2.1: Understanding Decimals
- Page ID
- 7086
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Numbers written with decimals is another way of expressing fractions. However, decimals have a distinct differentiation with fractions in that decimals are based on 10. In that, one decimal place to the right of the whole number indicates tenths, two decimal places to the right of the whole number indicates hundredths, three decimal places to the right of the whole number indicates thousandths, etc.
Example \(\PageIndex{1}\)
0.1 = tenths place
1.01 = hundredths place
0.001 = thousandths place
1.01 = ten thousandths place
Decimals can easily be written as fractions by using the number to the right of the decimal point as the numerator and then using a 10, 100, 1,000, 10,000, etc. as the denominator. Determining which “base-ten” number to use as the denominator is determined by the number of digits there are to the right of the decimal point.
Example \(\PageIndex{2}\)
\(\begin{array}{ll}
0.1 \rightarrow \dfrac{1}{10} & 1 \text { tenth } \\
0.01 \rightarrow \dfrac{1}{100} & 1 \text { hundredth } \\
0.001 \rightarrow \dfrac{1}{1,000} & 1 \text { thousandth } \\
0.0001 \rightarrow \dfrac{1}{10,000} & 1 \text { ten-thousandth }
\end{array}\)
Exercise 2.1
Write the following decimals as fractions and reduce if necessary.
- 0.2 =
- 0.103 =
- 0.13 =
- 0.02 =
- 0.1234 =
- 0.0023 =
- 0.0101 =
- 0.1010 =
- 0.020 =
- 0.0202 =
- 0.1000 =
- 0.4500 =
Exercise 2.1.1
Write the following expressions as decimals and fractions.
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Decimal |
Fraction |
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7 tenths |
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1,000 ten-thousandths |
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475 thousandths |
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32 hundredths |
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12 thousandths |
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2,345 ten-thousandths |
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3 hundredths |
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10 thousandths |
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132 ten-thousandths |
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4,002 ten-thousandths |
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Digits to the left of the decimal indicate a whole number. Whenever there is a whole number with a decimal fraction the expression is pronounced using “and.”
Example:
12.3 = Twelve and three tenths = \(12 \dfrac{3}{10}\)
100.07 = One hundred and seven hundredths = \(100 \dfrac{7}{100}\)
3,005.023 = Three thousand five and twenty-three thousandths = \(3,005 \dfrac{23}{1,000}\)
Exercise 2.1.2
Write the following expressions as a decimal and a fraction. Remember to reduce if necessary.
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Decimal |
Fraction |
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Two and five hundredths |
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Thirteen and two hundred thousandths |
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One thousand three and two ten-thousandths |
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Fifty-five and fifty hundredths |
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One and fourteen hundredth |
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Eleven thousand four and twelve thousandths |
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Ten and two tenths |
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Four hundred one and four ten-thousandths |
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Five thousand two hundred eight and one thousand ten-thousandths |
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