# Whole Numbers

## 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:  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 .
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.
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.

### 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 . 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 . 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.
 shows the number  modeled with  blocks.

### 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.
 shows how the number  is written in a place value chart.
1. The digit  is in the millions place. Its value is 
2. The digit  is in the hundred thousands place. Its value is 
3. The digit  is in the ten thousands place. Its value is 
4. The digit  is in the thousands place. Its value is 
5. The digit  is in the hundreds place. Its value is 
6. The digit  is in the tens place. Its value is 
7. The digit  is in the ones place. Its value is 

### 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.

### Use Place Value to Write Whole Numbers

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

### 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 . 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 .
How do we round  to the nearest ten. Find  in .
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 

### Key Concepts

1. Name a whole number in words.

2. Use place value to write a whole number.

3. Round a whole number to a specific place value.


### 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.
Use Place Value to Name Whole Numbers 
In the following exercises, name each number in words.
Use Place Value to Write Whole Numbers 
In the following exercises, write each number as a whole number using digits.
Round Whole Numbers 
In the following exercises, round to the indicated place value.

### Everyday Math


### Writing Exercises


### 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.