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: 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 and continue.

    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 is called the origin. The points are equally spaced to the right of and labeled with the counting numbers. When a number is paired with a point, it is called the coordinate of the point.

    The discovery of the number zero was a big step in the history of mathematics. Including zero with the counting numbers gives a new set of numbers called the whole numbers.

    Whole Numbers

    The whole numbers are the counting numbers and zero.

    We stopped at 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?

    Answer

    • ⓐ The counting numbers start at so is not a counting number. The numbers are all counting numbers.
    • ⓑ Whole numbers are counting numbers and The numbers are whole numbers.

    The numbers and 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?

    Try It 1.2

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

    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 has a different value than the number Even though they use the same digits, their value is different because of the different placement of the and the

    Money gives us a familiar model of place value. Suppose a wallet contains three

    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

    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 ones, and the hundreds square is made of tens, or 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 modeled with 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

    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
    hundreds
    tens
    ones

    Example 1.2

    Use place value notation to find the value of the number modeled by the 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 hundreds squares, which is

    There is tens rod, which is

    There are ones blocks, which is

    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
    hundreds
    tens
    ones

    The blocks model the number

    Try It 1.3

    Use place value notation to find the value of the number modeled by the 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 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 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 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 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 is in the millions place. Its value is
    • The digit is in the hundred thousands place. Its value is
    • The digit is in the ten thousands place. Its value is
    • The digit is in the thousands place. Its value is
    • The digit is in the hundreds place. Its value is
    • The digit is in the tens place. Its value is
    • The digit is in the ones place. Its value is

    Example 1.3

    In the number find the place value of each of the following digits:

    1. ⓐ
    2. ⓑ
    3. ⓒ
    4. ⓓ
    5. ⓔ
    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 is in the thousands place.
    • ⓑ The is in the ten thousands place.
    • ⓒ The is in the tens place.
    • ⓓ The is in the ten millions place.
    • ⓔ The is in the millions place.

    Try It 1.5

    For each number, find the place value of digits listed:

    1. ⓐ
    2. ⓑ
    3. ⓒ
    4. ⓓ
    5. ⓔ

    Try It 1.6

    For each number, find the place value of digits listed:

    1. ⓐ
    2. ⓑ
    3. ⓒ
    4. ⓓ
    5. ⓔ

    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 is closer to than to So rounded to the nearest ten is

    Now consider the number

    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 is closer to so rounded to the nearest ten is

    How do we round

    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 is exactly midway between and

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

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

    So, for example, to round 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 Because is greater than or equal to we increase the digit in the tens place by one. So the in the tens place becomes an Now, replace any digits to the right of the with zeros. So, rounds to

    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 to the nearest 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 Because is less than we keep the digit in the tens place the same and replace the digits to the right of it with zero. So rounded to the nearest ten is

    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 unless the digit immediately to the left is 9, in which case it may. (See Step 3.)
    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
      • Yes—add 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.
    4. Step 4. Replace all digits to the right of the given place value with zeros.

    Example 1.8

    Round 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:

    Try It 1.16

    Round to the nearest ten:

    Example 1.9

    Round each number to the nearest hundred:

    1. ⓐ
    2. ⓑ
    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:

    Try It 1.18

    Round to the nearest hundred:

    Example 1.10

    Round each number to the nearest thousand:

    1. ⓐ
    2. ⓑ
    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 thousand to the thousands, the total is thousands. We regroup this as ten thousand and thousands. We add the ten thousand to the ten thousands and put a in the thousands place.

    Try It 1.19

    Round to the nearest thousand:

    Try It 1.20

    Round to the nearest thousand:

    Media

    ACCESS ADDITIONAL ONLINE RESOURCES

    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.

    2.

    3.

    4.

    Model Whole Numbers

    In the following exercises, use place value notation to find the value of the number modeled by the 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.

    1. ⓐ 9
    2. ⓑ 6
    3. ⓒ 0
    4. ⓓ 7
    5. ⓔ 5
    10.

    1. ⓐ 9
    2. ⓑ 3
    3. ⓒ 2
    4. ⓓ 8
    5. ⓔ 7
    11.

    1. ⓐ 8
    2. ⓑ 6
    3. ⓒ 4
    4. ⓓ 7
    5. ⓔ 0
    12.

    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.

    14.

    15.

    16.

    17.

    18.

    19.

    20.

    21.

    The height of Mount Ranier is feet.

    22.

    The height of Mount Adams is feet.

    23.

    Seventy years is hours.

    24.

    One year is minutes.

    25.

    The U.S. Census estimate of the population of Miami-Dade county was

    26.

    The population of Chicago was

    27.

    There are projected to be college and university students in the US in five years.

    28.

    About twelve years ago there were registered automobiles in California.

    29.

    The population of China is expected to reach in

    30.

    The population of India is estimated at as of July

    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. ⓐ
    2. ⓑ
    44.

    Round to the nearest ten:

    1. ⓐ
    2. ⓑ
    45.

    Round to the nearest hundred:

    1. ⓐ
    2. ⓑ
    46.

    Round to the nearest hundred:

    1. ⓐ
    2. ⓑ
    47.

    Round to the nearest ten:

    1. ⓐ
    2. ⓑ
    48.

    Round to the nearest thousand:

    1. ⓐ
    2. ⓑ
    49.

    Round to the nearest hundred:

    1. ⓐ
    2. ⓑ
    50.

    Round to the nearest thousand:

    1. ⓐ
    2. ⓑ

    Everyday Math

    51.

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

    52.

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

    53.

    Buying a Car Jorge bought a car for 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 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 in 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 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.

    This page titled 1.2: Introduction to Whole Numbers was last modified on Sun, 28 May 2023 07:57:59 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax.