1.3: Decimals
- Page ID
- 152023
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)After completing this section, you should be able to:
- Name decimals
- Write decimals
- Convert decimals to fractions or mixed numbers
- Round 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. Table 1.3.1 shows the counting numbers.
Counting number | Name |
---|---|
One | |
Ten | |
One hundred | |
One thousand | |
Ten thousand |
How are decimals related to fractions? Table 1.3.2 shows the relation.
Decimal | Fraction | Name |
---|---|---|
One tenth | ||
One hundredth | ||
One thousandth | ||
One ten-thousandth |
When we name a whole number, the name corresponds to the place value based on the powers of ten. We know to read as ten thousand. Likewise, the names of the decimal places correspond to their fraction values. Notice how the place value names in Figure 1.3.1 relate to the names of the fractions from Table 1.3.2.
Notice two important facts shown in Figure 1.3.1.
- 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.
- 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 Figure 1.3.2, we write the amount on a check in both words and numbers.
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 |
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 1.3.1
Name each decimal: ⓐ ⓑ ⓒ ⓓ
- Answer
-
ⓐ 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 ⓑ 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 ⓒ 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 ⓓ 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
Your Turn 1.3.1
Name each decimal:
ⓐ ⓑ ⓒ ⓓ
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 1.3.2
Write fourteen and thirty-seven hundredths as a decimal.
- Answer
-
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.
Your Turn 1.3.2
Write as a decimal: thirteen and sixty-eight hundredths.
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 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 1.3.3
Write twenty-four thousandths as a decimal.
- Answer
-
twenty-four thousandths Look for the word "and". There is no "and" so start with 0
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
Your Turn 1.3.3
Write as a decimal: fifty-eight thousandths.
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.
How To
Convert a decimal number to a fraction or mixed number.
- 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.
- Determine the place value of the final digit.
- Write the fraction.
- numerator—the ‘numbers’ to the right of the decimal point
- denominator—the place value corresponding to the final digit
- Simplify the fraction, if possible.
Example 1.3.4
Write each of the following decimal numbers as a fraction or a mixed number:
ⓐ ⓑ ⓒ
- Answer
-
ⓐ 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?
ⓑ 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. ⓒ −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.
Your Turn 1.3.4
Write as a fraction or mixed number. Simplify the answer if possible.
ⓐ ⓑ ⓒ
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. 0.31 and 0.310 are equivalent decimals. They have the same value.
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.
ⓑ We see that is closer to than So we say that rounded to the nearest tenth is
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.
- Locate the given place value and mark it with an arrow.
- Underline the digit to the right of the given place value.
- Is this digit 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
- Rewrite the number, removing all digits to the right of the given place value.
Example 1.3.5
Round to the nearest hundredth.
- Answer
-
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 is 18.379 rounded to the nearest hundredth.
Your Turn 1.3.5
Round to the nearest hundredth:
Example 1.3.6
Round to the nearest ⓐ tenth ⓑ whole number.
- Answer
-
ⓐ Round 18.379 to the nearest tenth. 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. So, 18.379 rounded to the nearest tenth is 18.4. ⓑ Round 18.379 to the nearest whole number. 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. So 18.379 rounded to the nearest whole number is 18.
Your Turn 1.3.6
Round to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.