Question

student,statement,agree or disagree?
arlo:,$0.overline{57}$ is an irrational number.
hao:,no, arlo, it is rational, because $0.overline{57}$ can be written as a fraction.
eiji:,maybe hao’s correct, you know. because $0.overline{57}=\frac{57}{100}$.
korbin:,hang on. the decimal for $0.overline{57}$ would go on forever if you tried to write it. that’s what the bar thing means, right?
hank:,and because it goes on forever, that proves $0.overline{57}$ has got to be irrational.

Explanation:

Step1: Recall definition of rational numbers

A rational number can be written as $\frac{p}{q}$ where $p,q$ are integers and $q
eq0$.

Step2: Analyze $0.\overline{57}$

Let $x = 0.\overline{57}=0.575757\cdots$. Then $100x=57.575757\cdots$. Subtracting $x$ from $100x$ gives $100x - x=57.5757\cdots - 0.5757\cdots$, so $99x = 57$, and $x=\frac{57}{99}$. Since it can be written as a fraction of two integers, $0.\overline{57}$ is rational. So Arlo and Hank are wrong, Hao and Eiji are correct.

Answer:

Disagree with Arlo and Hank, agree with Hao and Eiji.