To show that when a rational number is written in decimal form it either terminates in 0s or eventually repeats, we first clarify what a rational number is. A rational number can be expressed in the form of \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
Termination or Repetition
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Termination: A rational number's decimal representation terminates if the denominator \( q \) can be expressed in the form \( 2^m \cdot 5^n \) (where \( m \) and \( n \) are non-negative integers). For instance, \( \frac{1}{8} = \frac{1}{2^3} \) which equals 0.125 (terminating).
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Repetition: If \( q \) contains any prime factors other than 2 or 5, then the decimal representation will repeat. For instance, \( \frac{1}{3} = 0.3333\ldots \) or \( \frac{1}{7} = 0.142857142857\ldots \).
Long Division for 6715
Converting 6715 into a decimal using long division:
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We start with \( 6715 \div 1000 \) since we want to convert it to a decimal (which is equivalent to finding how many times 1000 fits into 6715).
\[ 6715 \div 1000 = 6.715 \]
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To get two decimal places, we can express this in terms of \( 100 \):
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Divide 6715 by 100: \[ 6715 \div 100 = 67.15 \]
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This indicates that for a mathematical solution the approximate part to two decimal places: \[ 67.15 \rightarrow \approx 6.47 \]
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Thus, the decimal representation of \( 6715 \) divided by \( 1000 \) results in \( \approx 6.715\), which can be further approximated if we need a specific precision.
Final approximation based on context, considering hypothetical division through exact whole integers or required decimal placement, leads to the rational number being written as \( 6.47 \) when rounded appropriately based on the significant digits required (i.e., if truncating, rounding, or other mathematical constraints apply). The precise reading yields \( 6.715 \) based on step simplifications.
Conclusion
Thus, when a rational number is written in decimal form, it either terminates in 0s (in a non-fraction resulting status) or eventually repeats. In this particular context, \( 6715 \) when analyzed simply through division results in \( 6.715 \), leading to \( \approx 6.47\) under specific truncation.