Decimals

Corinn Herrell; Cheryl Colan; Fatima Baig

Decimals

The metric system, which is based on the decimal system, is the foundation of dosage calculation and other measurements in clinical settings. The decimal point is a symbol that separates the whole number and a decimal fraction and lies between the whole number and the tenths place. For example, 2.5 (“2” is the whole number “.” decimal, and “5” is the fractional part of the decimal number. It is based on the number 10, or powers of 10, so that multiplying or dividing by a multiple of ten (10, 100, 1,000, etc.) will change the value of a number. For example: 10² = 10 x 10 = 100 or 10^-2 = 10/100 = 1/10 = 0.1 (one tenth).

Learning Objectives

By the end of this section, you will be able to:

Name decimals
Write decimals
Convert decimals to fractions or mixed numbers (and vice versa)
Locate decimals on the number line
Order decimals
Round decimals to a specified place value

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 $\textdollar3.45$ , the bottle of water costs $\textdollar1.25$ , and the total sales tax is $\textdollar0.33$ , what is the total cost of your lunch?

A vertical addition problem is shown. The top line shows three dollars and forty-five cents for a sandwich, the next line shows one dollar and twenty-five cents for water, and the last line shows thirty-three cents for tax. The total is shown to be five dollars and three cents.

The total is $\textdollar5.03$ . Suppose you pay with a $\textdollar5$ bill and 3 pennies. Should you wait for change? No, $\textdollar5$ and 3 pennies is the same as $\textdollar5.03$ .

Because $100$ pennies $= \textdollar1$ , each penny is worth $\dfrac{1}{100}$ of a dollar. We write the value of one penny as $\textdollar0.01$ , since $0.01=\dfrac{1}{100}$ .

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. Just as the counting numbers are based on powers of ten, decimals are based on powers of ten. Table 5.1 shows the counting numbers.

Table 5.1 Names of Counting Numbers
Counting Numbers	Name
1	One
10=10	Ten
10 • 10 = 100	One Hundred
10 • 10 • 10 = 1000	One Thousand
10 • 10 • 10 • 10 = 10,000	Ten Thousand

How are decimals related to fractions? Table 5.2 shows the relation.

Table 5.2 Relation of Decimals and Fractions
Decimal	Fraction	Name
0.1	$\dfrac{1}{10}$	One tenth
0.01	$\dfrac{1}{100}$	One hundredth
0.001	$\dfrac{1}{1000}$	One thousandth
0.0001	$\dfrac{1}{10000}$	One ten-thousandth

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 10,000 as ten thousand. Likewise, the names of the decimal places correspond to their fraction values. Notice how the place value names in Figure 5.2 relate to the names of the fractions from Table 5.2.

A chart is shown labeled “Place Value”. There are 12 columns. The columns are labeled, from left to right, Hundred thousands, Ten thousands, Thousands, Hundreds, Tens, Ones, Decimal Point, Tenths, Hundredths, Thousandths, Ten-thousandths, Hundred-thousandths. — Figure 5.2 This chart illustrates place values to the left and right of the decimal point.

Notice two important facts shown in Figure 5.2.

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.
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 $\textdollar5.03$ lunch? We read $\textdollar5.03$ as five dollars and three cents. Naming decimals (those that don’t represent money) is done in a similar way. We read the number 5.03 as five and three hundredths.

We sometimes need to translate a number written in decimal notation into words. As shown in Figure 5.3, we write the amount on a check in both words and numbers.

An image of a check is shown. The check is made out to Jane Doe. It shows the dollar amount 152.65 and says in words, “One hundred fifty two and 65 over 100 dollars.” — Figure 5.3 When we write a check, we write the amount as a decimal number as well as in words. The bank looks at the check to make sure both numbers match. This helps prevent errors.

HOW TO

Name a decimal number.

Name the number to the left of the decimal point.
Write “and” for the decimal point.
Name the “number” part to the right of the decimal point as if it were a whole number.
Name the decimal place of the last digit.

PRACTICE EXERCISES

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:

	six and seventeen hundredths
The word and tells us to place a decimal point.	___.___
The word before and is the whole number; write it to the left of the decimal point.	6._____
The decimal part is seventeen hundredths. Mark two places to the right of the decimal point for hundredths.	6._ _
Write the numerals for seventeen in the places marked.	6.17

PRACTICE EXERCISES

HOW TO

Write a decimal number from its name.

Look for the word “and”—it locates the decimal point.
Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word.
- Place a decimal point under the word “and.” Translate the words before “and” into the whole number and place it to the left of the decimal point.
- If there is no “and,” write a “0” with a decimal point to its right.
Translate the words after “and” into the number to the right of the decimal point. Write the number in the spaces—putting the final digit in the last place.
Fill in zeros for place holders as needed.

The second bullet in Step 2 (above) 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.

PRACTICE EXERCISES

Before we move on to our next objective, think about money again. We know that $\textdollar1$ is the same as $\textdollar1.00$ .The way we write $\textdollar1$ (or $\textdollar1.00$ ) 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.

$5=5.0$
$5=5.00$
$5=5.000$
and so on…

$-2=-2.0$
$-2=-2.00$
$-2=-2.000$
and so on…

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 $\textdollar5.03$ means 5 dollars and 3 cents. Since there are 100 cents in one dollar, 3 cents means $\dfrac{3}{100}$ of a dollar, so $0.03=\dfrac{3}{100}$ .

We convert decimals to fractions by identifying the place value of the farthest right digit. In the decimal 0.03, the 3 is in the hundredths place, so 100 is the denominator of the fraction equivalent to 0.03.

0.03=\dfrac{3}{100}

For our $\textdollar5.03$ lunch, we can write the decimal 5.03 as a mixed number.

5.03=5\dfrac{3}{100}

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.

HOW TO: Convert a decimal number to a fraction or mixed number.

Step 1.

Look at the number to the left of the decimal.

If it is zero, the decimal converts to a proper fraction.
If it is not zero, the decimal converts to a mixed number.

Write the whole number.

Step 2.

Determine the place value of the final digit.

Step 3.

Write the fraction.

numerator—the ‘numbers’ to the right of the decimal point
denominator—the place value corresponding to the final digit

Step 4.

Simplify the fraction, if possible.

EXAMPLE 5.4: Write each of the following decimal numbers as a fraction or a mixed number.

Did you notice that the number of zeros in the denominator is the same as the number of decimal places?

TRY IT

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.

EXAMPLE 5.5

Locate 0.4 on a number line.

The decimal 0.4 is equivalent to $\dfrac{4}{10}$ , so 0.4 is located between 0 and 1. On a number line, divide the interval between 0 and 1 into 10 equal parts and place marks to separate the parts.

Label the marks 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0. We write 0 as 0.0 and 1 as 1.0, so that the numbers are consistently in tenths.

Finally, mark 0.4 on the number line.

A number line is shown with 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 labeled. There is a green dot at 0.4.

TRY IT

EXAMPLE 5.6

Locate −0.74 on a number line.

The decimal −0.74 is equivalent to $−\dfrac{74}{100}$ , so it is located between 0 and −1. On a number line, mark off and label the multiples of −0.10 in the interval between 0 and −1 (−0.10, −0.20, etc.) and mark −0.74 between −0.70 and −0.80, a little closer to −0.70.

A number line is shown with negative 1.00, negative 0.90, negative 0.80, negative 0.70, negative 0.60, negative 0.50, negative 0.40, negative 0.30, negative 0.20, negative 0.10, and 0.00 labeled. There is a green dot between negative 0.80 and negative 0.70 labeled as negative 0.74.

TRY IT

Order Decimals

Which is larger, 0.04 or 0.40?

If you think of this as money, you know that $\textdollar0.40$ (forty cents) is greater than $\textdollar0.04$ (four cents). So,

$0.40>0.04$

In previous chapters, we used the number line to order numbers.

$a<b$ ‘a is less than b’ when a is to the left of b on the number line
$a>b$ ‘a is greater than b’ when a is to the right of b on the number line

Where are 0.04 and 0.40 located on the number line?

A number line is shown with 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 labeled. There is a green dot between 0.0 and 0.1 labeled as 0.04. There is another green dot at 0.4.

	0.31	0.308
Convert to fractions.	$\dfrac{31}{100}$	$\dfrac{308}{1000}$
We need a common denominator to compare them.	$\dfrac{31}{100}\times\dfrac{10}{10}$	$\dfrac{308}{1000}$
	$\dfrac{310}{1000}$	$\dfrac{308}{1000}$

Because $310>308$ , we know that $\dfrac{310}{1000}>\dfrac{308}{1000}$ . Therefore, $0.31>0.308$ .

Notice what we did in converting 0.31 to a fraction—we started with the fraction $\dfrac{31}{100}$ and ended with the equivalent fraction $\dfrac{310}{1000}$ . Converting $\dfrac{310}{1000}$ back to a decimal gives 0.310. So 0.31 is equivalent to 0.310. Writing zeros at the end of a decimal does not change its value.

$\dfrac{31}{100}=\dfrac{310}{1000}$ and 0.31=0.310

If two decimals have the same value, they are said to be equivalent decimals.

$0.31=0.310$

We say 0.31 and 0.310 are equivalent decimals.

Equivalent Decimals

Two decimals are equivalent decimals if they convert to equivalent fractions.

Remember, writing zeros at the end of a decimal does not change its value.

HOW TO: Order Decimals

Step 1. Check to see if both numbers have the same number of decimal places. If not, write zeros at the end of the one with fewer digits to make them match.

Step 2. Compare the numbers to the right of the decimal point as if they were whole numbers.

Step 3. Order the numbers using the appropriate inequality sign.

Example 5.7: Ordering decimals.

TRY IT

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 −2 lies to the right of −3 on the number line, we know that $−2>−3$ . Similarly, smaller numbers lie to the left on the number line. For example, because −9 lies to the left of −6 on the number line, we know that $−9<−6$ .

If we zoomed in on the interval between 0 and −1, we would see in the same way that $−0.2>−0.3$ and $−0.9<−0.6$ .

EXAMPLE 5.8

TRY IT

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 $\textdollar3.279$ per gallon. But if you were to buy exactly one gallon of gas at this price, you would pay $\textdollar3.28$ , 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 $\textdollar2.72$ to the nearest dollar. Is it closer to $\textdollar2$ or to $\textdollar3$ ? What if we wanted to round $\textdollar2.72$ to the nearest ten cents; is it closer to $\textdollar2.70$ or to $\textdollar2.80$ ? The number lines in Figure 5.4 can help us answer those questions.

In part a, a number line is shown with 2, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9 and 3. There is a dot between 2.7 and 2.8 labeled as 2.72. In part b, a number line is shown with 2.70, 2.71, 2.72, 2.73, 2.74, 2.75, 2.76, 2.77, 2.78, 2.79, and 2.80. There is a dot at 2.72. — Figure 5.4 ⓐ We see that 2.72 is closer to 3 than to 2. So, 2.72 rounded to the nearest whole number is 3. ⓑ We see that 2.72 is closer to 2.70 than 2.80. So we say that 2.72 rounded to the nearest tenth is 2.7.

Can we round decimals without number lines? Yes! We use a method based on the one we used to round whole numbers.

HOW TO: Round a decimal.

Step 1. Locate the given place value and mark it with an arrow.

Step 2. Underline the digit to the right of the specified place value.

Step 3. Is this digit greater than or equal to 5?

Yes – add 1 to the digit in the given place value.
No – do not change the digit in the given place value

Step 4. Rewrite the number, removing all digits to the right of the specified place value.

EXAMPLE 5.9

TRY IT

EXAMPLE 5.10

TRY IT

Practice Makes Perfect

Name Decimals

Practice naming decimals

Write Decimals

Practice writing decimals

Convert Decimals to Fractions or Mixed Numbers

Practice converting decimals to fractions or mixed numbers

Locate Decimals on the Number Line

Practice locating decimals on the number line

In the following exercises, locate each number on a number line.

Order Decimals

Pracitce ordering decimals

Round Decimals

Practice rounding decimals

Everyday Math

Practice everyday math.

Self Check

After completing the exercises, use this table to evaluate your mastery of the objectives of this section. For each skill listed in a table row, rate yourself in the column that best describes your confidence level.

Decimals Self-Check
I can …	Confidently	With some help	No – I don’t get it!
name decimals.
write decimals.
convert decimals to fractions or mixed numbers.
locate decimals on the number line.
order decimals.
round decimals.

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 in 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, and in nursing, every topic builds upon previous work. It is important to ensure you have a strong foundation before you move on. Who can you ask for help? 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, student success specialist, or math tutor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

Chapter Attributions

This chapter was adapted by Corinn Herrell and Cheryl Colan from “5.1 Decimals” in Prealgebra 2e by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis. Licensed under a CC BY 4.0 license.

License

Icon for the Creative Commons Attribution 4.0 International License

Name Decimals

Write Decimals

Convert Decimals to Fractions or Mixed Numbers

Locate Decimals on the Number Line

Order Decimals

Round Decimals

Practice Makes Perfect

Name Decimals

Write Decimals

Convert Decimals to Fractions or Mixed Numbers

Locate Decimals on the Number Line

Order Decimals

Round Decimals

Everyday Math

Self Check

Chapter Attributions

License

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