5. Find the quotient of 90 and 7. 6. Martin teaches American Sign Language classes for $55 per student for a 7-week session. He collects $1430 from the group of students. Find how many students are in the group. Math 300 M-G 4e Chapter 1; Rev: Mar 2011 Page 11 of 22 1.8 An Introduction to Problem Solving Problem-Solving Steps 1. UNDERSTAND the problem. (Read and reread the problem. Know what you are looking for.) 2. TRANSLATE the problem. (Write the problem as an equation.) 3. SOLVE the problem. (Work out the problem and check to see if the answer you get is reasonable.) 4. INTERPRET the result. (Check your work and state your answer.) Key words or phrases use to help indicate which operation to use Addition sum Subtraction Multiplication difference product Division Equality quotient equals plus minus times divide is equal to added to subtract multiply is/was more than less than multiply by shared equally among increased by total decreased by less of divided by double/triple divided into yields Solve. 1. What is the product of 12 and 9? 2. 78 decreased by 12 is what number? Math 300 M-G 4e Chapter 1; Rev: Mar 2011 Page 12 of 22 1.8 An Introduction to Problem Solving (cont) Solve. 3. A parking lot in the shape of a rectangle measures 100 feet by 150 feet. a. What is the perimeter of the lot? b. What is the area of the parking lot? 4. Three people dream of equally sharing a $147 million lottery. How much would each person receive if they have a winning ticket? 5. Find the total cost of 10 computers at $2100 each and 7 boxes of diskettes at $12 each. Math 300 M-G 4e Chapter 1; Rev: Mar 2011 Page 13 of 22 1.9 Exponents, Square Roots, and Order of Operations Exponential Notation Exponential notation is a shorter notation for writing products that occur often. Example: 3 3 3 3 can be shorten to the exponential notation of 3 4 Note: The 3 is called the base number and the 4 is called the exponent. The exponent tells you how many times to multiply the base times itself Writing Exponential Notation Example: Write exponential notation for 10 10 10 10 10 5 Answer: 10 Example: Write exponential notation for 2 2 2 3 Answer: 2 Evaluating Exponential Notation To evaluate exponential notation, we rewrite it as a product and compute the product. Example: Evaluate 10 3 3 Answer: 10 3 = 10 10 10 = 1,000 not 10 = 10 3 = 30 Math 300 M-G 4e Chapter 1; Rev: Mar 2011 Page 14 of 22 1.9 Exponents, Square Roots, and Order of Operations (cont) 4 Example: Evaluate 5 4 Answer: 5 = 5 5 5 5 = 625 Evaluating Square Roots A square root of a number is one of two identical factors of the number. Example: 7 7 49 , so a square root of 49 is 7. We use a radical sign for finding square roots. Example: Find each square root. 1. 25 5 because 5 5 25 2. 81 9 because 9 9 81 3. 0 0 because 0 0 0 Make sure to understand the difference between squaring a number and finding the square root of a number. Squaring a number: 9 2 9 9 81 Find the square root of a number: 9 3 * A list of perfect squares can be found on page 767. * Math 300 M-G 4e Chapter 1; Rev: Mar 2011 Page 15 of 22 1.9 Exponents, Square Roots, and Order of Operations (cont) Finding the Area of a Square Formula for finding the area of a square Area of a square = side 2 or Area of a square = side side Example: Find the area of a square whose side measures 5 inches. Answer: A = side 2 A = 5 2 A = 25 square inches Example: Find the area of this square. 8 cm Answer: A = side 2 A = 8 2 A = 64 square cm Math 300 M-G 4e Chapter 1; Rev: Mar 2011 Page 16 of 22 1.9 Exponents, Square Roots, and Order of Operations (cont) “Please Excuse My Dear Aunt Sally” P stands for parentheses…………………( ), [ ], { }, Always work the innermost set of parentheses first! Think inside-out!! 2 4 E stands for exponents……………….… 2 , 3 , 45 Be sure to evaluate exponents correctly! The exponent (small number) tell you how many times to multiply the base (big number) to itself!! M stands for multiplication………….… , , *, D stands for division……………….….… /, Multiplication and division go together! It must be worked from left to right!! A stands for addition…………….……... + S stands for subtraction……………..… Addition and subtraction go together! It must be worked from left to right!! Math 300 M-G 4e Chapter 1; Rev: Mar 2011 Page 17 of 22 1.9 Exponents, Square Roots, and Order of Operations (cont) Write exponential notation. 1. 5 5 5 Evaluate. 2. 63 Find the square root. 3. 36 Simplify. 4. 42 7 6 5. 512 7 4 5 2 18 Math 300 M-G 4e Chapter 1; Rev: Mar 2011 Page 18 of 22 1.9 Exponents, Square Roots, and Order of Operations (cont) Simplify. 2 2 6. 35 3 9 7 2 10 3 7. 3 25 2 81 8. 8 x 9 – (12 – 8)/4 – (10 – 7) 9. 72/6 – {2 x [9 – (4 x 2)]} Find the area of each square. 10. 4 meters Math 300 M-G 4e Chapter 1; Rev: Mar 2011 Page 19 of 22 Math 300 Chapter 1 Glossary Digits - 0, 1, 2, 3, 4, 5, 6, 7. 8, 9 Whole Numbers - 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ... Place Value - Position of a number Period - Beginning at the right of the number, groups of three digits separated by a comma Standard Form - Write the number using periods separated by a comma. For example, 1,234. Expanded Form - Shows the number with each digit in its place value with zeros to the right. For example the standard form number 1234 in expanded form is 1000+200+30+4 Table - Display that organizes and shows facts about a set of numbers. Addend - Any number being added to another number Sum - The result of an addition operation. Polygon - Any flat figure that is created by joining straight lines. Perimeter - The distance around a polygon. Minuend - Number from which a number is subtracted. Subtrahend - Number that is subtracted from another. Difference - Result when one number is subtracted from another. Rounding - Approximating a number to a certain accuracy. Factor - When two number multiply to get a result, the numbers are called factors of the result. Product - When two number are multiplied together, the result is called the product. Distribute - to disperse over an area. (See Distributive Property) Area - Measure of a surface. (Result is expressed in square units such as sq ft or sq mi) Dividend - The number begin divided. Divisor - The number by which the dividend is divided. Quotient - The result when the divisor is divided into the dividend Average - A special application of division and addition. It is the result when a list of number is added and the sum is divided by the number of numbers. For example, the average of 3, 6, 9, 14 is 5+6+10+11 = 32 divided by 4 = 8. 8 is the average. Exponent - a value that is placed above and after a base value to denote the power to which the base is to be raised. Base - the value that is to be raised to a power indicated by the exponent. Exponential notation - This notation is shown with the exponent using a smaller font type and it is raised above the base value. Perfect squares - the product of any whole number multiplied by itself. For example, 1=1x1, 4=2x2, 144=12x12 Square root - A square root of a number is one of two identical factors. For example, the square root of 144 is 12 since 12x12=144. The square root of 256 is 16 since 16x16 = 256. Math 300 M-G 4e Chapter 1; Rev: Mar 2011 Page 20 of 22 Math 300 Chapter 1 Glossary (cont) Properties Additive property of 0 - The sum of 0 and any number is that number. 0 + 9 = 9 Commutative Property of Addition - Changing the order of two addends does not change their sum. For example, 5 + 6 = 11 and 6 + 5 = 11 Associative Property of Addition - Changing the grouping of addends does not change their sum. For example, 1 + ( 2 + 3 ) = 1 + 5 = 6 and (1 + 2) + 3 = 3 + 3 = 6 Subtraction Properties of 0 The difference of any number and that same number is 0. For example 24 -24 = 0 The difference of any number and 0 is that same number. For example 24 - 0 = 24 Multiplication Property of 0 - The product of 0 and any number is 0. For example, 5 x 0 =0 or 0 x 5 =0 Multiplication Property of 1 - The product of 1 and any number is the same number. For example, 24 x 1 = 24 or 1 x 24 = 24 Commutative Property of Multiplication - Changing the order of two factors does not change their product. For example 24 x 1 = 24 and 1 x 24 = 24 Associative Property of Multiplication - Changing the grouping of factors does not change their product. For example, (5x6)x7= 5x(6x7) Distributive Property - Multiplication distributes over addition. For example, 4(5+6) = 4x5 + 4x6 = 20+24 = 44 Division Properties of 1 The quotient of any number and that same number is 1. For example, 9/9 = 1 The quotient of any number and 1 is that same number. For example, 12/1=12 Division Properties of 0 The quotient of any number and 0 (except 0) is 0. For example, 0/12=0 The quotient of any number and 0 is not a number, we say it is undefined. For example, 12/0 - undefined Math 300 M-G 4e Chapter 1; Rev: Mar 2011 Page 21 of 22 Math 300 Chapter 1 Hints (cont) A comma may or may not be inserted in a four-digit number. For example, 1234 and 1,234 are both acceptable ways of writing this whole number. Rounding Locate the digit to the right of the given place value If the digit in that place value is 5 or greater, add 1 to the digit in the given place value and replace each digit to the right with a 0 If the digit in the place value is less than 5, replace each digit to the right with a 0 Multiply by whole numbers ending in 0 - The product is found by adding the number of ending 0’s to the other factor. For example, 23x100 = 2300 and 45x1,000,000 = 45,000,000 Area of a rectangle (or square) - Multiply the length times width with the answer specified in square units. Area = length x width. Opposite operations Since addition and subtraction are opposite operations, you can check addition by doing a subtraction. You can check a subtraction by doing an addition. Since multiplication and division are opposite operations, you can check multiplication by doing a division. You can check a division by doing a multiplication. Problem solving steps Understand the problem. Read the problem at least twice. Draw a sketch or graph to be sure you know what is being asked. Look for “magic” keywords (see table). Translate the problem from words into numbers and symbols that indicate what math operation you need to perform. Solve the problem created in the prior step Interpret the result. Verify that the solution satisfies the stated problem. Be sure to put your answer in the proper units of measure. The units of measure will multiply and divide just like numbers so the units of your answer may give you a hint if you have approached the problem properly. Exponents An exponent only applies to its base. Evaluation of an exponent is calculated by multiplying the base times itself “exponent” times. It is not base x exponent. Order of operations Perform all operations within grouping symbols first. The grouping symbols are parenthesis (), brackets [] and other grouping symbols. Evaluate any expressions with exponents. Multiply or divide beginning at the left and work right across the expression Add or subtract beginning at the left and work right across the expression Math 300 M-G 4e Chapter 1; Rev: Mar 2011 Page 22 of 22