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Introduction to Whole Numbers

Module by: First Last. E-mail the author

Summary: By the end of this section, you will be able to:

  • Identify counting numbers and whole numbers
  • Model whole numbers
  • Identify the place value of a digit
  • Use place value to name whole numbers
  • Use place value to write whole numbers
  • Round whole numbers

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Identify Counting Numbers and Whole Numbers

Learning algebra is similar to learning a language. You start with a basic vocabulary and then add to it as you go along. You need to practice often until the vocabulary becomes easy to you. The more you use the vocabulary, the more familiar it becomes.

Algebra uses numbers and symbols to represent words and ideas. Let’s look at the numbers first. The most basic numbers used in algebra are those we use to count objects: 1,2,3,4,5,1,2,3,4,5, and so on. These are called the counting numbers. The notation “…” is called an ellipsis, which is another way to show “and so on”, or that the pattern continues endlessly. Counting numbers are also called natural numbers.

Note:

Doing the Manipulative Mathematics activity Number Line-Part 1 will help you develop a better understanding of the counting numbers and the whole numbers.

Note: Counting Numbers:

The counting numbers start with 11 and continue.

1,2,3,4,5…1,2,3,4,5…

Counting numbers and whole numbers can be visualized on a number line as shown in Figure 1.

Figure 1: The numbers on the number line increase from left to right, and decrease from right to left.
An image of a number line from 0 to 6 in increments of one. An arrow above the number line pointing to the right with the label “larger”. An arrow pointing to the left with the label “smaller”.

The point labeled 00 is called the origin. The points are equally spaced to the right of 00 and labeled with the counting numbers. When a number is paired with a point, it is called the coordinate of the point.

The discovery of the number zero was a big step in the history of mathematics. Including zero with the counting numbers gives a new set of numbers called the whole numbers.

Note: Whole Numbers:

The whole numbers are the counting numbers and zero.

0,1,2,3,4,5…0,1,2,3,4,5…

We stopped at 55 when listing the first few counting numbers and whole numbers. We could have written more numbers if they were needed to make the patterns clear.

Example 1

Problem 1

Which of the following are counting numbers? whole numbers?

0,14,3,5.2,15,1050,14,3,5.2,15,105

Solution: Solution
  • The counting numbers start at 1,1, so 00 is not a counting number. The numbers 3,15,and1053,15,and105 are all counting numbers.
  • Whole numbers are counting numbers and 0.0. The numbers 0,3,15,and1050,3,15,and105 are whole numbers.

The numbers 1414 and 5.25.2 are neither counting numbers nor whole numbers. We will discuss these numbers later.

Note:

Exercise 1

Which of the following are counting numbers whole numbers?

0,23,2,9,11.8,241,3760,23,2,9,11.8,241,376

Solution
  • 2, 9, 241, 376
  • 0, 2, 9, 241, 376

Note:

Exercise 2

Which of the following are counting numbers whole numbers?

0,53,7,8.8,13,2010,53,7,8.8,13,201

Solution
  • 7, 13, 201
  • 0, 7, 13, 201

Model Whole Numbers

Our number system is called a place value system because the value of a digit depends on its position, or place, in a number. The number 537537 has a different value than the number 735.735. Even though they use the same digits, their value is different because of the different placement of the 33 and the 77 and the 5.5.

Money gives us a familiar model of place value. Suppose a wallet contains three $100$100 bills, seven $10$10 bills, and four $1$1 bills. The amounts are summarized in Figure 2. How much money is in the wallet?

Figure 2
An image of three stacks of American currency. First stack from left to right is a stack of 3 $100 bills, with label “Three $100 bills, 3 times $100 equals $300”. Second stack from left to right is a stack of 7 $10 bills, with label “Seven $10 bills, 7 times $10 equals $70”. Third stack from left to right is a stack of 4 $1 bills, with label “Four $1 bills, 4 times $1 equals $4”.

Find the total value of each kind of bill, and then add to find the total. The wallet contains $374.$374.

An image of “$300 + $70 +$4” where the “3” in “$300”, the “7” in “$70”, and the “4” in “$4” are all in red instead of black like the rest of the expression. Below this expression there is the value “$374”. An arrow points from the red “3” in the expression to the “3” in “$374”, an arrow points to the red “7” in the expression to the “7” in “$374”, and an arrow points from the red “4” in the expression to the “4” in “$374”.

Base-10 blocks provide another way to model place value, as shown in Figure 3. The blocks can be used to represent hundreds, tens, and ones. Notice that the tens rod is made up of 1010 ones, and the hundreds square is made of 1010 tens, or 100100 ones.

Figure 3
An image with three items. The first item is a single block with the label “A single block represents 1”. The second item is a horizontal rod consisting of 10 blocks, with the label “A rod represents 10”. The third item is a square consisting of 100 blocks, with the label “A square represents 100”. The square is 10 blocks tall and 10 blocks wide.

Figure 4 shows the number 138138 modeled with base-10base-10 blocks.

Figure 4: We use place value notation to show the value of the number 138.138.
An image consisting of three items. The first item is a square of 100 blocks, 10 blocks wide and 10 blocks tall, with the label “1 hundred”. The second item is 3 horizontal rods containing 10 blocks each, with the label “3 tens”. The third item is 8 individual blocks with the label “8 ones”.
An image of “100 + 30 +8” where the “1” in “100”, the “3” in “30”, and the “8” are all in red instead of black like the rest of the expression. Below this expression there is the value “138”. An arrow points from the red “1” in the expression to the “1” in “138”, an arrow points to the red “3” in the expression to the “3” in “138”, and an arrow points from the red “8” in the expression to the “8” in 138.
Digit Place value Number Value Total value
11 hundreds 11 100100 100100
33 tens 33 1010 3030
88 ones 88 11 +8+8
    Sum = 138Sum = 138

Example 2

Problem 1

Use place value notation to find the value of the number modeled by the base-10base-10 blocks shown.

An image consisting of three items. The first item is two squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is one horizontal rod containing 10 blocks. The third item is 5 individual blocks.
Solution: Solution

There are 22 hundreds squares, which is 200.200.

There is 11 tens rod, which is 10.10.

There are 55 ones blocks, which is 5.5.
An image of “200 + 10 + 5” where the “2” in “200”, the “1” in “10”, and the “5” are all in red instead of black like the rest of the expression. Below this expression there is the value “215”. An arrow points from the red “2” in the expression to the “2” in “215”, an arrow points to the red “1” in the expression to the “1” in “215”, and an arrow points from the red “5” in the expression to the “5” in 215.

Digit Place value Number Value Total value
22 hundreds 22 100100 200200
11 tens 11 1010 1010
55 ones 55 11 +5+5
    215215

The base-10base-10 blocks model the number 215.215.

Note:

Exercise 3

Use place value notation to find the value of the number modeled by the base-10base-10 blocks shown.

An image consisting of three items. The first item is a square of 100, 10 blocks wide and 10 blocks tall. The second item is 7 horizontal rods containing 10 blocks each. The third item is 6 individual blocks.
Solution

176

Note:

Exercise 4

Use place value notation to find the value of the number modeled by the base-10base-10 blocks shown.

An image consisting of three items. The first item is two squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is three horizontal rods containing 10 blocks each. The third item is 7 individual blocks.
Solution

237

Note:

Doing the Manipulative Mathematics activity “Model Whole Numbers” will help you develop a better understanding of place value of whole numbers.

Identify the Place Value of a Digit

By looking at money and base-10base-10 blocks, we saw that each place in a number has a different value. A place value chart is a useful way to summarize this information. The place values are separated into groups of three, called periods. The periods are ones, thousands, millions, billions, trillions, and so on. In a written number, commas separate the periods.

Just as with the base-10base-10 blocks, where the value of the tens rod is ten times the value of the ones block and the value of the hundreds square is ten times the tens rod, the value of each place in the place-value chart is ten times the value of the place to the right of it.

Figure 5 shows how the number 5,278,1945,278,194 is written in a place value chart.

Figure 5
A chart titled 'Place Value' with fifteen columns and 4 rows, with the columns broken down into five groups of three. The header row shows Trillions, Billions, Millions, Thousands, and Ones. The next row has the values 'Hundred trillions', 'Ten trillions', 'trillions', 'hundred billions', 'ten billions', 'billions', 'hundred millions', 'ten millions', 'millions', 'hundred thousands', 'ten thousands', 'thousands', 'hundreds', 'tens', and 'ones'. The first 8 values in the next row are blank. Starting with the ninth column, the values are '5', '2', '7', '8', '1', '9', and '4'.
  • The digit 55 is in the millions place. Its value is 5,000,000.5,000,000.
  • The digit 22 is in the hundred thousands place. Its value is 200,000.200,000.
  • The digit 77 is in the ten thousands place. Its value is 70,000.70,000.
  • The digit 88 is in the thousands place. Its value is 8,000.8,000.
  • The digit 11 is in the hundreds place. Its value is 100.100.
  • The digit 99 is in the tens place. Its value is 90.90.
  • The digit 44 is in the ones place. Its value is 4.4.

Example 3

Problem 1

In the number 63,407,218;63,407,218; find the place value of each of the following digits:

  1. 77
  2. 00
  3. 11
  4. 66
  5. 33
Solution: Solution

Write the number in a place value chart, starting at the right.
A figure titled “Place Values” with fifteen columns and 2 rows, with the colums broken down into five groups of three. The first row has the values “Hundred trillions”, “Ten trillions”, “trillions”, “hundred billions”, “ten billions”, “billions”, “hundred millions”, “ten millions”, “millions”, “hundred thuosands”, “ten thousands”, “thousands”, “hundreds”, “tens”, and “ones”. The first 7 values in the second row are blank. Starting with eighth column, the values are “6”, “3”, “4”, “0”, “7”, “2”, “1” and “8”. The first group is labeled “trillions” and contains the first row values of “Hundred trillions”, “ten trillions”, and “trillions”. The second group is labeled “billions” and contains the first row values of “Hundred billions”, “ten billions”, and “billions”. The third group is labeled “millions” and contains the first row values of “Hundred millions”, “ten millions”, and “millions”. The fourth group is labeled “thousands” and contains the first row values of “Hundred thousands”, “ten thousands”, and “thousands”. The fifth group is labeled “ones” and contains the first row values of “Hundreds”, “tens”, and “ones”.

  • The 77 is in the thousands place.
  • The 00 is in the ten thousands place.
  • The 11 is in the tens place.
  • The 66 is in the ten millions place.
  • The 33 is in the millions place.

Note:

Exercise 5

For each number, find the place value of digits listed: 27,493,61527,493,615

  1. 22
  2. 11
  3. 44
  4. 77
  5. 55
Solution
  • ten millions
  • tens
  • hundred thousands
  • millions
  • ones

Note:

Exercise 6

For each number, find the place value of digits listed: 519,711,641,328519,711,641,328

  1. 99
  2. 44
  3. 22
  4. 66
  5. 77
Solution
  • billions
  • ten thousands
  • tens
  • hundred thousands
  • hundred millions

Use Place Value to Name Whole Numbers

When you write a check, you write out the number in words as well as in digits. To write a number in words, write the number in each period followed by the name of the period without the ‘s’ at the end. Start with the digit at the left, which has the largest place value. The commas separate the periods, so wherever there is a comma in the number, write a comma between the words. The ones period, which has the smallest place value, is not named.

An image with three values separated by commas. The first value is “37” and has the label “millions”. The second value is “519” and has the label thousands. The third value is “248” and has the label ones. Underneath, the value “37” has an arrow pointing to “Thirty-seven million”, the value “519” has an arrow pointing to “Five hundred nineteen thousand”, and the value “248” has an arrow pointing to “Two hundred forty-eight”.

So the number 37,519,24837,519,248 is written thirty-seven million, five hundred nineteen thousand, two hundred forty-eight.

Notice that the word and is not used when naming a whole number.

Note: Name a whole number in words.:

  1. Step 1. Starting at the digit on the left, name the number in each period, followed by the period name. Do not include the period name for the ones.
  2. Step 2. Use commas in the number to separate the periods.

Example 4

Problem 1

Name the number 8,165,432,098,7108,165,432,098,710 in words.

Solution: Solution
Table 3
Begin with the leftmost digit, which is 8. It is in the trillions place. eight trillion
The next period to the right is billions. one hundred sixty-five billion
The next period to the right is millions. four hundred thirty-two million
The next period to the right is thousands. ninety-eight thousand
The rightmost period shows the ones. seven hundred ten

An image with five values separated by commas. The first value is “8” and has the label “trillions”. The second value is “165” and has the label “bilions”. The third value is “432” and has the label “millions”. The fourth value is “098” and has the label “thousands”. The fifth value is “710” and has the label “ones”. Underneath, the value “8” has an arrow pointing to “Eight trillion”, the value “165” has an arrow pointing to “One hundred sixty-five billion”, the value “432” has an arrow pointing to “Four hundred thirty-two million”, the value “098” has an arrow pointing to “Ninety-eight thousand”, and the value “710” has an arrow pointing to “seven hundred ten”.

Putting all of the words together, we write 8,165,432,098,7108,165,432,098,710 as eight trillion, one hundred sixty-five billion, four hundred thirty-two million, ninety-eight thousand, seven hundred ten.

Note:

Exercise 7

Name each number in words: 9,258,137,904,0619,258,137,904,061

Solution

nine trillion, two hundred fifty-eight billion, one hundred thirty-seven million, nine hundred four thousand, sixty-one

Note:

Exercise 8

Name each number in words: 17,864,325,619,00417,864,325,619,004

Solution

seventeen trillion, eight hundred sixty-four billion, three hundred twenty-five million, six hundred nineteen thousand, four

Example 5

Problem 1

A student conducted research and found that the number of mobile phone users in the United States during one month in 20142014 was 327,577,529.327,577,529. Name that number in words.

Solution: Solution

Identify the periods associated with the number.
An image with three values separated by commas. The first value is “327” and has the label “millions”. The second value is “577” and has the label “thousands”. The third value is “529” and has the label “ones”.

Name the number in each period, followed by the period name. Put the commas in to separate the periods.

Millions period: three hundred twenty-seven million

Thousands period: five hundred seventy-seven thousand

Ones period: five hundred twenty-nine

So the number of mobile phone users in the Unites States during the month of April was three hundred twenty-seven million, five hundred seventy-seven thousand, five hundred twenty-nine.

Note:

Exercise 9

The population in a country is 316,128,839.316,128,839. Name that number.

Solution

three hundred sixteen million, one hundred twenty-eight thousand, eight hundred thirty nine

Note:

Exercise 10

One year is 31,536,00031,536,000 seconds. Name that number.

Solution

thirty one million, five hundred thirty-six thousand

Use Place Value to Write Whole Numbers

We will now reverse the process and write a number given in words as digits.

Note: Use place value to write a whole number.:

  1. Step 1. Identify the words that indicate periods. (Remember the ones period is never named.)
  2. Step 2. Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.
  3. Step 3. Name the number in each period and place the digits in the correct place value position.

Example 6

Problem 1

Write the following numbers using digits.

  • fifty-three million, four hundred one thousand, seven hundred forty-two
  • nine billion, two hundred forty-six million, seventy-three thousand, one hundred eighty-nine
Solution: Solution

Identify the words that indicate periods.

Except for the first period, all other periods must have three places. Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.

Then write the digits in each period.
An image with three blocks of text pointing to numerical values. The first block of text is “fifty-three million”, has the label “millions”, and points to value 53. The second block of text is “four hundred one thousand”, has the label “thousands”, and points to value 401. The third block of text is “seven hundred forty-two”, has the label “ones”, and points to value 742.

Put the numbers together, including the commas. The number is 53,401,742.53,401,742.

Identify the words that indicate periods.

Except for the first period, all other periods must have three places. Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.

Then write the digits in each period.
An image with four blocks of text pointing to numerical values. The first block of text is “nine billion”, has the label “billions”, and points to value 9. The second block of text is “two hundred forty-six million”, has the label “millions”, and points to value 246. The third block of text is “seventy-three thousand”, has the label “thousands”, and points to value 742. The fourth block of text is “one hundred eighty-nine”, has the label “ones”, and points to the value 189.

The number is 9,246,073,189.9,246,073,189.

Notice that in part , a zero was needed as a place-holder in the hundred thousands place. Be sure to write zeros as needed to make sure that each period, except possibly the first, has three places.

Note:

Exercise 11

Write each number in standard form:

fifty-three million, eight hundred nine thousand, fifty-one.

Solution

53,809,051

Note:

Exercise 12

Write each number in standard form:

two billion, twenty-two million, seven hundred fourteen thousand, four hundred sixty-six.

Solution

2,022,714,466

Example 7

Problem 1

A state budget was about $77$77 billion. Write the budget in standard form.

Solution: Solution

Identify the periods. In this case, only two digits are given and they are in the billions period. To write the entire number, write zeros for all of the other periods.
An image with four blocks of text pointing to numerical values. The first block of text is “77 billion”, has the label “billions”, and points to value “77”. The second block of text is null, has the label “millions”, and points to value “000”. The third block of text is null, has the label “thousands”, and points to value “000”. The fourth block of text is null, has the label “ones”, and points to the value “000”.

So the budget was about $77,000,000,000.$77,000,000,000.

Note:

Exercise 13

Write each number in standard form:

The closest distance from Earth to Mars is about 3434 million miles.

Solution

34,000,000 miles

Note:

Exercise 14

Write each number in standard form:

The total weight of an aircraft carrier is 204204 million pounds.

Solution

204,000,000 pounds

Round Whole Numbers

In 2013,2013, the U.S. Census Bureau reported the population of the state of New York as 19,651,12719,651,127 people. It might be enough to say that the population is approximately 2020 million. The word approximately means that 2020 million is not the exact population, but is close to the exact value.

The process of approximating a number is called rounding. Numbers are rounded to a specific place value depending on how much accuracy is needed. Saying that the population of New York is approximately 2020 million means we rounded to the millions place. The place value to which we round to depends on how we need to use the number.

Using the number line can help you visualize and understand the rounding process. Look at the number line in Figure 7. Suppose we want to round the number 7676 to the nearest ten. Is 7676 closer to 7070 or 8080 on the number line?

Figure 7: We can see that 7676 is closer to 8080 than to 70.70. So 7676 rounded to the nearest ten is 80.80.
An image of a number line from 70 to 80 with increments of one. All the numbers on the number line are black except for 70 and 80 which are red. There is an orange dot at the value “76” on the number line.

Now consider the number 72.72. Find 7272 in Figure 8.

Figure 8: We can see that 7272 is closer to 70,70, so 7272 rounded to the nearest ten is 70.70.
An image of a number line from 70 to 80 with increments of one. All the numbers on the number line are black except for 70 and 80 which are red. There is an orange dot at the value “72” on the number line.

How do we round 7575 to the nearest ten. Find 7575 in Figure 9.

Figure 9: The number 7575 is exactly midway between 7070 and 80.80.
An image of a number line from 70 to 80 with increments of one. All the numbers on the number line are black except for 70 and 80 which are red. There is an orange dot at the value “75” on the number line.

So that everyone rounds the same way in cases like this, mathematicians have agreed to round to the higher number, 80.80. So, 7575 rounded to the nearest ten is 80.80.

Now that we have looked at this process on the number line, we can introduce a more general procedure. To round a number to a specific place, look at the number to the right of that place. If the number is less than 5,5, round down. If it is greater than or equal to 5,5, round up.

So, for example, to round 7676 to the nearest ten, we look at the digit in the ones place.

An image of value “76”. The text “tens place” is in blue and points to number 7 in “76”. The text “is greater than 5” is in red and points to the number 6 in “76”.

The digit in the ones place is a 6.6. Because 66 is greater than or equal to 5,5, we increase the digit in the tens place by one. So the 77 in the tens place becomes an 8.8. Now, replace any digits to the right of the 88 with zeros. So, 7676 rounds to 80.80.

An image of the value “76”. The “6” in “76” is crossed out and has an arrow pointing to it which says “replace with 0”. The “7” has an arrow pointing to it that says “add 1”. Under the value “76” is the value “80”.

Let’s look again at rounding 7272 to the nearest 10.10. Again, we look to the ones place.

An image of value “72”. The text “tens place” is in blue and points to number 7 in “72”. The text “is less than 5” is in red and points to the number 2 in “72”.

The digit in the ones place is 2.2. Because 22 is less than 5,5, we keep the digit in the tens place the same and replace the digits to the right of it with zero. So 7272 rounded to the nearest ten is 70.70.

An image of the value “72”. The “2” in “72” is crossed out and has an arrow pointing to it which says “replace with 0”. The “7” has an arrow pointing to it that says “do not add 1”. Under the value “72” is the value “70”.

Note: Round a whole number to a specific place value.:

  1. Step 1. Locate the given place value. All digits to the left of that place value do not change.
  2. Step 2. Underline the digit to the right of the given place value.
  3. Step 3. Determine if this digit is greater than or equal to 5.5.
    • Yes—add 11 to the digit in the given place value.
    • No—do not change the digit in the given place value.
  4. Step 4. Replace all digits to the right of the given place value with zeros.

Example 8

Problem 1

Round 843843 to the nearest ten.

Solution: Solution
Table 4
Locate the tens place. The number 843 with the label “tens place” pointed at the 4 in 843.
Underline the digit to the right of the tens place. The number 843 with the 3 underlined.
Since 3 is less than 5, do not change the digit in the tens place. The number 843 with the 3 underlined.
Replace all digits to the right of the tens place with zeros. The number 840 with the 0 underlined.
  Rounding 843 to the nearest ten gives 840.

Note:

Exercise 15

Round to the nearest ten: 157.157.

Solution

160

Note:

Exercise 16

Round to the nearest ten: 884.884.

Solution

880

Example 9

Problem 1

Round each number to the nearest hundred:

  1. 23,65823,658
  2. 3,9783,978
Solution: Solution
 
Locate the hundreds place. ..
The digit of the right of the hundreds place is 5. Underline the digit to the right of the hundreds place. ..
Since 5 is greater than or equal to 5, round up by adding 1 to the digit in the hundreds place. Then replace all digits to the right of the hundreds place with zeros. ..
So 23,658 rounded to the nearest hundred is 23,700.
 
Locate the hundreds place. ..
Underline the digit to the right of the hundreds place. ..
The digit to the right of the hundreds place is 7. Since 7 is greater than or equal to 5, round up by added 1 to the 9. Then place all digits to the right of the hundreds place with zeros. ..
So 3,978 rounded to the nearest hundred is 4,000.

Note:

Exercise 17

Round to the nearest hundred: 17,852.17,852.

Solution

17,900

Note:

Exercise 18

Round to the nearest hundred: 4,951.4,951.

Solution

5,000

Example 10

Problem 1

Round each number to the nearest thousand:

  1. 147,032147,032
  2. 29,50429,504
Solution: Solution
 
Locate the thousands place. Underline the digit to the right of the thousands place. ..
The digit to the right of the thousands place is 0. Since 0 is less than 5, we do not change the digit in the thousands place. ..
We then replace all digits to the right of the thousands pace with zeros. ..
So 147,032 rounded to the nearest thousand is 147,000.
 
Locate the thousands place. ..
Underline the digit to the right of the thousands place. ..
The digit to the right of the thousands place is 5. Since 5 is greater than or equal to 5, round up by adding 1 to the 9. Then replace all digits to the right of the thousands place with zeros. ..
So 29,504 rounded to the nearest thousand is 30,000.

Notice that in part , when we add 11 thousand to the 99 thousands, the total is 1010 thousands. We regroup this as 11 ten thousand and 00 thousands. We add the 11 ten thousand to the 33 ten thousands and put a 00 in the thousands place.

Note:

Exercise 19

Round to the nearest thousand: 63,921.63,921.

Solution

64,000

Note:

Exercise 20

Round to the nearest thousand: 156,437.156,437.

Solution

156,000

Note: ACCESS ADDITIONAL ONLINE RESOURCES:

Key Concepts

Figure 10
A chart titled 'Place Value' with fifteen columns and 4 rows, with the columns broken down into five groups of three. The header row shows Trillions, Billions, Millions, Thousands, and Ones. The next row has the values 'Hundred trillions', 'Ten trillions', 'trillions', 'hundred billions', 'ten billions', 'billions', 'hundred millions', 'ten millions', 'millions', 'hundred thousands', 'ten thousands', 'thousands', 'hundreds', 'tens', and 'ones'. The first 8 values in the next row are blank. Starting with the ninth column, the values are '5', '2', '7', '8', '1', '9', and '4'.
  • Name a whole number in words.
    1. Step 1. Starting at the digit on the left, name the number in each period, followed by the period name. Do not include the period name for the ones.
    2. Step 2. Use commas in the number to separate the periods.
  • Use place value to write a whole number.
    1. Step 1. Identify the words that indicate periods. (Remember the ones period is never named.)
    2. Step 2. Draw three blanks to indicate the number of places needed in each period.
    3. Step 3. Name the number in each period and place the digits in the correct place value position.
  • Round a whole number to a specific place value.
    1. Step 1. Locate the given place value. All digits to the left of that place value do not change.
    2. Step 2. Underline the digit to the right of the given place value.
    3. Step 3. Determine if this digit is greater than or equal to 5. If yes—add 1 to the digit in the given place value. If no—do not change the digit in the given place value.
    4. Step 4. Replace all digits to the right of the given place value with zeros.

Practice Makes Perfect

Identify Counting Numbers and Whole Numbers

In the following exercises, determine which of the following numbers are counting numbers whole numbers.

Exercise 21

0,23,5,8.1,1250,23,5,8.1,125

Solution
  1. 0, 5, 125
  2. 0, 5, 125

Exercise 22

0,710,3,20.5,3000,710,3,20.5,300

Exercise 23

0,49,3.9,50,2210,49,3.9,50,221

Solution
  1. 50, 221
  2. 0, 50, 221

Exercise 24

0,35,10,303,422.60,35,10,303,422.6

Model Whole Numbers

In the following exercises, use place value notation to find the value of the number modeled by the base-10base-10 blocks.

Exercise 26

An image consisting of three items. The first item is three squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is eight horizontal rods containing 10 blocks each. The third item is 4 individual blocks.

Exercise 28

An image consisting of two items. The first item is six squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is 2 horizontal rods with 10 blocks each.

Identify the Place Value of a Digit

In the following exercises, find the place value of the given digits.

Exercise 29

579,601579,601

  1. 9
  2. 6
  3. 0
  4. 7
  5. 5
Solution
  1. thousands
  2. hundreds
  3. tens
  4. ten thousands
  5. hundred thousands

Exercise 30

398,127398,127

  1. 9
  2. 3
  3. 2
  4. 8
  5. 7

Exercise 31

56,804,37956,804,379

  1. 8
  2. 6
  3. 4
  4. 7
  5. 0
Solution
  1. hundred thousands
  2. millions
  3. thousands
  4. tens
  5. ten thousands

Exercise 32

78,320,46578,320,465

  1. 8
  2. 4
  3. 2
  4. 6
  5. 7

Use Place Value to Name Whole Numbers

In the following exercises, name each number in words.

Exercise 33

1,0781,078

Solution

One thousand, seventy-eight

Exercise 34

5,9025,902

Exercise 35

364,510364,510

Solution

Three hundred sixty-four thousand, five hundred ten

Exercise 36

146,023146,023

Exercise 37

5,846,1035,846,103

Solution

Five million, eight hundred forty-six thousand, one hundred three

Exercise 38

1,458,3981,458,398

Exercise 39

37,889,00537,889,005

Solution

Thirty seven million, eight hundred eighty-nine thousand, five

Exercise 40

62,008,46562,008,465

Exercise 41

The height of Mount Ranier is 14,41014,410 feet.

Solution

Fourteen thousand, four hundred ten

Exercise 42

The height of Mount Adams is 12,27612,276 feet.

Exercise 43

Seventy years is 613,200613,200 hours.

Solution

Six hundred thirteen thousand, two hundred

Exercise 44

One year is 525,600525,600 minutes.

Exercise 45

The U.S. Census estimate of the population of Miami-Dade county was 2,617,176.2,617,176.

Solution

Two million, six hundred seventeen thousand, one hundred seventy-six

Exercise 46

The population of Chicago was 2,718,782.2,718,782.

Exercise 47

There are projected to be 23,867,00023,867,000 college and university students in the US in five years.

Solution

Twenty three million, eight hundred sixty-seven thousand

Exercise 48

About twelve years ago there were 20,665,41520,665,415 registered automobiles in California.

Exercise 49

The population of China is expected to reach 1,377,583,1561,377,583,156 in 2016.2016.

Solution

One billion, three hundred seventy-seven million, five hundred eighty-three thousand, one hundred fifty-six

Exercise 50

The population of India is estimated at 1,267,401,8491,267,401,849 as of July 1,2014.1,2014.

Use Place Value to Write Whole Numbers

In the following exercises, write each number as a whole number using digits.

Exercise 51

four hundred twelve

Solution

412

Exercise 52

two hundred fifty-three

Exercise 53

thirty-five thousand, nine hundred seventy-five

Solution

35,975

Exercise 54

sixty-one thousand, four hundred fifteen

Exercise 55

eleven million, forty-four thousand, one hundred sixty-seven

Solution

11,044,167

Exercise 56

eighteen million, one hundred two thousand, seven hundred eighty-three

Exercise 57

three billion, two hundred twenty-six million, five hundred twelve thousand, seventeen

Solution

3,226,512,017

Exercise 58

eleven billion, four hundred seventy-one million, thirty-six thousand, one hundred six

Exercise 59

The population of the world was estimated to be seven billion, one hundred seventy-three million people.

Solution

7,173,000,000

Exercise 60

The age of the solar system is estimated to be four billion, five hundred sixty-eight million years.

Exercise 61

Lake Tahoe has a capacity of thirty-nine trillion gallons of water.

Solution

39,000,000,000,000

Exercise 62

The federal government budget was three trillion, five hundred billion dollars.

Round Whole Numbers

In the following exercises, round to the indicated place value.

Exercise 63

Round to the nearest ten:

  1. 386386
  2. 2,9312,931

Solution
  1. 390
  2. 2,930

Exercise 64

Round to the nearest ten:

  1. 792792
  2. 5,6475,647

Exercise 65

Round to the nearest hundred:

  1. 13,74813,748
  2. 391,794391,794

Solution
  1. 13,700
  2. 391,800

Exercise 66

Round to the nearest hundred:

  1. 28,16628,166
  2. 481,628481,628

Exercise 67

Round to the nearest ten:

  1. 1,4921,492
  2. 1,4971,497
Solution
  1. 1,490
  2. 1,500

Exercise 68

Round to the nearest thousand:

  1. 2,3912,391
  2. 2,7952,795

Exercise 69

Round to the nearest hundred:

  1. 63,99463,994
  2. 63,94963,949

Solution

  1. 64,00064,000
  2. 63,90063,900

Exercise 70

Round to the nearest thousand:

  1. 163,584163,584
  2. 163,246163,246

Everyday Math

Exercise 71

Writing a Check Jorge bought a car for $24,493.$24,493. He paid for the car with a check. Write the purchase price in words.

Solution

Twenty four thousand, four hundred ninety-three dollars

Exercise 72

Writing a Check Marissa’s kitchen remodeling cost $18,549.$18,549. She wrote a check to the contractor. Write the amount paid in words.

Exercise 73

Buying a Car Jorge bought a car for $24,493.$24,493. Round the price to the nearest:

  1. ten dollars
  2. hundred dollars
  3. thousand dollars
  4. ten-thousand dollars
Solution
  1. $24,490
  2. $24,500
  3. $24,000
  4. $20,000

Exercise 74

Remodeling a Kitchen Marissa’s kitchen remodeling cost $18,549.$18,549. Round the cost to the nearest:

  1. ten dollars
  2. hundred dollars
  3. thousand dollars
  4. ten-thousand dollars

Exercise 75

Population The population of China was 1,355,692,5441,355,692,544 in 2014.2014. Round the population to the nearest:

  1. billion people
  2. hundred-million people
  3. million people
Solution
  1. 1,000,000,0001,000,000,000
  2. 1,400,000,0001,400,000,000
  3. 1,356,000,0001,356,000,000

Exercise 76

Astronomy The average distance between Earth and the sun is 149,597,888149,597,888 kilometers. Round the distance to the nearest:

  1. hundred-million kilometers
  2. ten-million kilometers
  3. million kilometers

Writing Exercises

Exercise 77

In your own words, explain the difference between the counting numbers and the whole numbers.

Solution

Answers may vary. The whole numbers are the counting numbers with the inclusion of zero.

Exercise 78

Give an example from your everyday life where it helps to round numbers.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

If most of your checks were...

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

Glossary

coordinate:
A number paired with a point on a number line is called the coordinate of the point.
counting numbers:
The counting numbers are the numbers 1, 2, 3, ….
number line:
A number line is used to visualize numbers. The numbers on the number line get larger as they go from left to right, and smaller as they go from right to left.
origin:
The origin is the point labeled 0 on a number line.
place value system:
Our number system is called a place value system because the value of a digit depends on its position, or place, in a number.
rounding:
The process of approximating a number is called rounding.
whole numbers:
The whole numbers are the numbers 0, 1, 2, 3, ….

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