Decimals

5.1 Decimals

The topics covered in this section are:

  1. Name decimals
  2. Write decimals
  3. Convert decimals to fractions or mixed numbers
  4. Locate decimals on the number line
  5. Order decimals
  6. Round decimals

5.1.1 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 $ \$ 3.45$, the bottle of water costs $ \$ 1.25$, and the total sales tax is $ \$ 0.33$, what is the total cost of your lunch?

A vertical addition problem is shown. The top line shows $3.45 for a sandwich, the next line shows $1.25 for water, and the last line shows $0.33 for tax. The total is shown to be $5.03.

The total is $ \$ 5.03$.. Suppose you pay with a $ \$ 5$ bill and $3$ pennies. Should you wait for change? No, $ \$ 5$ and $3$ pennies is the same as $ \$ 5.03$.

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

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. The table below shows the counting numbers.

Counting numberName
$1$One
$10 = 10$Ten
$10 \cdot 10 = 100$One hundred
$10 \cdot 10 \cdot 10 = 1,000$One thousand
$10 \cdot 10 \cdot 10 \cdot 10 = 10,000$Ten thousand

How are decimals related to fractions? The table below shows the relation.

DecimalFractionName
$0.1$$\frac{1}{10}$One tenth
$0.01$$\frac{1}{100}$One hundredth
$0.001$$\frac{1}{1,000}$One thousandth
$0.0001$$\frac{1}{10,000}$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 the figure below relate of the names of the fractions from the figure above.

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.
This chart illustrates place values to the left and right of the decimal point.

Notice two important facts shown in the figure above.

  • 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 $ \$ 5.03$ lunch? We read $ \$ 5.03$ 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 show in the figure below, 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 number $152.65 and says in words, “One hundred fifty two and 65 over 100 dollars.”
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.
Let’s try naming a decimal, such as $15.68$.
We start by naming the number to the left of the decimal.fifteen______
We use the word “and” to indicate the decimal point.fifteen and_____
Then we name the number to the right of the decimal point as if it were a whole number.fifteen and sixty-eight_____
Last, name the decimal place of the last digit.fifteen and sixty-eight hundredths

The number $15.68$ is read fifteen and sixty-eight hundredths.

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.

Example 1

Name each decimal:

  1. $4.3$
  2. $2.45$
  3. $0.009$
  4. $-15.571$
Solution

Part 1.
$4.3$
Name the number to the left of the decimal point.four_____
Write “and” for the decimal point.four and_____
Name the number to the right of the decimal point as if it were a whole number.four and three_____
Name the decimal place of the last digit.four and three tenths
Part 2.
$2.45$
Name the number to the left of the decimal point.two_____
Write “and” for the decimal point.two and_____
Name the number to the right of the decimal point as if it were a whole number.two and forty-five_____
Name the decimal place of the last digit.two and forty-five hundredths
Part 3.
$0.009$
Name the number to the left of the decimal point.Zero is the number to the left of the decimal; it is not included in the name.
Name the number to the right of the decimal point as if it were a whole number.nine_____
Name the decimal place of the last digit.nine thousandths
Part 4.
$-15.571$
Name the number to the left of the decimal point.negative fifteen
Write “and” for the decimal point.negative fifteen and_____
Name the number to the right of the decimal point as if it were a whole number.negative fifteen and five hundred seventy-one_____
Name the decimal place of the last digit.negative fifteen and five hundred seventy-one thousandths

5.1.2 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.$6.17$

Example 2

Write fourteen and thirty-seven hundredths as a decimal.

Solution

fourteen and thirty-seven hundredths
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.$14.$_________
Mark two places to the right of the decimal point for “hundredths”.$14.$__ __
Translate the words after “and” and write the number to the right of the decimal point.$14.37$
Fourteen and thirty-seven hundredths is written $14.37$.

HOW TO: Write a decimal number from its name.

  1. Look for the word “and”—it locates the decimal point.
  2. 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.
  3. 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.
  4. Fill in zeros for place holders as needed.

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.

Example 3

Write twenty-four thousandths as a decimal.

Solution

twenty-four thousandths
Look for the word “and”.There is no “and” so start with $0$.
To the right of the decimal point, put three decimal places for thousandths..
Write the number $24$ with the $4$ in the thousandths place..
Put zeros as placeholders in the remaining decimal places.$0.024$
So, twenty-four thousandths is written $0.024$

Before we move on to our next objective, think about money again. We know that $ \$ 1$ is the same as $ \$ 1.00$. The way we write $ \$ 1$ (or $ \$ 1.00$) 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$$-2=-2.0$
$5=5.00$$-2=-2.00$
$5=5.000$$-2=-2.000$

and so on…

5.1.3 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 $ \$ 5.03$ means $5$ dollars and $3$ cents. Since there are $100$ cents in one dollar, $3$ cents means $\frac{3}{100}$ of a dollar, so $0.03 = \frac{3}{100}$.

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 $0.03$.

$\large 0.03 = \frac{3}{100}$

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

$\large 5.03 = 5 \frac{3}{100}$

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.

  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.
  2. Determine the place value of the final digit.
  3. Write the fraction.
    • numerator—the ‘numbers’ to the right of the decimal point
    • denominator—the place value corresponding to the final digit
  4. Simplify the fraction, if possible.

Example 4

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

  1. $4.09$
  2. $3.7$
  3. $-0.286$
Solution

Part 1.
$4.09$
There is a $4$ to the left of the decimal point.
Write “$4$” as the whole number part of the mixed number.
.
Determine the place value of the final digit..
Write the fraction.
Write $9$ in the numerator as it is the number to the right of the decimal point.
.
Write $100$ in the denominator as the place value of the final digit, $9$, is hundredth..
The fraction is in simplest form..

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

Part 2.
$3.7$
There is a $3$ to the left of the decimal point.
Write “$3$” as the whole number part of the mixed number.
.
Determine the place value of the final digit..
Write the fraction.
Write $7$ in the numerator as it is the number to the right of the decimal point.
.
Write $10$ in the denominator as the place value of the final digit, $7$, is tenths..
The fraction is in simplest form..
Part 3.
$-0.286$
There is a $0$ to the left of the decimal point.
Write a negative sign before the fraction.
.
Determine the place value of the final digit and write it in the denominator..
Write the fraction.
Write $286$ in the numerator as it is the number to the right of the decimal point.
Write $1,000$ in the denominator as the place value of the final digit, $6$, is thousandths.
.
We remove a common factor of $2$ to simplify the fraction..

5.1.4 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

Locate $0.4$ on a number line.

Solution

The decimal $0.4$ is equivalent to $\frac{4}{10}$, 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 marks to separate the parts.

Label the marks $0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.9, 1.0$. 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 red dot at 0.4.

Example 6

Locate $-0.74$ on a number line.

Solution

The decimal $-0.74$ is equivalent to $- \frac{74}{100}$, 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 red dot between negative 0.80 and negative 0.70 labeled as negative 0.74.

5.1.5 Order Decimals

Which is larger, $0.04$ or $0.40$?

If you think of this as money, you know that $ \$0.40$ (forty cents) is greater than $ \$ 0.04$ (four cents). So,

$\large 0.40 > 0.04$

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 red dot between 0.0 and 0.1 labeled as 0.04. There is another red dot at 0.4.

We see that $0.40$ is to the right of $0.04$. So we know $0.40>0.04$.

How does $0.31$ compare to $0.308$? This doesn’t translate into money to make the comparison easy. But if we convert $0.31$ and $0.308$ to fractions, we can tell which is larger.

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

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

Notice what we did in converting $0.31$ to a fractions – we started with the fraction $\frac{31}{100}$ and ended with the equivalent fraction $\frac{310}{1000}$. Converting $\frac{310}{1000}$ 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.

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

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

$\large 0.31 = 0.310$

We say $0.310$ 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.

  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.
  2. Compare the numbers to the right of the decimal point as if they were whole numbers.
  3. Order the numbers using the appropriate inequality sign.

Example 7

Order the following decimals using $<$ or $>$:

Solution

Part 1.
$0.64$ ____ $0.6$
Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of $0.6$.$0.64$ ____ $0.60$
Compare the numbers to the right of the decimal point as if they were whole numbers.$64 > 60$
Order the numbers using the appropriate inequality sign.$0.64 > 0.60$
$0.64 > 0.6$
Part 2.
$0.83$ ____ $0.803$
Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of $0.83$.$0.830$ ____ $0.803$
Compare the numbers to the right of the decimal point as if they were whole numbers.$830 > 803$
Order the numbers using the appropriate inequality sign.$0.830 > 0.803$
$0.830 > 0.803$

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 $-9<-6$.

A number line is shown with integers from negative 10 to 0. Blue dots are placed on negative nine and negative six. Red dots are placed at negative two and negative three.

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 8

Use $<$ or $>$ to order. $-0.1$ ____ $-0.8$.

Solution

$-0.1$ ____ $-0.8$
Write the numbers one under the other, lining up the decimal points.$-0.1$
$-0.8$
They have the same number of digits.
Since $-1 > -8$, $-1$ tenth is greater than $-8$ tenths.$-0.1 > -0.8$

5.1.6 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 $ \$3.279$ per gallon. But if you were to buy exactly one gallon of gas at this price, you would pay $ \$ 3.28$, 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 $ \$ 2.72$ to the nearest dollar. Is it closer to $ \$ 2$ or to $ \$ 3$? What if we wanted to round $ \$ 2.72$ to the nearest ten cents; is it closer to $ \$ 2.70$ or to $ \$ 2.80$? The number lines in the figure below 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.
$(a)$ We see that $2.72$ is closer to $3$ than to $2$. So, $2.72$ rounded to the nearest whole number is $3$. $(b)$ 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.

  1. Locate the given place value and mark it with an arrow.
  2. Underline the digit to the right of the given place value.
  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
  4. Rewrite the number, removing all digits to the right of the given place value.

Example 9

Round $18.379$ to the nearest hundredth.

Solution

$18.379$
Locate the hundredths place and mark it with an arrow..
Underline the digit to the right of the $7$..
Because $9$ is greater than or equal to $5$, add $1$ to the $7$..
Rewrite the number, deleting all digits to the right of the hundredths place.$18.38$
$18.38$ is $18.379$ rounded to the nearest hundredth.

Example 10

Round $18.379$ to the nearest…

  1. tenth
  2. whole number
Solution

Part 1. Round $18.379$ to the nearest tenth.
$18.379$
Locate the tenths place and mark it with an arrow..
Underline the digit to the right of the tenths digit..
Because $7$ is greater than or equal to $5$, add $1$ to the $3$..
Rewrite the number, deleting all digits to the right of the tenths place.$18.4$
$18.379$ rounded to the nearest tenth is $18.4$.
Part 2. Round $18.379$ to the nearest whole number.
$18.379$
Locate the ones place and mark it with an arrow..
Underline the digit to the right of the ones place..
Since $3$ is not greater than or equal to $5$, do not add $1$ to the $8$..
Rewrite the number, deleting all digits to the right of the ones place.$18$
So, $18.379$ rounded to the whole number is $18$.
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