No, the number \(0.343434...\) is not an irrational number; it is a rational number. The notation \(0.343434...\) represents a repeating decimal, which can be expressed as a fraction.
To convert \(0.343434...\) into a fraction, you can set \(x = 0.343434...\). Then, multiply both sides by 100 (since the decimal repeats every two digits):
\[ 100x = 34.343434... \]
Now, subtract the original equation from this new equation:
\[ 100x - x = 34.343434... - 0.343434... \]
This simplifies to:
\[ 99x = 34 \]
Now, divide both sides by 99:
\[ x = \frac{34}{99} \]
Since \(34\) and \(99\) are both integers and \(99\) is not zero, \(0.343434...\) is represented as the fraction \(\frac{34}{99}\), which confirms that it is a rational number.
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