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 $q\ne 0$. 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.