Question

practice & problem solving
is 5.787787778... a rational or irrational number? explain.

Explanation:

Step1: Recall rational number definition

A rational number can be expressed as a fraction $\frac{p}{q}$ (where $p,q$ are integers, $q
eq0$) or has a repeating or terminating decimal expansion. An irrational number has a non - repeating, non - terminating decimal expansion.

Step2: Analyze the given decimal

The decimal $5.787787778\ldots$ has a non - repeating pattern (the number of 7s between 8s is increasing: 1 seven between 8s, then 2 sevens, then 3 sevens, etc.). So its decimal expansion is non - repeating and non - terminating.

Answer:

$5.787787778\ldots$ is an irrational number because its decimal expansion is non - repeating and non - terminating, and thus it cannot be expressed as a fraction of two integers.