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5. Check whether the decimal form of rational number will terminate or non-terminate

If a rational number

\frac{p}{q}, q \neq 0

can be expressed in the form of

\frac{p}{2^{m} \times 5^{n}}

, where

p \in ℤ

and

m, n \in W

, the rational number will have terminating decimal expansion. Otherwise, the rational number will have non-terminating and recurring decimal expansion.

Example:

Verify the following have terminating decimals or non-terminating and recurring decimal expansion.

\frac{1}{20}

\frac{1}{20} = \frac{1}{4 \times 5} = \frac{1}{2^{2} \times 5^{1}}

Thus, the given rational number follows the pattern

\frac{p}{2^{m} \times 5^{n}}

Therefore the number

\frac{1}{20}

have terminating decimal expansion.

\frac{- 42}{625}

\frac{- 42}{625} = \frac{- 42}{5^{4}} = \frac{- 42}{2^{0} \times 5^{4}}

Thus, the given rational number follows the pattern

\frac{p}{2^{m} \times 5^{n}}

Therefore the number

\frac{- 42}{625}

have terminating decimal expansion.

\frac{19}{225}

\frac{19}{225} = \frac{19}{9 \times 25} = \frac{19}{3^{2} \times 5^{2}}

Thus, the given rational number does not follow the pattern

\frac{p}{2^{m} \times 5^{n}}

Therefore the number

\frac{19}{225}

does not have terminating decimal expansion. It has a non-terminating and recurring decimal expansion.

\frac{- 37}{350}

\frac{- 37}{350} = \frac{- 37}{2 \times 25 \times 7}

\frac{- 37}{350} = \frac{- 37}{2^{1} \times 5^{2} \times 7^{1}}

Thus, the given rational number does not follow the pattern

\frac{p}{2^{m} \times 5^{n}}

Therefore the number

\frac{- 37}{350}

does not have terminating decimal expansion. But it has a non-terminating and recurring decimal expansion.

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5. Check whether the decimal form of rational number will terminate or non-terminate

Theory: