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,…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.

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.

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 1.2.

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 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.

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 1.1

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

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

TRY IT 1.1

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

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

TRY IT 1.2

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

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

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 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 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.$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 1010 ones, and the hundreds square is made of 1010 tens, or 100100 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 138138 modeled with base-10base-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 138.138.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.

DigitPlace valueNumberValueTotal value
1111100100100100
333310103030
888811+8+8
Sum =138Sum =138

EXAMPLE 1.2

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.

TRY IT 1.3

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.

TRY IT 1.4

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.

MANIPULATIVE MATHEMATICS

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 1.6 shows how the number 5,278,1945,278,194 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 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 1.3

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
TRY IT 1.5

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
TRY IT 1.6

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

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.

HOW TO

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 1.4

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

TRY IT 1.7

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

TRY IT 1.8

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

EXAMPLE 1.5

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

TRY IT 1.9

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

TRY IT 1.10

One year is 31,536,00031,536,000 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.

  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 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
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 $77$77 billion. Write the budget in standard form.

TRY IT 1.13

Write each number in standard form:

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

TRY IT 1.14

Write each number in standard form:

The total weight of an aircraft carrier is 204204 million 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. 2020 million was achieved by rounding to the millions place. Had we rounded to the one hundred thousands place, we would have 19,700,00019,700,000 as a result. Had we rounded to the ten thousands place, we would have 19,650,00019,650,000

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 7676 to the nearest ten. Is 7676 closer to 7070 or 8080 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 1.7 We can see that 7676 is closer to 8080 than to 70.70. So 7676 rounded to the nearest ten is 80.80.

Now consider the number 72.72. Find 7272 in 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 7272 is closer to 70,70, so 7272 rounded to the nearest ten is 70.70.

How do we round 7575 to the nearest ten. Find 7575 in Figure 1.9.

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.

Figure 1.9 The number 7575 is exactly midway between 7070 and 80.80.

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

HOW TO

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 1.8

Round 843843 to the nearest ten.

TRY IT 1.15

Round to the nearest ten: 157.157.

TRY IT 1.16

Round to the nearest ten: 884.884.

EXAMPLE 1.9

Round each number to the nearest hundred:

  1. ⓐ 23,65823,658
  2. ⓑ 3,9783,978
TRY IT 1.17

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

TRY IT 1.18

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

EXAMPLE 1.10

Round each number to the nearest thousand:

  1. ⓐ 147,032147,032
  2. ⓑ 29,50429,504
TRY IT 1.19

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

TRY IT 1.20

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

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.

1.

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

2.

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

3.

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

4.

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.

5.

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.

6.

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.

7.

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.

8.

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.

9.

579,601579,601

  1. ⓐ 9
  2. ⓑ 6
  3. ⓒ 0
  4. ⓓ 7
  5. ⓔ 5

10.

398,127398,127

  1. ⓐ 9
  2. ⓑ 3
  3. ⓒ 2
  4. ⓓ 8
  5. ⓔ 7

11.

56,804,37956,804,379

  1. ⓐ 8
  2. ⓑ 6
  3. ⓒ 4
  4. ⓓ 7
  5. ⓔ 0

12.

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.

13.

1,0781,078

14.

5,9025,902

15.

364,510364,510

16.

146,023146,023

17.

5,846,1035,846,103

18.

1,458,3981,458,398

19.

37,889,00537,889,005

20.

62,008,46562,008,465

21.

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

22.

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

23.

Seventy years is 613,200613,200 hours.

24.

One year is 525,600525,600 minutes.

25.

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

26.

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

27.

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

28.

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

29.

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

30.

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.

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:

  1. ⓐ 386386
  2. ⓑ 2,9312,931

44.

Round to the nearest ten:

  1. ⓐ 792792
  2. ⓑ 5,6475,647

45.

Round to the nearest hundred:

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

46.

Round to the nearest hundred:

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

47.

Round to the nearest ten:

  1. ⓐ 1,4921,492
  2. ⓑ 1,4971,497

48.

Round to the nearest thousand:

  1. ⓐ 2,3912,391
  2. ⓑ 2,7952,795

49.

Round to the nearest hundred:

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

50.

Round to the nearest thousand:

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

Everyday Math

51.

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.

52.

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.

53.

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

54.

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

55.

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

56.

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

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.