Section 1.1 – Place Value 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

Copyright © 2014 – Luis Soto-Ortiz 10

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.

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

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

Example 1.1.1

millions

hundred

thousands ten

thousands

thousands hundreds tens

ones


The amount $29,632 is written as

“Twenty-nine thousand, six hundred thirty-two dollars.”

2 9 , 6 3 2

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 “seventy-

eight 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.

Example 1.1.2

ten

thousands thousands

hundreds tens

ones


Write the following numbers using a hyphen where appropriate:

A)

Answer: “Seventy-five billion, nine hundred six million, one hundred eighty-

two thousand, five hundred thirty-four.”

B)

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.

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

7 5 , 9 0 6 , 1 8 2 , 5 3 4

9 1 6 , 0 0 0 , 2 6 3 , 8 5 1

Classwork 1.1

Example 1.1.3


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


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


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


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.



a. Tens place

b. Thousands place

c. Ones place

d. Ten thousands place

e. Hundreds place


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


a. 9

b. 3

c. 7

d. 2

e. 5

f. 6

g. 8

Homework 1.1


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


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


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.


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.


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.

Original number: $ 3 4, 7 8 1

Rounded to the nearest ten: $ 3 4, 7 8 0

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.

Original number: $ 4 9, 7 5 0, 2 6 9

Rounded to the nearest hundred thousand: $ 4 9, 8 0 0, 0 0 0

Example 1.2.1

Example 1.2.2

Place digit

Place digit


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.

Original number: $ 6 3 7

Rounded to the nearest hundred: $ 6 0 0

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.

Original number: $ 5, 9 3 0

Rounded to the nearest thousand: $ 6, 0 0 0

Example 1.2.3

Example 1.2.4

Place digit

Place digit


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.

Original number: $ 3 9 8, 7 2 4

Rounded to the nearest ten thousand: $ 4 0 0, 0 0 0

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.

Original number: $ 2, 3 9 9, 9 5 4

Rounded to the nearest hundred: $ 2, 4 0 0, 0 0 0

Example 1.2.5

Example 1.2.6

Place digit

Place digit


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.

Original number: $ 3 8 = $ 0 3 8

Rounded to the nearest hundred: $ 0 0 0 = $ 0

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.

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

Example 1.2.7

Example 1.2.8

Place digit

Place digit


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


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

Classwork 1.2



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


a. hundred

b. hundred million

c. ten

d. ten billion

e. million


a. hundred trillion

b. ten million

c. hundred thousand

d. hundred

e. ten thousand

f. billion


a. ten billion


b. thousand

c. ten million

d. billion

e. trillion


a. hundred

b. hundred thousand

c. billion

d. ten thousand

e. ten million

f. ten


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



a. thousand

b. hundred thousand

c. ten

d. million

e. hundred

f. ten thousand


a. hundred

b. ten

c. hundred thousand

d. thousand

e. million


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

Homework 1.2



a. thousand

b. hundred thousand

c. billion

d. ten thousand

e. hundred million

f. hundred


a. thousand

b. hundred million

c. trillion

d. million

e. ten trillion

f. billion


a. hundred

b. ten

c. hundred billion

d. hundred thousand

e. million


a. hundred thousand

b. billion

c. hundred

d. million

e. thousand

f. ten


a. million


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


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

Documents

Section 1.1 – Place Value 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