Question

determine if 0.212112111211112111112... is rational or irrational and give a reason for your answer. answer attempt 2 out of 3 the number 0.212112111211112111112... is rational because it is a decimal that does not repeat or terminate

Explanation:

Step1: Recall Rational/Irrational Definition

Rational numbers are numbers that can be expressed as a fraction $\frac{p}{q}$ (where $p,q$ are integers, $q
eq0$), and their decimal expansions are either terminating or repeating (with a repeating block). Irrational numbers have decimal expansions that are non - terminating and non - repeating.

Step2: Analyze the Given Decimal

The decimal $0.212112111211112111112\ldots$: Let's look at the pattern. The number of $1$s between the $2$s is increasing: first one $1$ (after the first $2$), then two $1$s (after the second $2$), then three $1$s (after the third $2$), and so on. There is no fixed repeating block of digits. So its decimal expansion is non - terminating and non - repeating.

Answer:

The number $0.212112111211112111112\ldots$ is irrational because its decimal expansion is non - terminating and non - repeating (it does not have a repeating block of digits, and it does not terminate).