To determine if the number \(0.796535535535535...\) is rational or irrational, we need to analyze its decimal representation.
The decimal goes as follows:
- The beginning part \(0.7965\) does not immediately indicate a repetition.
- However, after \(0.7965\), the digits "3535" begin to repeat indefinitely.
This means the decimal can be expressed as: \[ 0.7965\overline{3535} \] where the overline indicates that "3535" repeats indefinitely.
A number is rational if it can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). Decimal expansions that terminate (like \(0.5\)) or that repeat (like \(0.333...\)) signify that the number is rational.
Since \(0.7965\overline{3535}\) has a repeating decimal, we conclude that it is a rational number.
Thus, the number \(0.796535535535535...\) is rational because it can be represented with a repeating decimal.