1.1 Introduction to 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: $1, 2, 3, 4, 5, and so on$ 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.

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”. — **Figure 1.2** The numbers on the number line increase from left to right, and decrease from right to left.

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.

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 $$100$ bills, seven $$10$ bills, and four $$1$ bills. The amounts are summarized in Figure 1.3. How much money is in the wallet?

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”. — **Figure 1.3**

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

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 1.4**

Figure 1.5 shows the number $138$ modeled with $base-10$ blocks.

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”. — **Figure 1.5** We use place value notation to show the value of the number $138 .$

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
$1$	hundreds	$1$	$100$	$100$
$3$	tens	$3$	$10$	$30$
$8$	ones	$8$	$1$	$plus 8$
				$Sum = 138$

Example 2

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

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	$2$	$100$	$200$
$1$	tens	$1$	$10$	$10$
$5$	ones	$5$	$1$	$plus 5$
				$215$

The $base-10$ blocks model the number $215 .$

Identify the Place Value of a Digit

By looking at money and $base-10$ 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 $base-10$ 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.

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'. — **Figure 1.6**

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 $4 .$

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”. — **Figure 1.7**

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 4

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

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 $8,165,432,098,710$ as eight trillion, one hundred sixty-five billion, four hundred thirty-two million, ninety-eight thousand, seven hundred ten.

Use Place Value to Write Whole Numbers

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

Example 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

ⓐ 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 $53,401,742.$

ⓑ 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 $9,246,073,189.$

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.

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 1.8. Suppose we want to round the number $76$ to the nearest ten. Is $76$ closer to $70$ or $80$ on the number line?

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. — **Figure 1.8** We can see that $76$ is closer to $80$ than to $70 .$ So $76$ rounded to the nearest ten is $80 .$

Now consider the number $72 .$ Find $72$ in Figure 1.9.

How do we round $75$ to the nearest ten. Find $75$ in Figure 1.10.

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.

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 $6 .$ 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 $80 .$

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 $72$ to the nearest $10 .$ 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 $2 .$ 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 $70 .$

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”.

Example 8

Round $843$ to the nearest ten.

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.

Example 9

Round each number to the nearest hundred:

ⓐ $23,658$
ⓑ $3,978$

ⓐ
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.

Example 10

Round each number to the nearest thousand:

ⓐ $147,032$
ⓑ $29,504$

ⓐ
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.

Key Concepts

Name a whole number in words.
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. Use commas in the number to separate the periods.
Use place value to write a whole number.
1. Identify the words that indicate periods. (Remember the ones period is never named.)
2. Draw three blanks to indicate the number of places needed in each period.
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. 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.).
2. Underline the digit to the right of the given place value.
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 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. If no—do not change the digit in the given place value.
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.

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

ⓐ 5, 125
ⓑ 0, 5, 125

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

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

ⓐ 50, 221
ⓑ 0, 50, 221

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

Model Whole Numbers

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

An image consisting of three items. The first item is five squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is six horizontal rods containing 10 blocks each. The third item is 1 individual block.

561

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.

An image consisting of two items. The first item is four squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is 7 individual blocks.

407

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.

$579,601$

ⓐ 9
ⓑ 6
ⓒ 0
ⓓ 7
ⓔ 5

ⓐ thousands
ⓑ hundreds
ⓒ tens
ⓓ ten thousands
ⓔ hundred thousands

$398,127$

ⓐ 9
ⓑ 3
ⓒ 2
ⓓ 8
ⓔ 7

$56,804,379$

ⓐ 8
ⓑ 6
ⓒ 4
ⓓ 7
ⓔ 0

ⓐ hundred thousands
ⓑ millions
ⓒ thousands
ⓓ tens
ⓔ ten thousands

$78,320,465$

ⓐ 8
ⓑ 4
ⓒ 2
ⓓ 6
ⓔ 7

Use Place Value to Name Whole Numbers

In the following exercises, name each number in words.

$1,078$

One thousand, seventy-eight

$5,902$

$364,510$

Three hundred sixty-four thousand, five hundred ten

$146,023$

$5,846,103$

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

$1,458,398$

$37,889,005$

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

$62,008,465$

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

Fourteen thousand, four hundred ten

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

Seventy years is $613,200$ hours.

Six hundred thirteen thousand, two hundred

One year is $525,600$ minutes.

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

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

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

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

Twenty three million, eight hundred sixty-seven thousand

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

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

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

The population of India is estimated at $1,267,401,849$ 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.

four hundred twelve

412

two hundred fifty-three

thirty-five thousand, nine hundred seventy-five

35,975

sixty-one thousand, four hundred fifteen

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

11,044,167

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

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

3,226,512,017

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.

7,173,000,000

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

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

39,000,000,000,000

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:

ⓐ $386$
ⓑ $2,931$

ⓐ 390
ⓑ 2,930

Round to the nearest ten:

ⓐ $792$
ⓑ $5,647$

Round to the nearest hundred:

ⓐ $13,748$
ⓑ $391,794$

ⓐ 13,700
ⓑ 391,800

Round to the nearest hundred:

ⓐ $28,166$
ⓑ $481,628$

Round to the nearest ten:

ⓐ $1,492$
ⓑ $1,497$

ⓐ 1,490
ⓑ 1,500

Round to the nearest thousand:

ⓐ $2,391$
ⓑ $2,795$

Round to the nearest hundred:

ⓐ $63,994$
ⓑ $63,949$

ⓐ $64,000$
ⓑ $63,900$

Round to the nearest thousand:

ⓐ $163,584$
ⓑ $163,246$

Everyday Math

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

Twenty four thousand, four hundred ninety-three dollars

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

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

ⓐ ten dollars
ⓑ hundred dollars
ⓓ ten-thousand dollars

ⓐ $24,490
ⓑ $24,500
ⓒ $24,000
ⓓ $20,000

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

ⓐ ten dollars
ⓑ hundred dollars
ⓓ ten-thousand dollars

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

ⓐ billion people
ⓑ hundred-million people

ⓐ $1,000,000,000$
ⓑ $1,400,000,000$

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

ⓐ hundred-million kilometers
ⓑ ten-million kilometers

Writing Exercises

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

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

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.

A self-assessment chart for students to rate their understanding of whole numbers, place value, and rounding skills with options: Confidently, With some help, or No-I don't get it!. — **Figure 1.12**

ⓑ 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.