Question

Determine if 0, point, 7, 9, 6, 5, 3, 5, 5, 3, 5, 5, 3, 5, 5, 3, 5, point, point, .0.796535535535535... is rational or irrational and give a reason for your answer.

Answer 1

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.