Section 1.1 Place Value Whole numbers appear in everyday situations. We encounter whole numbers in ATM machines when we withdraw money, conduct an inventory, carry out a census count, and when counting the number of votes a politician received in an election campaign. Hence, it is important to develop number sense. Whole numbers consist of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 and each digit has a place value depending on its place or position. The following example serves to illustrate the concept of place and place value. Example 1.1.1 The amount $1,845,027 is written as One million, eight hundred forty-five thousand, twenty-seven dollars. 1, 8 4 5, 0 2 7 millions ones hundred thousands ten thousands thousands hundreds tens The digit 1 is in the millions place, and so its place value is 1,000,000. The digit 8 is in the hundred thousands place, and so its place value is 800,000. The digit 4 is in the ten thousands place, and so its place value is 40,000. The digit 5 is in the thousands place, and its place value is 5,000. The digit 0 is in the hundreds place, and its place value is 0. The digit 2 is in the tens place, and its place value is 20. The digit 7 is in the ones place and its place value is 7. This means that $1,845,027 = $1,000,000 + $800,000 + $40,000 + $5,000 + $0 + $20 + $7 Copyright 2014 Luis Soto-Ortiz 10
Example 1.1.2 The amount $29,632 is written as Twenty-nine thousand, six hundred thirty-two dollars. 2 9, 6 3 2 ten thousands thousands hundreds tens ones The 2 on the left is in the ten thousands place, and so its place value is 20,000. The 9 is in the thousands place, and its place value is 9,000. The 6 is in the hundreds place, and its place value is 600. The 3 is in the tens place, and its place value is 30. The 2 on the right is in the ones place and its place value is 2. This means that $29,632 = $20,000 + $9,000 + $600 + $30 + $2 Note: We do not use the word and when we talk about whole numbers. For example, for the amount $1,845,027 we do not say one million, eight hundred forty-five thousand, and twenty-seven dollars. We use the word and when we have decimal numbers. The correct way to say the amount $78.35 is seventyeight dollars and thirty-five cents. We use the word and to represent the decimal point. We will discuss decimals in greater detail in Chapter 5. You should memorize the names of the various places (tens, hundreds, thousands, etc.) in which each digit of a whole number is located. Knowing the place name will allow you to determine the corresponding place value of each digit. Knowing the place names will be helpful in Section 1.2 when you are asked to round a whole number to a specific place. Copyright 2014 Luis Soto-Ortiz 11
Example 1.1.3 Write the following numbers using a hyphen where appropriate: A) 7 5, 9 0 6, 1 8 2, 5 3 4 Answer: Seventy-five billion, nine hundred six million, one hundred eightytwo thousand, five hundred thirty-four. B) 9 1 6, 0 0 0, 2 6 3, 8 5 1 Answer: Nine hundred sixteen billion, two hundred sixty-three thousand, eight hundred fifty-one. Notice that you are supposed to write the hyphen when you write two numbers one after the other, like fifty-one, sixty-three, eighty-two, but not when you write sixteen, two hundred, etc. Classwork 1.1 1. For the number 25,963,407 determine which digit is in the a. Hundreds place b. Ones place c. Ten thousands place d. Millions place e. Tens place 2. For the number 9,148 determine which digit is in the a. Thousands place b. Tens place c. Ones place d. Hundreds place Copyright 2014 Luis Soto-Ortiz 12
3. For the number 638,941 determine the place of each digit a. 3 b. 8 c. 1 d. 9 e. 4 f. 6 4. For the number 86,430,127 determine the place value of each digit a. 1 b. 6 c. 7 d. 0 e. 4 f. 2 5. Write each number. Use a hyphen when necessary. a. 37,821 b. 860,000,703,892,004 c. 109,000,763 6. For the number 836,945,067,219 determine which digit is in the a. Billions place b. Thousands place c. Hundred thousands d. Millions place e. Hundred billions place 7. For the number 645,009,378, 521,693 determine which digit is in the a. Ten millions place b. Hundreds place c. Ones place d. Ten trillions place Copyright 2014 Luis Soto-Ortiz 13
e. Hundred thousands place 8. For the number 56,847,935,290,515 determine the place of each digit a. 3 b. 8 c. 2 d. 6 e. 0 f. 1 9. For the number 973,603,518 determine the place value of each digit a. 9 b. 8 c. 7 d. 5 e. 0 f. 6 10.Write each number. Use a hyphen when necessary. a. 56,747,900,270 b. 213,756,206,887,954 c. 612,000, 040,845 d. 45,000,009 e. 368,208,540 f. 98,037,216,000 g. 783,965 h. 176,007,362,000,845 Copyright 2014 Luis Soto-Ortiz 14
CW 1.1 Solutions: 1. a. 4 b. 7 c. 6 d. 5 e. 0 2. a. 9 b. 4 c. 8 d. 1 3. a. Ten thousands b. Thousands c. Ones d. Hundreds e. Tens f. Hundred thousands 4. a. 100 b. 30,000 c. 6,000,000 d. 7 e. 0 f. 400,000 g. 20 5. a. Thirty-seven thousand, eight hundred twenty-one. b. Eight hundred sixty trillion, seven hundred three million, eight hundred ninety-two thousand, four. c. One hundred nine million, seven hundred sixty-three. 6. a. 6 b. 7 c. 0 d. 5 e. 8 7. a. 7 b. 6 c. 3 d. 4 e. 5 8. a. ten millions b. hundred billions c. hundred thousands d. trillions e. thousands f. ones 9. a. 900,000,000 b. 8 c. 70,000,000 d. 500 e. 0 f. 600,000 10. a. Fifty-six billion, seven hundred forty-seven million, nine hundred thousand, two hundred seventy. b. Two hundred thirteen trillion, seven hundred fifty-six billion, two hundred six million, eight hundred eighty-seven, nine hundred fifty-four. c. Six hundred twelve billion, forty thousand, eight hundred forty-five. d. Forty-five million, nine e. Three hundred sixty-eight million, two hundred eight thousand, five hundred forty. f. Ninety-eight billion, thirty-seven million, two hundred sixteen thousand. g. Seven hundred eighty-three thousand, nine hundred sixty-five. h. One hundred seventy-six trillion, seven billion, three hundred sixty-two million, eight hundred forty-five. Copyright 2014 Luis Soto-Ortiz 15
Homework 1.1 1. For the number 53,784 determine which digit is in the a. Tens place b. Thousands place c. Ones place d. Ten thousands place e. Hundreds place 2. For the number 380,157 determine which digit is in the a. Thousands place b. Hundreds place c. Hundred thousands place d. Tens place 3. For the number 9,207,586 determine the place of each digit a. 5 b. 0 c. 7 d. 6 e. 2 f. 9 4. For the number 76,192,835 determine the place value of each digit a. 9 b. 3 c. 7 d. 2 e. 5 f. 6 g. 8 Copyright 2014 Luis Soto-Ortiz 16
5. Write each number. Use a hyphen when necessary. a. 872,559 b. 42,213,099 c. 780,019,000,365 6. For the number 145,453,698,072 determine which digit is in the a. Tens place b. Hundred thousands place c. Ten billions place d. Millions place e. Hundreds place 7. For the number 396,008,572,481,202 determine which digit is in the a. Hundred billions place b. Ones place c. Ten millions place d. Hundred trillions place 8. For the number 723,409,817,586 determine the place of each digit a. 4 b. 0 c. 3 d. 6 e. 2 f. 5 9. For the number 360,010,992, 590,478 determine the place value of each digit a. 4 b. 2 c. 7 d. 8 e. 3 Copyright 2014 Luis Soto-Ortiz 17
f. 6 g. 1 10.Write each number. Use a hyphen when necessary. a. 830,461,000,312 b. 571,992,412,019,720 c. 981,020,478,556,008 d. 417,372,900,430,112 e. 782,355,415,050 f. 178,346 g. 79,467,000,317 h. 688,940,763,947 Copyright 2014 Luis Soto-Ortiz 18
HW 1.1 Solutions: 1. a. 8 b. 3 c. 4 d. 5 e. 7 2. a. 0 b. 1 c. 3 d. 5 3. a. Hundreds b. Ten thousands c. Thousands d. Ones e. Hundred thousands f. Millions 4. a. 90,000 b. 30 c. 70,000,000 d. 2,000 e. 5 f. 6,000,000 g. 800 5. a. Eight hundred seventy-two thousand, five hundred fifty-nine. b. Forty-two million, two hundred thirteen thousand, ninety-nine. c. Seven hundred eighty billion, nineteen million, three hundred sixty-five. 6. a. 7 b. 6 c. 4 d. 3 e. 0 7. a. 0 b. 2 c. 7 d. 3 8. a. hundred millions b. ten millions c. billions d. ones e. ten billions f. hundreds 9. a. 400 b. 2,000,000 c. 70 d. 8 e. 300,000,000,000,000 f. 60,000,000,000,000 g. 10,000,000,000 10. a. Eight hundred thirty billion, four hundred sixty-one million, three hundred twelve. b. Five hundred seventy-one trillion, nine hundred ninety-two billion, four hundred twelve million, nineteen thousand, seven hundred twenty. c. Nine hundred eighty-one trillion, twenty billion, four hundred seventy-eight million, five hundred fifty-six thousand, eight. d. Four hundred seventeen trillion, three hundred seventy-two billion, nine hundred million, four hundred thirty thousand, one hundred twelve. e. Seven hundred eighty-two billion, three hundred fifty-five million, four hundred fifteen thousand, fifty. f. One hundred seventy-eight thousand, three hundred forty-six. g. Seventy-nine billion, four hundred sixty-seven million, three hundred seventeen. h. Six hundred eighty-eight billion, nine hundred forty million, seven hundred sixty-three thousand, nine hundred forty-seven. Copyright 2014 Luis Soto-Ortiz 19
Section 1.2 Rounding Whole Numbers Rounding a whole number is often useful because it gives us an estimate of cost, expense, or earnings. Suppose that last year you had your patio remodeled for $5,647 and that you also bought a new car for $36,825. You may want to write down a quick estimate of your expenses from last year. For example, you may decide to write the cost of remodeling the patio as $5,650 and the cost of the new car as $36,800. This estimation is called rounding. To round a whole number in a particular way, you first need to determine to which place you want to round the amount. Do you want to round the amount to the nearest ten, to the nearest hundred, to the nearest thousand, etc.? Before continuing reading, please review Section 1.1 which explains place and place value if you are not familiar with these terms. Method to Round Whole Numbers: Step 1: Determine to which place (tens, hundreds, thousands, etc.) you want to round a given whole number. Step 2: Find the digit that is located in that place. We call this digit the place digit. Then draw an imaginary wall to the right of the place digit. Step 3: Look at the digit immediately to the right of the wall you drew in step 2. If this digit is 5 or more (5, 6, 7, 8 or 9), increase the place digit in step 2 by 1 and change all the digits to the right of the wall to zero. Otherwise, if this digit is less than 5 (0, 1, 2, 3, 4), leave the place digit in step 2 as it is, and make all the digits to the right of the wall zero. We will now illustrate this method to round whole numbers with some examples. Copyright 2014 Luis Soto-Ortiz 20
Example 1.2.1 Suppose you want to round $34,781 to the nearest ten. Since 8 is in the tens place, we call 8 the place digit. We draw a wall to the right of the place digit. Since the digit to the right of the wall is 1 (it is less than 5), we leave the number 3478 as is and make the 1 on the right of the wall into a zero. Therefore, $34,781 rounded to the nearest ten is $34,780. This means that $34,781 is closer to $34,780 than to $34,790 when we count by tens. Place digit Original number: $ 3 4, 7 8 1 Rounded to the nearest ten: $ 3 4, 7 8 0 Example 1.2.2 Suppose you want to round $49,750,269 to the nearest hundred thousand. Since 7 is in the hundred thousands place, we call 7 the place digit. We draw a wall to the right of the place digit. Since the digit to the right of the wall is 5 (it is 5 or greater), we add 1 to 497, which now becomes 498, and make all the digits to the right of the wall zero. Therefore, $49,750,269 rounded to the nearest hundred thousand is $49,800,000. This means that $49,750,269 is closer to $49,800,000 than to $49,700,000 when we count by hundred thousands. Place digit Original number: $ 4 9, 7 5 0, 2 6 9 Rounded to the nearest hundred thousand: $ 4 9, 8 0 0, 0 0 0 Copyright 2014 Luis Soto-Ortiz 21
Example 1.2.3 If we want to round $637 to the nearest hundred, we first notice that the digit 6 is in the hundreds place. Thus, we call 6 the place digit and draw a wall to the right of the place digit. Since the digit to the right of the wall is 3 (it is less than 5), we leave the 6 as is and make all the digits to the right of the wall into zeros. Therefore, $637 rounded to the nearest hundred is $600. This means that $637 is closer to $600 than to $700 when we count by hundreds. Place digit Original number: $ 6 3 7 Rounded to the nearest hundred: $ 6 0 0 Example 1.2.4 Let s round $5,930 to the nearest thousand. We notice that the digit 5 is in the thousands place. Thus, we call 5 the place digit and draw a wall to the right of the place digit. Since the digit to the right of the wall is 9 (it is 5 or greater), we add 1 to the 5, which now becomes 6, and make all the digits to the right of the wall zero. Therefore, $5,930 rounded to the nearest thousand is $6,000. This means that $5,930 is closer to $6,000 than to $5,000 when we count by thousands. Place digit Original number: $ 5, 9 3 0 Rounded to the nearest thousand: $ 6, 0 0 0 Copyright 2014 Luis Soto-Ortiz 22
Example 1.2.5 Let s round $398,724 to the nearest ten thousand. We notice that in the ten thousands place there is a 9. Thus, we call 9 the place digit and draw a wall to the right of the place digit. Since the digit to the right of the wall is an 8 (it is 5 or greater), we add 1 to the 39, which now becomes 40. We then turn all the digits to the right of the wall into zeros. Therefore, $398,724 rounded to the nearest ten thousand is $400,000. This means that $399,724 is closer to $400,000 than to $390,000 when we count by ten thousands. Place digit Original number: $ 3 9 8, 7 2 4 Rounded to the nearest ten thousand: $ 4 0 0, 0 0 0 Example 1.2.6 Let s round $2,399,954 to the nearest hundred. We notice that in the hundreds place there is a 9. Thus, we call 9 the place digit and draw a wall to the right of the place digit. Since the digit to the right of the wall is a 5 (it is 5 or greater), we add 1 to 23999, which now becomes 24000. We then turn all the digits to the right of the wall into zeros. Therefore, $2,399,954 rounded to the nearest hundred is $2,400,000. This means that $2,399,954 is closer to $2,400,000 than to $2,399,900 when we count by hundreds. Place digit Original number: $ 2, 3 9 9, 9 5 4 Rounded to the nearest hundred: $ 2, 4 0 0, 0 0 0 Copyright 2014 Luis Soto-Ortiz 23
Example 1.2.7 Let s round $38 to the nearest hundred. We notice that in the hundreds place there is no digit. When this happens, we assume there is a zero in the hundreds place. Thus, we call 0 the place digit and draw a wall to the right of the place digit. Since the digit to the right of the wall is a 3 (it is less than 5), we leave the zero in the hundreds place as it is. We then turn all the digits to the right of the wall into zeros. Therefore, $38 rounded to the nearest hundred is $000 = $0. This means that $38 is closer to $0 than to $100 when we count by hundreds. Place digit Original number: $ 3 8 = $ 0 3 8 Rounded to the nearest hundred: $ 0 0 0 = $ 0 Example 1.2.8 Let s round $53,414 to the nearest hundred thousand. We notice that in the hundred thousands place there is no digit. When this happens, we assume there is a zero in the hundred thousands place. Thus, we call 0 the place digit and draw a wall to the right of the place digit. Since the digit to the right of the wall is a 5 (it is 5 or greater), we add 1 to the zero in the hundred thousands place, which now becomes 1. We then turn all the digits to the right of the wall into zeros. Therefore, $53,414 rounded to the nearest hundred thousand is $100,000. This means that $53,414 is closer to $100,000 than to $0 when we count by hundred thousands. Place digit Original number: $ 5 3, 4 1 4 = $ 0 5 3, 4 1 4 Rounded to the nearest hundred thousand: $ 1 0 0, 0 0 0 Copyright 2014 Luis Soto-Ortiz 24
Classwork 1.2 1. Round the number 38,946,018 to the nearest a. thousand b. hundred thousand c. ten d. million e. hundred f. ten thousand 2. Round the number 457,195 to the nearest a. hundred b. ten c. hundred thousand d. thousand e. million 3. Round the number 865,974,096 to the nearest a. hundred million b. hundred thousand c. hundred d. million e. ten thousand f. billion 4. Round the number 832,000,759,299 to the nearest a. ten billion b. hundred thousand c. million d. billion e. trillion Copyright 2014 Luis Soto-Ortiz 25
5. Round the number 209,995,184 to the nearest a. thousand b. hundred thousand c. billion d. ten thousand e. hundred million f. hundred 6. Round the number 946,772,062,834,561 to the nearest a. trillion b. ten million c. hundred d. hundred billion e. ten f. billion 7. Round the number 679,999,538,407 to the nearest a. hundred b. hundred million c. ten d. ten billion e. million 8. Round the number 307,412,885,809,945 to the nearest a. hundred trillion b. ten million c. hundred thousand d. hundred e. ten thousand f. billion 9. Round the number 383,000,943,458 to the nearest a. ten billion Copyright 2014 Luis Soto-Ortiz 26
b. thousand c. ten million d. billion e. trillion 10. Round the number 499,126,008 to the nearest a. hundred b. hundred thousand c. billion d. ten thousand e. ten million f. ten Copyright 2014 Luis Soto-Ortiz 27
CW 1.2 Solutions: 1. a. 38,946,000 b. 38,900,000 c. 38,946,020 d. 39,000,000 e. 38,946,000 f. 38,950,000 2. a. 457,200 b. 457,200 c. 500,000 d. 457,000 e. 0 3. a. 900,000,000 b. 866,000,000 c. 865,974,100 d. 866,000,000 e. 865,970,000 f. 1,000,000,000 4. a. 830,000,000,000 b. 832,000,800,000 c. 832,001,000,000 d. 832,000,000,000 e. 1,000,000,000,000 5. a. 209,995,000 b. 210,000,000 c. 0 d. 210,000,000 e. 200,000,000 f. 209,995,200 6. a. 947,000,000,000,000 b. 946,772,060,000,000 c. 946,772,062,834,600 d. 946,800,000,000,000 e. 946,772,062,834,560 f. 946,772,000,000,000 7. a. 679,999,538,400 b. 680,000,000,000 c. 679,999,538,410 d. 680,000,000,000 e. 680,000,000,000 8. a. 300,000,000,000,000 b. 307,412,890,000,000 c. 307,412,885,800,000 d. 307,412,885,809,900 e. 307,413,000,000,000 9. a. 380,000,000,000 b. 383,000,943,000 c. 383,000,000,000 d. 383,000,000,000 e. 0 10. a. 499,126,000 b. 499,100,000 c. 0 d. 499,130,000 e. 500,000,000 f. 499,126,010 Copyright 2014 Luis Soto-Ortiz 28
Homework 1.2 1. Round the number 68,905,745 to the nearest a. thousand b. hundred thousand c. ten d. million e. hundred f. ten thousand 2. Round the number 8,997,095 to the nearest a. hundred b. ten c. hundred thousand d. thousand e. million 3. Round the number 502,915,746 to the nearest a. hundred million b. hundred thousand c. hundred d. million e. ten thousand f. ten 4. Round the number 767,452 to the nearest a. million b. ten c. ten thousand d. hundred e. ten million Copyright 2014 Luis Soto-Ortiz 29
5. Round the number 963,702,491 to the nearest a. thousand b. hundred thousand c. billion d. ten thousand e. hundred million f. hundred 6. Round the number 45,895,004,279,366 to the nearest a. thousand b. hundred million c. trillion d. million e. ten trillion f. billion 7. Round the number 477,936,500,034 to the nearest a. hundred b. ten c. hundred billion d. hundred thousand e. million 8. Round the number 275,999,958,624 to the nearest a. hundred thousand b. billion c. hundred d. million e. thousand f. ten 9. Round the number 378,566,921 to the nearest a. million Copyright 2014 Luis Soto-Ortiz 30
b. hundred million c. ten d. hundred e. hundred thousand 10.Round the number 732,491,230,445 to the nearest a. ten million b. hundred thousand c. billion d. trillion e. hundred million f. hundred Copyright 2014 Luis Soto-Ortiz 31
HW 1.2 Solutions: 1. a. 68,906,000 b. 68,900,000 c. 68,905,750 d. 69,000,000 e. 68,905,700 f. 68,910,000 2. a. 8,997,100 b. 8,997,100 c. 9,000,000 d. 8,997,000 e. 9,000,000 3. a. 500,000,000 b. 502,900,000 c. 502,915,700 d. 503,000,000 e. 502,920,000 f. 502,915,750 4. a. 1,000,000 b. 767,450 c. 770,000 d. 767,500 e. 0 5. a. 963,702,000 b. 963,700,000 c. 1,000,000,000 d. 963,700,000 e. 1,000,000,000 f. 963,702,500 6. a. 45,895,004,279,000 b. 45,895,000,000,000 c. 46,000,000,000,000 d. 45,895,004,000,000 e. 50,000,000,000,000 f. 45,895,000,000,000 7. a. 477,936,500,000 b. 477,936,500,030 c. 480,000,000,000 d. 477,936,500,000 e. 477,937,000,000 8. a. 276,000,000,000 b. 276,000,000,000 c. 275,999,958,600 d. 276,000,000,000 e. 275,999,959,000 f. 275,999,958,620 9. a. 379,000,000 b. 400,000,000 c. 378,566,920 d. 378,566,900 e. 378,600,000 10. a. 732,490,000,000 b. 732,491,200,000 c. 732,000,000,000 d. 1,000,000,000,000 e. 732,500,000,000 f. 732,491,230,400 Copyright 2014 Luis Soto-Ortiz 32