1.2: Introduction to Whole Numbers

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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.

$1, 2, 3, 4, 5…$

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.

Whole Numbers

The whole numbers are the counting numbers and zero.

$0, 1, 2, 3, 4, 5…$

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?

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

Answer

ⓐ The counting numbers start at $1,$ so $0$ is not a counting number. The numbers $3, 15, and 105$ are all counting numbers.
ⓑ Whole numbers are counting numbers and $0 .$ The numbers $0, 3, 15, and 105$ 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 $$100 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	$8$	$1$	$+ 8$
				$Sum = 138$

Example 1.2

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

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

Answer

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

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

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

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

Digit	Place value	Number	Value	Total value
$2$	hundreds	$2$	$100$	$200$
$1$	tens	$1$	$10$	$10$
$5$	ones	$5$	$1$	$+ 5$
				$215$

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

Try It 1.3

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

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

Try It 1.4

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

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

Manipulative Mathematics

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

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 $4 .$

Example 1.3

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

ⓐ $7$
ⓑ $0$
ⓒ $1$
ⓓ $6$
ⓔ $3$

Answer

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 $7$ 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: $27,493,615$

ⓐ $2$
ⓑ $1$
ⓒ $4$
ⓓ $7$
ⓔ $5$

Try It 1.6

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

ⓐ $9$
ⓑ $4$
ⓒ $2$
ⓓ $6$
ⓔ $7$

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.

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 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 $72 . Figure 1.8.$

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. — Figure 1.8 We can see that $72$ is closer to $70,$ so $72$ rounded to the nearest ten is $70 .$

How do we round $75 Figure 1.9.$

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.

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

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 $70 .$

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

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 digit immediately to the left is 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.

Answer

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.

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:

ⓐ $23,658$
ⓑ $3,978$

Answer

ⓐ
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 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.

Try It 1.17

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

Try It 1.18

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

Example 1.10

Round each number to the nearest thousand:

ⓐ $147,032$
ⓑ $29,504$

Answer

ⓐ
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: $63,921 .$

Try It 1.20

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

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Section 1.1 Exercises

Practice Makes Perfect

Identify Counting Numbers and Whole Numbers

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

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

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

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

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

Model Whole Numbers

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

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

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

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

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

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

10.

$398,127$

ⓐ 9
ⓑ 3
ⓒ 2
ⓓ 8
ⓔ 7

11.

$56,804,379$

ⓐ 8
ⓑ 6
ⓒ 4
ⓓ 7
ⓔ 0

12.

$78,320,465$

ⓐ 8
ⓑ 4
ⓒ 2
ⓓ 6
ⓔ 7

Use Place Value to Name Whole Numbers

In the following exercises, name each number in words.

13.

$1,078$

14.

$5,902$

15.

$364,510$

16.

$146,023$

17.

$5,846,103$

18.

$1,458,398$

19.

$37,889,005$

20.

$62,008,465$

21.

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

22.

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

23.

Seventy years is $613,200$ hours.

24.

One year is $525,600$ minutes.

25.

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

26.

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

27.

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

28.

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

29.

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

30.

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.

31.

four hundred twelve

32.

two hundred fifty-three

33.

thirty-five thousand, nine hundred seventy-five

34.

sixty-one thousand, four hundred fifteen

35.

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

36.

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

37.

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

38.

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

39.

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

40.

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

41.

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

42.

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

Round Whole Numbers

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

43.

Round to the nearest ten:

ⓐ $386$
ⓑ $2,931$

44.

Round to the nearest ten:

ⓐ $792$
ⓑ $5,647$

45.

Round to the nearest hundred:

ⓐ $13,748$
ⓑ $391,794$

46.

Round to the nearest hundred:

ⓐ $28,166$
ⓑ $481,628$

47.

Round to the nearest ten:

ⓐ $1,492$
ⓑ $1,497$

48.

Round to the nearest thousand:

ⓐ $2,391$
ⓑ $2,795$

49.

Round to the nearest hundred:

ⓐ $63,994$
ⓑ $63,949$

50.

Round to the nearest thousand:

ⓐ $163,584$
ⓑ $163,246$

Everyday Math

51.

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

52.

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

53.

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

ⓐ ten dollars
ⓑ hundred dollars
ⓓ ten-thousand dollars

54.

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

ⓐ ten dollars
ⓑ hundred dollars
ⓓ ten-thousand dollars

55.

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

ⓐ billion people
ⓑ hundred-million people

56.

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

57.

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

58.

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.

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Counting Numbers

Whole Numbers

Example 1.1

Try It 1.1

Try It 1.2

Example 1.2

Try It 1.3

Try It 1.4

Manipulative Mathematics

Identify the Place Value of a Digit

Example 1.3

Try It 1.5

Try It 1.6

Use Place Value to Name Whole Numbers

How To

Round a whole number to a specific place value.

Example 1.8

Try It 1.15

Try It 1.16

Example 1.9

Try It 1.17

Try It 1.18

Example 1.10

Try It 1.19

Try It 1.20

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Section 1.1 Exercises

Practice Makes Perfect

Everyday Math

Writing Exercises

Self Check