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Q:
The characteristic of the equivalence classes of $\mathbb{Q}$ under a set of equivalence relations.
Suppose that $\mathbb{Q}$ is the set of rational numbers and let $R$ be an equivalence relation on it. Let $a$ be an irrational number, determine a rational number $r_0\in \mathbb{Q}$ such that $a\in R[r_0]$.
my attempt
let $a=\sqrt2$ and the equivalence relation is $R= \{(x, y) : x\frac{y}{x} \in \mathbb{Q}\}$
there exists rational number such that $a \frac{1}{a} = \frac{a}{a^2}\in \mathbb{Q}$
so $r_0 = \frac{a}{a}=1 $
then this fact is true, and I don’t know if this is the correct way to prove this.
could you help to check if this is correct, and if not, could you tell me where is the problem?
A:
The problem is that $\frac{a}{a}=1$ is not a rational number. That’s not very visible since it has the same numerator and denominator as $1$, but if we take the prime factorization of $1$, we see that the only rational factors are the trivial ones.
Edit: An obvious way to reach this conclusion would be to write $a=p_n^k$ for some positive integer $n,k$ and note that $p_n$ is a prime number, so for any $a$ we have $a=p_n^k$, $p_n$ divides $a$ and $a/p_n$ is irrational.
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