# Decimals

## Decimals


### Name Decimals

You probably already know quite a bit about decimals based on your experience with money. Suppose you buy a sandwich and a bottle of water for lunch. If the sandwich costs , the bottle of water costs , and the total sales tax is , what is the total cost of your lunch?
The total is  Suppose you pay with a  bill and  pennies. Should you wait for change? No,  and  pennies is the same as 
Because  each penny is worth  of a dollar. We write the value of one penny as  since 
Writing a number with a decimal is known as decimal notation. It is a way of showing parts of a whole when the whole is a power of ten. In other words, decimals are another way of writing fractions whose denominators are powers of ten. Just as the counting numbers are based on powers of ten, decimals are based on powers of ten.  shows the counting numbers.
How are decimals related to fractions?  shows the relation.
When we name a whole number, the name corresponds to the place value based on the powers of ten. In Whole Numbers, we learned to read  as ten thousand. Likewise, the names of the decimal places correspond to their fraction values. Notice how the place value names in  relate to the names of the fractions from .
Notice two important facts shown in .
1. The “th” at the end of the name means the number is a fraction. “One thousand” is a number larger than one, but “one thousandth” is a number smaller than one.
2. The tenths place is the first place to the right of the decimal, but the tens place is two places to the left of the decimal.
Remember that  lunch? We read  as five dollars and three cents. Naming decimals (those that don’t represent money) is done in a similar way. We read the number  as five and three hundredths.
We sometimes need to translate a number written in decimal notation into words. As shown in , we write the amount on a check in both words and numbers.
The number  is read fifteen and sixty-eight hundredths.

### Write Decimals

Now we will translate the name of a decimal number into decimal notation. We will reverse the procedure we just used.
Let’s start by writing the number six and seventeen hundredths:
The second bullet in Step 2 is needed for decimals that have no whole number part, like ‘nine thousandths’. We recognize them by the words that indicate the place value after the decimal – such as ‘tenths’ or ‘hundredths.’  Since there is no whole number, there is no ‘and.’ We start by placing a zero to the left of the decimal and continue by filling in the numbers to the right, as we did above.
Before we move on to our next objective, think about money again. We know that  is the same as  The way we write  depends on the context. In the same way, integers can be written as decimals with as many zeros as needed to the right of the decimal.

### Convert Decimals to Fractions or Mixed Numbers

We often need to rewrite decimals as fractions or mixed numbers. Let’s go back to our lunch order to see how we can convert decimal numbers to fractions. We know that  means  dollars and  cents. Since there are  cents in one dollar,  cents means  of a dollar, so 
We convert decimals to fractions by identifying the place value of the farthest right digit. In the decimal  the  is in the hundredths place, so  is the denominator of the fraction equivalent to 
For our  lunch, we can write the decimal  as a mixed number.
Notice that when the number to the left of the decimal is zero, we get a proper fraction. When the number to the left of the decimal is not zero, we get a mixed number.

### Locate Decimals on the Number Line

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.

### Order Decimals

Which is larger,  or 
If you think of this as money, you know that  (forty cents) is greater than  (four cents). So,
In previous chapters, we used the number line to order numbers.
Where are  and  located on the number line?
We see that  is to the right of  So we know 
How does  compare to  This doesn’t translate into money to make the comparison easy. But if we convert  and  to fractions, we can tell which is larger.
Because  we know that  Therefore, 
Notice what we did in converting  to a fraction—we started with the fraction  and ended with the equivalent fraction  Converting  back to a decimal gives  So  is equivalent to  Writing zeros at the end of a decimal does not change its value.
If two decimals have the same value, they are said to be equivalent decimals.
We say  and  are equivalent decimals.
Remember, writing zeros at the end of a decimal does not change its value.

When we order negative decimals, it is important to remember how to order negative integers. Recall that larger numbers are to the right on the number line. For example, because  lies to the right of  on the number line, we know that  Similarly, smaller numbers lie to the left on the number line. For example, because  lies to the left of  on the number line, we know that 
If we zoomed in on the interval between  and  we would see in the same way that 

### Round Decimals

In the United States, gasoline prices are usually written with the decimal part as thousandths of a dollar. For example, a gas station might post the price of unleaded gas at  per gallon. But if you were to buy exactly one gallon of gas at this price, you would pay , because the final price would be rounded to the nearest cent. In Whole Numbers, we saw that we round numbers to get an approximate value when the exact value is not needed. Suppose we wanted to round  to the nearest dollar. Is it closer to  or to  What if we wanted to round  to the nearest ten cents; is it closer to  or to  The number lines in  can help us answer those questions.
Can we round decimals without number lines? Yes! We use a method based on the one we used to round whole numbers.

### Key Concepts

1. Name a decimal number.

2. Write a decimal number from its name.

3. Convert a decimal number to a fraction or mixed number.

4. Order decimals.

5. Round a decimal.


### Practice Makes Perfect

Name Decimals 
In the following exercises, name each decimal.
Write Decimals 
In the following exercises, translate the name into a decimal number.
Convert Decimals to Fractions or Mixed Numbers 
In the following exercises, convert each decimal to a fraction or mixed number.
Locate Decimals on the Number Line 
In the following exercises, locate each number on a number line.
Order Decimals 
In the following exercises, order each of the following pairs of numbers, using 
Round Decimals 
In the following exercises, round each number to the nearest tenth.
In the following exercises, round each number to the nearest hundredth.
In the following exercises, round each number to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.

### 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.