1.1 Introduction to Whole Numbers

Learning objectives

Learning objectives

• 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: $1,2,3,4,5,\ldots$ 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 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.

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

Solution

1. Define both sets. Counting numbers are $1,2,3$ and onward, while whole numbers are $0,1,2,3$ and onward.
2. Test zero. Zero is not a counting number because counting numbers begin at $1$, but zero is a whole number.
3. Test the positive integers. The values $3,15,$ and $105$ are positive integers, so each one is both a counting number and a whole number.
4. Reject the non-integers. The fraction $\frac{1}{4}$ and the decimal $5.2$ are not counting numbers and are not whole numbers.
5. Report both sets. The counting numbers are $3,15,$ and $105$, while the whole numbers are $0,3,15,$ and $105.

Takeaway: Counting numbers exclude zero, whereas whole numbers include zero along with every counting number.

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 $\text{\textdollar 100}$ bills, seven $\text{\textdollar 10}$ bills, and four $\text{\textdollar 1}$ 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 $\text{\textdollar 374}.$

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 $\mathrm{base-10}$ blocks.

Example 1.2

Solution

1. Count the hundreds squares. There are $2$ hundreds squares, and each square represents $100$, so their combined value is $2(100)=200$.
2. Count the tens rods. There is $1$ tens rod, and each rod represents $10$, so this block contributes $1(10)=10$.
3. Count the ones blocks. There are $5$ ones blocks, and each block represents $1$, so their combined value is $5(1)=5$.
4. Combine all place values. Add the hundreds, tens, and ones contributions as $200+10+5$, which equals $215$.
5. State the modeled number. The base-ten blocks therefore model the whole number $215$.

Takeaway: The value of a base-ten model is the sum of its hundreds, tens, and ones contributions, giving $215$ here.

Identify the Place Value of a Digit

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

• The digit $5$ is in the millions place. Its value is $5,000,000.$

• The digit $2$ is in the hundred thousands place. Its value is $200,000.$

• The digit $7$ is in the ten thousands place. Its value is $70,000.$

• The digit $8$ is in the thousands place. Its value is $8,000.$

• 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

Solution

1. Place the number in a chart. Write $63,407,218$ so that the rightmost digit $8$ occupies the ones place, and group the remaining digits into the millions, thousands, and ones periods.
2. Locate the digit $7$. The $7$ is the rightmost digit of the thousands period, so its place value is thousands.
3. Locate the digit $0$. The $0$ is the rightmost digit of the ten thousands period, so its place value is ten thousands even though its value is zero.
4. Locate the digit $1$. The $1$ is in the tens place of the ones period, so its place value is tens.
5. Locate the remaining digits. The $6$ is in the ten millions place and the $3$ is in the millions place, completing all five requested place values.

Takeaway: Counting positions from the rightmost ones place gives the place values of $7$, $0$, $1$, $6$, and $3$ in $63,407,218$.

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 $37,519,248$ 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.

Example 1.4

Solution

1. Separate the number into periods. Group the digits as $8$, $165$, $432$, $098$, and $710$, working from the trillions period on the left to the ones period on the right.
2. Name the trillions period. The leftmost group $8$ is read as eight trillion.
3. Name the billions period. The next group $165$ is read as one hundred sixty-five billion.
4. Name the remaining periods. The groups $432$, $098$, and $710$ are four hundred thirty-two million, ninety-eight thousand, and seven hundred ten, respectively.
5. Combine the period names. Join the five names in order to obtain eight trillion, one hundred sixty-five billion, four hundred thirty-two million, ninety-eight thousand, seven hundred ten.

Takeaway: Naming each three-digit period from trillions to ones produces the complete words for $8,165,432,098,710$.

Example 1.5

Solution

1. Identify the periods in the number. Separate $327,577,529$ into the millions period $327$, the thousands period $577$, and the ones period $529$.
2. Name the millions period. Read $327$ as three hundred twenty-seven and append million, giving three hundred twenty-seven million.
3. Name the thousands period. Read $577$ as five hundred seventy-seven and append thousand, giving five hundred seventy-seven thousand.
4. Name the ones period. Read $529$ as five hundred twenty-nine, with no additional period name.
5. Combine the period names. Join the three parts in order to state three hundred twenty-seven million, five hundred seventy-seven thousand, five hundred twenty-nine.

Takeaway: The number $327,577,529$ is named by reading its millions, thousands, and ones periods from left to right.

Use Place Value to Write Whole Numbers

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

Example 1.6

Solution

1. Identify the periods in part a. The words in part a specify a millions group, a thousands group, and an ones group: $53$, $401$, and $742$.
2. Write part a in standard form. Place those groups in order and include a comma between periods, producing $53,401,742$.
3. Identify the periods in part b. The words in part b specify a billions group, a millions group, a thousands group, and an ones group: $9$, $246$, $73$, and $189$.
4. Fill the missing place in part b. Pad the thousands group $73$ with a zero on the left to make $073$, because every period after the first has three places.
5. Write part b in standard form. Combine the four groups as $9$, $246$, $073$, and $189$, producing $9,246,073,189$.
6. Compare the two results. Both standard-form numbers follow the three-place rule, and part b includes the required placeholder zero in the hundred-thousands place.

Takeaway: Converting period names to digits requires three places in every period after the first, including placeholder zeros.

Example 1.7

Solution

1. Identify the magnitude of the budget. The phrase $77$ billion shows that the two given digits belong in the billions period.
2. Keep the billions digits. Write the leading part of the number as $77$ because $77$ billion equals $77$ groups of one billion.
3. Fill the lower periods with zeros. The millions, thousands, and ones periods must each contain three zero places, producing $000$, $000$, and $000$ after the billions period.
4. Assemble the standard form. Combining the billions digits with the three lower periods gives $77,000,000,000$ dollars.

Takeaway: Writing $77$ billion in standard form requires the three lower periods to be zeros, giving $77,000,000,000$.

Round Whole Numbers

In $2013,$ the U.S. Census Bureau reported the population of the state of New York as $19,651,127$ 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 $19,700,000$ as a result. Had we rounded to the ten thousands place, we would have $19,650,000$ 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 $70.$

Example 1.8

Solution

1. Identify the tens place. In $843$, the tens digit is $4$, as indicated by the diagram labeled “tens place.”
2. Compare the ones digit with five. The ones digit is $3$, which is less than $5$, so the tens digit stays $4$; the diagram underlines this $3$ to show the deciding digit.
3. Round to the nearest ten. Replace the ones digit with $0$, giving $840$, which is also the value shown in the final diagram.

Takeaway: When rounding to the nearest ten, look only at the ones digit and leave all digits to its left unchanged.

Example 1.9

Solution

1. Round $23,658$ to the nearest hundred. The hundreds digit is $6$, and the next digit is $5$, so increase $6$ to $7$ and replace the tens and ones with zeros, obtaining $23,700$. The diagrams show the $6$, the underlined $5$, and the replacement of $58$ by $00$.
2. Round $3,978$ to the nearest hundred. The hundreds digit is $9$, and the next digit is $7$, so increase $9$ to $10$. This carries $1$ into the thousands place, changing $3$ to $4$.
3. Finish the second rounding. Replace the hundreds, tens, and ones with zeros, giving $4,000$, which agrees with the final diagram.

Takeaway: A rounding digit of $5$ or more rounds upward, and carrying past a $9$ can increase the place to its left.

Example 1.10

Solution

1. Round $147,032$ to the nearest thousand. The thousands digit is $7$, while the hundreds digit is $0$. Because $0<5$, the thousands digit stays $7$, and the last three digits become zeros, giving $147,000$. The diagrams highlight the $7$ and the underlined $0$.
2. Round $29,504$ to the nearest thousand. The thousands digit is $9$, and the hundreds digit is $5$, so the rounding digit requires an increase. Adding $1$ to $9$ thousands produces $10$ thousands, which regroup as $1$ ten thousand and $0$ thousands.
3. Complete the regrouping. Increase the $2$ ten thousands to $3$ and replace the final three digits with zeros. The result shown in the final diagram is $30,000$.

Takeaway: When the rounding digit is $5$ or greater, increase the target place and replace all less significant digits with zeros.

Section 1.1 Exercises

Section 1.1 Exercises

Identify Counting Numbers and Whole Numbers

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

Model Whole Numbers

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

Identify the Place Value of a Digit

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

Use Place Value to Name Whole Numbers

In the following exercises, name each number in words.

Use Place Value to Write Whole Numbers

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

Round Whole Numbers

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

Everyday Math

Writing Exercises

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.

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

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

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

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

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

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

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

3. For each number, find the place value of digits listed: $27,493,615$

1. ⓐ $2$

2. ⓑ $1$

3. ⓒ $4$

4. ⓓ $7$

5. ⓔ $5$

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

1. ⓐ $9$

2. ⓑ $4$

3. ⓒ $2$

4. ⓓ $6$

5. ⓔ $7$

2. Name each number in words: $9,258,137,904,061$

3. Name each number in words: $17,864,325,619,004$

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

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

3. Write each number in standard form:

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

1. Write each number in standard form:

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

2. Write each number in standard form:

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

3. Write each number in standard form:

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

1. Round to the nearest hundred: $17,852.$

2. Round to the nearest hundred: $4,951.$

3. Round to the nearest thousand: $63,921.$

1. Round to the nearest thousand: $156,437.$

2. $0,\frac{2}{3},5,8.1,125$

3. $0,\frac{4}{9},3.9,50,221$

1. $579,601$

1. ⓐ 9

2. ⓑ 6

3. ⓒ 0

4. ⓓ 7

5. ⓔ 5

2. $56,804,379$

1. ⓐ 8

2. ⓑ 6

3. ⓒ 4

4. ⓓ 7

5. ⓔ 0

3. $1,078$

1. $364,510$

2. $5,846,103$

3. $37,889,005$

1. The height of Mount Rainier is $14,410$ feet.

2. Seventy years is $613,200$ hours.

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

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

2. The population of China is expected to reach $1,377,583,156$ in $2016.$

3. thirty-five thousand, nine hundred seventy-five

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

2. Round to the nearest ten:

1. ⓐ $386$

2. ⓑ $2,931$

3. Round to the nearest hundred:

1. ⓐ $13,748$

2. ⓑ $391,794$

1. Round to the nearest ten:

1. ⓐ $1,492$

2. ⓑ $1,497$

2. Round to the nearest hundred:

1. ⓐ $63,994$

2. ⓑ $63,949$

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

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

1. ⓐ ten dollars

2. ⓑ hundred dollars

3. ⓒ thousand dollars

4. ⓓ ten-thousand dollars

2. Population The population of China was $1,355,692,544$ in $2014.$ Round the population to the nearest:

1. ⓐ billion people

2. ⓑ hundred-million people

3. ⓒ million people

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