2.1: Understanding Decimals
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 |
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 |