## Learning Objectives

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

## 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,\dots $ 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.

## Counting Numbers

The counting numbers start with $1$ and continue.

Counting numbers and whole numbers can be visualized on a number line as shown in Figure 1.2.

The point labeled $0$ is called the origin. The points are equally spaced to the right of $0$ 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.

## Whole Numbers

The whole numbers are the counting numbers and zero.

We stopped at $5$ 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 1.1

Which of the following are ⓐ counting numbers? ⓑ whole numbers?

$0,\frac{1}{4},3,5.2,15,105$

### Solution

- ⓐ The counting numbers start at $1,$ so $0$ is not a counting number. The numbers $3,15,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}105$ are all counting numbers.
- ⓑ Whole numbers are counting numbers and $0.$ The numbers $0,3,15,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}105$ are whole numbers.

The numbers $\frac{1}{4}$ and $5.2$ are neither counting numbers nor whole numbers. We will discuss these numbers later.

## Try It 1.1

Which of the following are ⓐ counting numbers ⓑ whole numbers?

$0,\frac{2}{3},2,9,11.8,241,376$

## Try It 1.2

Which of the following are ⓐ counting numbers ⓑ whole numbers?

$0,\frac{5}{3},7,8.8,13,201$

## 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 $537$ has a different value than the number $735.$ Even though they use the same digits, their value is different because of the different placement of the $7$ and the $5.$

Money gives us a familiar model of place value. Suppose a wallet contains three $\text{\$100}$ bills, seven $\text{\$10}$ bills, and four $\text{\$1}$ bills. The amounts are summarized in Figure 1.3. How much money is in the wallet?

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

Base-10 blocks provide another way to model place value, as shown in Figure 1.4. The blocks can be used to represent hundreds, tens, and ones. Notice that the tens rod is made up of $10$ ones, and the hundreds square is made of $10$ tens, or $100$ ones.

Figure 1.5 shows the number $138$ modeled with $\text{base-10}$ blocks.

Digit | Place value | Number | Value | Total value |
---|---|---|---|---|

$1$ | hundreds | $1$ | $100$ | $100\phantom{\rule{1em}{0ex}}$ |

$3$ | tens | $3$ | $10$ | $30\phantom{\rule{1em}{0ex}}$ |

$8$ | ones | $8$ | $1$ | $+\phantom{\rule{0.5em}{0ex}}8\phantom{\rule{1em}{0ex}}$ |

$\text{Sum =}138\phantom{\rule{1em}{0ex}}$ |

## Example 1.2

Use place value notation to find the value of the number modeled by the $\text{base-10}$ blocks shown.

### Solution

There are $2$ hundreds squares, which is $200.$

There is $1$ tens rod, which is $10.$

There are $5$ ones blocks, which is $5.$

Digit | Place value | Number | Value | Total value |
---|---|---|---|---|

$2$ | hundreds | $2$ | $100$ | $200\phantom{\rule{1em}{0ex}}$ |

$1$ | tens | $1$ | $10$ | $10\phantom{\rule{1em}{0ex}}$ |

$5$ | ones | $5$ | $1$ | $+\phantom{\rule{0.5em}{0ex}}5\phantom{\rule{1em}{0ex}}$ |

$215\phantom{\rule{1em}{0ex}}$ |

The $\text{base-10}$ blocks model the number $215.$

## Try It 1.3

Use place value notation to find the value of the number modeled by the $\text{base-10}$ blocks shown.

## Try It 1.4

Use place value notation to find the value of the number modeled by the $\text{base-10}$ blocks shown.

## Manipulative Mathematics

## Identify the Place Value of a Digit

By looking at money and $\text{base-10}$ 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 $\text{base-10}$ 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 1.6 shows how the number $\mathrm{5,278,194}$ is written in a place value chart.

- The digit $5$ is in the millions place. Its value is $\mathrm{5,000,000}.$
- The digit $2$ is in the hundred thousands place. Its value is $\mathrm{200,000}.$
- The digit $7$ is in the ten thousands place. Its value is $\mathrm{70,000}.$
- The digit $8$ is in the thousands place. Its value is $\mathrm{8,000}.$
- The digit $1$ is in the hundreds place. Its value is $100.$
- The digit $9$ is in the tens place. Its value is $90.$
- The digit $4$ is in the ones place. Its value is $4.$

## Example 1.3

In the number $\mathrm{63,407,218};$ find the place value of each of the following digits:

- ⓐ $7$
- ⓑ $0$
- ⓒ $1$
- ⓓ $6$
- ⓔ $3$

### Solution

Write the number in a place value chart, starting at the right.

- ⓐ The $7$ is in the thousands place.
- ⓑ The $0$ is in the ten thousands place.
- ⓒ The $1$ is in the tens place.
- ⓓ The $6$ is in the ten millions place.
- ⓔ The $3$ is in the millions place.

## Try It 1.5

For each number, find the place value of digits listed: $\mathrm{27,493,615}$

- ⓐ $2$
- ⓑ $1$
- ⓒ $4$
- ⓓ $7$
- ⓔ $5$

## Try It 1.6

For each number, find the place value of digits listed: $\mathrm{519,711,641,328}$

- ⓐ $9$
- ⓑ $4$
- ⓒ $2$
- ⓓ $6$
- ⓔ $7$

## 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.

So the number $\mathrm{37,519,248}$ 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.

## How To

### Name a whole number in words.

- 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.
- Step 2. Use commas in the number to separate the periods.

## Example 1.4

Name the number $\mathrm{8,165,432,098,710}$ in words.

### Solution

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 |

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

## Try It 1.7

Name each number in words: $\mathrm{9,258,137,904,061}$

## Try It 1.8

Name each number in words: $\mathrm{17,864,325,619,004}$

## Example 1.5

A student conducted research and found that the number of mobile phone users in the United States during one month in $2014$ was $\mathrm{327,577,529}.$ Name that number in words.

### Solution

Identify the periods associated with the number.

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.

## Try It 1.9

The population in a country is $\mathrm{316,128,839}.$ Name that number.

## Try It 1.10

One year is $\mathrm{31,536,000}$ seconds. Name that number.

## Use Place Value to Write Whole Numbers

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

## How To

### Use place value to write a whole number.

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

## Example 1.6

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

ⓐ 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.

Put the numbers together, including the commas. The number is $\mathrm{53,401,742}.$

ⓑ 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.

The number is $\mathrm{9,246,073,189}.$

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.

## Try It 1.11

Write each number in standard form:

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

## Try It 1.12

Write each number in standard form:

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

## Example 1.7

A state budget was about $\text{\$77}$ billion. Write the budget in standard form.

### 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.

So the budget was about $\text{\$77,000,000,000.}$

## Try It 1.13

Write each number in standard form:

The closest distance from Earth to Mars is about $34$ million miles.

## Try It 1.14

Write each number in standard form:

The total weight of an aircraft carrier is $204$ million pounds.

## Round Whole Numbers

In $2013,$ the U.S. Census Bureau reported the population of the state of New York as $\mathrm{19,651,127}$ people. It might be enough to say that the population is approximately $20$ million. The word *approximately* means that $20$ 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. $20$ million was achieved by rounding to the millions place. Had we rounded to the one hundred thousands place, we would have $\mathrm{19,700,000}$ as a result. Had we rounded to the ten thousands place, we would have $\mathrm{19,650,000}$ as a result, and so on. 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 1.7. Suppose we want to round the number $76$ to the nearest ten. Is $76$ closer to $70$ or $80$ on the number line?

Now consider the number $72.$ Find $72$ in Figure 1.8.

How do we round $75$ to the nearest ten. Find $75$ in Figure 1.9.

So that everyone rounds the same way in cases like this, mathematicians have agreed to round to the higher number, $80.$ So, $75$ rounded to the nearest ten is $80.$

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 $5,$ round down. If it is greater than or equal to $5,$ round up.

So, for example, to round $76$ to the nearest ten, we look at the digit in the ones place.

The digit in the ones place is a $6.$ Because $6$ is greater than or equal to $5,$ we increase the digit in the tens place by one. So the $7$ in the tens place becomes an $8.$ Now, replace any digits to the right of the $8$ with zeros. So, $76$ rounds to $80.$

Let’s look again at rounding $72$ to the nearest $10.$ Again, we look to the ones place.

The digit in the ones place is $2.$ Because $2$ is less than $5,$ we keep the digit in the tens place the same and replace the digits to the right of it with zero. So $72$ rounded to the nearest ten is $70.$

## How To

### Round a whole number to a specific place value.

- Step 1. Locate the given place value. All digits to the left of that place value do not change unless the given place value is a 9, in which case it may. (See Step 3.)
- Step 2. Underline the digit to the right of the given place value.
- Step 3.
Determine if this digit is greater than or equal to $5.$
- Yes—add $1$ to the digit in the given place value. If that digit is 9, replace it with 0 and add 1 to the digit immediately to its left. If that digit is also a 9, repeat.
- No—do not change the digit in the given place value.

- Step 4. Replace all digits to the right of the given place value with zeros.

## Example 1.8

Round $843$ to the nearest ten.

### Solution

Locate the tens place. | |

Underline the digit to the right of the tens place. | |

Since 3 is less than 5, do not change the digit in the tens place. | |

Replace all digits to the right of the tens place with zeros. | |

Rounding 843 to the nearest ten gives 840. |

## Try It 1.15

Round to the nearest ten: $157.$

## Try It 1.16

Round to the nearest ten: $884.$

## Example 1.9

Round each number to the nearest hundred:

- ⓐ $\mathrm{23,658}$
- ⓑ $\mathrm{3,978}$

### Solution

ⓐ | |

Locate the hundreds place. | |

The digit to 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 adding 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. |

## Try It 1.17

Round to the nearest hundred: $\mathrm{17,852}.$

## Try It 1.18

Round to the nearest hundred: $\mathrm{4,951}.$

## Example 1.10

Round each number to the nearest thousand:

- ⓐ $\mathrm{147,032}$
- ⓑ $\mathrm{29,504}$

### 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 $1$ thousand to the $9$ thousands, the total is $10$ thousands. We regroup this as $1$ ten thousand and $0$ thousands. We add the $1$ ten thousand to the $2$ ten thousands and put a $0$ in the thousands place.

## Try It 1.19

Round to the nearest thousand: $\mathrm{63,921}.$

## Try It 1.20

Round to the nearest thousand: $\mathrm{156,437}.$

## Media

### ACCESS ADDITIONAL ONLINE RESOURCES

### Practice Makes Perfect

**Identify Counting Numbers and Whole Numbers**

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

$0,\frac{7}{10},3,20.5,300$

$0,\frac{3}{5},10,303,422.6$

**Model Whole Numbers**

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

**Identify the Place Value of a Digit**

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

$\mathrm{398,127}$

- ⓐ 9
- ⓑ 3
- ⓒ 2
- ⓓ 8
- ⓔ 7

$\mathrm{78,320,465}$

- ⓐ 8
- ⓑ 4
- ⓒ 2
- ⓓ 6
- ⓔ 7

**Use Place Value to Name Whole Numbers**

In the following exercises, name each number in words.

$\mathrm{5,902}$

$\mathrm{146,023}$

$\mathrm{1,458,398}$

$\mathrm{62,008,465}$

The height of Mount Adams is $\mathrm{12,276}$ feet.

One year is $\mathrm{525,600}$ minutes.

The population of Chicago was $\mathrm{2,718,782}.$

There are projected to be $\mathrm{23,867,000}$ college and university students in the US in five years.

About twelve years ago there were $\mathrm{20,665,415}$ registered automobiles in California.

The population of India is estimated at $\mathrm{1,267,401,849}$ as of July $1,2014.$

**Use Place Value to Write Whole Numbers**

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

two hundred fifty-three

sixty-one thousand, four hundred fifteen

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

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

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

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

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

**Round Whole Numbers**

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

Round to the nearest ten:

- ⓐ $792$
- ⓑ $\mathrm{5,647}$

Round to the nearest hundred:

- ⓐ $\mathrm{28,166}$
- ⓑ $\mathrm{481,628}$

Round to the nearest thousand:

- ⓐ $\mathrm{2,391}$
- ⓑ $\mathrm{2,795}$

Round to the nearest thousand:

- ⓐ $\mathrm{163,584}$
- ⓑ $\mathrm{163,246}$

### Everyday Math

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

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

**Buying a Car** Jorge bought a car for $\text{\$24,493}.$ Round the price to the nearest:

- ⓐ ten dollars
- ⓑ hundred dollars
- ⓒ thousand dollars
- ⓓ ten-thousand dollars

**Remodeling a Kitchen** Marissa’s kitchen remodeling cost $\text{\$18,549}.$ Round the cost to the nearest:

- ⓐ ten dollars
- ⓑ hundred dollars
- ⓒ thousand dollars
- ⓓ ten-thousand dollars

**Population** The population of China was $\mathrm{1,355,692,544}$ in $2014.$ Round the population to the nearest:

- ⓐ billion people
- ⓑ hundred-million people
- ⓒ million people

**Astronomy** The average distance between Earth and the sun is $\mathrm{149,597,888}$ kilometers. Round the distance to the nearest:

- ⓐ hundred-million kilometers
- ⓑ ten-million kilometers
- ⓒ million kilometers

### Writing Exercises

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. Whom 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.