Represent each of the following rational numbers in fraction form.
Standard 8.NS.1 requires students to "convert a decimal expansion which repeats eventually into a rational number." Despite this choice of wording, the numbers in this task are rational numbers regardless of choice of representation. For example, $0.33\overline{3}$ and $\frac13$ are two different ways of representing the same number.
So what is a rational number? Sometimes people define a rational number to be a ratio of integers, but to be consistent with the CCSSM, we would need to say a rational number is any number that is the value of a ratio of two integers. Sometimes people define a rational number based on how it can be represented; here is a typical definition: A rational number is any number that can be represented as $\frac{a}{b}$ where $a$ and $b$ are integers and $b\neq 0$. It is interesting to compare this with the definition of a rational number given in the Glossary of the CCSSM (as well as the more nuanced meaning developed in the standards themselves starting in grade 3 and beyond).
A more constructive definition for a rational number that does not depend on the way we represent it is:
A number is rational if it is a quotient $a\div b$ of two integers $a$ and $b$ where $b\neq 0$.or, equivalently,
A rational number is a number that satisfies an equation of the form $a=bx$, where $a$ and $b$ are integers and $b\neq 0$.
So $0.33\overline{3}$ is a rational number because it is the result we get when we divide 1 by 3, or equivalently, because it is a solution to $1=3x$. However, numbers like $\pi$ and $\sqrt{2}$ are not rational because neither of them satisfies an equation of the form $a=bx$ where $a$ and $b$ are integers. This is actually tricky to show and is an exercise left to high school or college.
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