Question

score 3/5 penalty: 0.25 off determine if number is rational / irrational question video examples determine if 0.929229222922229222229... is rational or irrational and give a reason for your answer. answer attempt 1 out of 2 the number 0.929229222922229222229... is because

Explanation:

Step1: Recall definitions

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

Step2: Analyze the given decimal

The decimal number $0.929229222922229222229\ldots$: Let's look at the pattern of digits. The number of 2s between the 9s is increasing: after the first 9, there is 1 two; after the second 9, there are 2 twos; after the third 9, there are 3 twos and so on. So, there is no repeating pattern in the decimal expansion. Since it is a non - repeating and non - terminating decimal, it cannot be expressed as a fraction of two integers.

Answer:

The number \(0.929229222922229222229\ldots\) is irrational because it is a non - repeating, non - terminating decimal (the pattern of digits does not repeat in a cyclic manner, the number of 2s between 9s keeps increasing, so there is no fixed repeating cycle).