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.
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?
Which of the following are ⓐ counting numbers ⓑ whole numbers?
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 bills, seven bills, and four bills. The amounts are summarized in Figure 1.3. How much money is in the wallet?
Find the total value of each kind of bill, and then add to find the total. The wallet contains
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.
Figure 1.5 shows the number modeled with blocks.
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.
Use place value notation to find the value of the number modeled by the blocks shown.
Use place value notation to find the value of the number modeled by the blocks shown.
Manipulative Mathematics
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.
- 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:
- ⓐ
- ⓑ
- ⓒ
- ⓓ
- ⓔ
For each number, find the place value of digits listed:
- ⓐ
- ⓑ
- ⓒ
- ⓓ
- ⓔ
For each number, find the place value of digits listed:
- ⓐ
- ⓑ
- ⓒ
- ⓓ
- ⓔ
Use Place Value to Name Whole Numbers
When you write a check, you write out the number in words as well as in digits. To write a number in words, write the number in each period followed by the name of the period without the ‘s’ at the end. Start with the digit at the left, which has the largest place value. The commas separate the periods, so wherever there is a comma in the number, write a comma between the words. The ones period, which has the smallest place value, is not named.
So the number 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.
- 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.
- Step 2. Use commas in the number to separate the periods.
Example 1.4
Name the number in words.
Name each number in words:
Name each number in words:
Example 1.5
A student conducted research and found that the number of mobile phone users in the United States during one month in was Name that number in words.
The population in a country is Name that number.
One year is 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.
- Step 1. Identify the words that indicate periods. (Remember the ones period is never named.)
- Step 2. Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.
- 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
Write each number in standard form:
fifty-three million, eight hundred nine thousand, fifty-one.
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 billion. Write the budget in standard form.
Write each number in standard form:
The closest distance from Earth to Mars is about million miles.
Write each number in standard form:
The total weight of an aircraft carrier is million pounds.
Round Whole Numbers
In the U.S. Census Bureau reported the population of the state of New York as people. It might be enough to say that the population is approximately million. The word approximately means that 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. million was achieved by rounding to the millions place. Had we rounded to the one hundred thousands place, we would have as a result. Had we rounded to the ten thousands place, we would have as a result, and so on. The place value to which we round to depends on how we need to use the number.
Using the number line can help you visualize and understand the rounding process. Look at the number line in Figure 1.7. Suppose we want to round the number to the nearest ten. Is closer to or on the number line?
Now consider the number Find in Figure 1.8.
How do we round to the nearest ten. Find in Figure 1.9.
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.
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
Let’s look again at rounding to the nearest Again, we look to the ones place.
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
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.
- 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
- Yes—add to the digit in the given place value.
- 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 to the nearest ten.
Round to the nearest ten:
Round to the nearest ten:
Example 1.9
Round each number to the nearest hundred:
- ⓐ
- ⓑ
Round to the nearest hundred:
Round to the nearest hundred:
Example 1.10
Round each number to the nearest thousand:
- ⓐ
- ⓑ
Round to the nearest thousand:
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.
Model Whole Numbers
In the following exercises, use place value notation to find the value of the number modeled by the blocks.
Identify the Place Value of a Digit
In the following exercises, find the place value of the given digits.
- ⓐ 9
- ⓑ 3
- ⓒ 2
- ⓓ 8
- ⓔ 7
- ⓐ 8
- ⓑ 4
- ⓒ 2
- ⓓ 6
- ⓔ 7
Use Place Value to Name Whole Numbers
In the following exercises, name each number in words.
The height of Mount Adams is feet.
One year is minutes.
The population of Chicago was
About twelve years ago there were registered automobiles in California.
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.
two hundred fifty-three
sixty-one thousand, four hundred fifteen
eighteen million, one hundred two thousand, seven hundred eighty-three
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.
The age of the solar system is estimated to be four billion, five hundred sixty-eight million years.
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:
- ⓐ
- ⓑ
Round to the nearest hundred:
- ⓐ
- ⓑ
Round to the nearest thousand:
- ⓐ
- ⓑ
Round to the nearest thousand:
- ⓐ
- ⓑ
Everyday Math
Writing a Check Jorge bought a car for He paid for the car with a check. Write the purchase price in words.
Writing a Check Marissa’s kitchen remodeling cost She wrote a check to the contractor. Write the amount paid in words.
Buying a Car Jorge bought a car for Round the price to the nearest:
- ⓐ ten dollars
- ⓑ hundred dollars
- ⓒ thousand dollars
- ⓓ ten-thousand dollars
Remodeling a Kitchen Marissa’s kitchen remodeling cost Round the cost to the nearest:
- ⓐ ten dollars
- ⓑ hundred dollars
- ⓒ thousand dollars
- ⓓ ten-thousand dollars
Population The population of China was in Round the population to the nearest:
- ⓐ billion people
- ⓑ hundred-million people
- ⓒ million people
Astronomy The average distance between Earth and the sun is kilometers. Round the distance to the nearest:
- ⓐ hundred-million kilometers
- ⓑ ten-million kilometers
- ⓒ million kilometers
Writing Exercises
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.