No, \( 3.6363636363636363... \) is not an irrational number. It is a repeating decimal, which can be expressed as a fraction. The repeating part is "63," and so the number can be represented as:
\[ 3.6363636363... = 3 + 0.6363636363... \]
To convert \( 0.6363636363... \) into a fraction, let \( x = 0.6363636363... \). Then, by multiplying both sides by 100 (since the repeating part has two digits), we have:
\[ 100x = 63.6363636363... \]
Now, subtract the original equation from this one:
\[ 100x - x = 63.6363636363... - 0.6363636363... \] \[ 99x = 63 \] \[ x = \frac{63}{99} \]
This fraction can be simplified:
\[ x = \frac{63 \div 9}{99 \div 9} = \frac{7}{11} \]
Thus,
\[ 0.6363636363... = \frac{7}{11} \]
Now, we can combine this back into our original equation:
\[ 3.6363636363... = 3 + \frac{7}{11} = \frac{33}{11} + \frac{7}{11} = \frac{40}{11} \]
Since \( \frac{40}{11} \) is a fraction and not a whole number or an irrational number, we conclude that \( 3.6363636363... \) is a rational number.
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