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I would like to say that at least to some extent we need the real numbers and not just rational approximations.

Suppose we were being completely formal and we asked the question, ``Is there some algebraic number $\beta>0$ such that for all $n$ there exist $p_n , q_n \in \mathbf{Z}_{> 0}$ with $\beta \ne \frac{p_n}{q_n}$ and there exists a fixed $\delta >1$ such that $|\beta - \frac{p_n}{q_n}| < \frac{1}{q_n^{1+\delta}}$'' ?

The answer of course is no by the Thue-Siegel-Roth Theorem, and this method is currently how we know that $\zeta(3)$ is not algebraic (more details in this discussion http://mathoverflow.net/questions/25630/major-mathematical-advances-past-age-fifty ). But moreover, we know that we can only discover a measure zero subset of the transcendental numbers in this way. So while we can approximate real numbers by rationals all day long, the amount of information we get out of a rational approximation can vary wildly.
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I could not come up with a high concept reason, aside from perhaps the offensively simple one that sometimes number theorists like to measure things like integers or rationals or algebraic numbers (at least the best that is possible).

I would however like to say that at least to some extent we need the real numbers and not just rational approximations.

Suppose we were being completely formal and we asked the question, ``Is there some algebraic number $\beta$ \beta>0$ such that for all $n$ there exist $p_n , q_n \in \mathbf{Z}_{> 0}$ with $\beta \ne \frac{p_n}{q_n}$ and there exists a fixed $\delta >1$ such that $|\beta - \frac{p_n}{q_n}| < \frac{1}{q_n^{1+\delta}}$'' ?

The answer of course is no by the Thue-Siegel-Roth Theorem, and this method is currently how we know that $\zeta(3)$ is not algebraic (more details in this discussion http://mathoverflow.net/questions/25630/major-mathematical-advances-past-age-fifty ).

But moreover, we know that we can only discover a measure zero subset of the transcendental numbers in this way. So while we can approximate real numbers by rationals all day long, the amount of information we get out of a rational approximation can vary wildly.
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I could not come up with a high concept reason, aside from perhaps the offensively simple one that sometimes number theorists like to measure things like integers or rationals or algebraic numbers (at least the best that is possible).

I would however like to say that at least to some extent we need the real numbers and not just rational approximations.

Suppose we were being completely formal and we asked the question, ``Is there some algebraic number $\beta$ such that for all $n$ there exist $p_n , q_n \in \mathbf{Z}_{> 0}$ with $\beta \ne \frac{p_n}{q_n}$ and there exists a fixed $\delta >1$ such that $|\beta - \frac{p_n}{q_n}| < \frac{1}{q_n^{1+\delta}}$'' ?

The answer of course is no by the Thue-Siegel-Roth Theorem, and this method is currently how we know that $\zeta(3)$ is not algebraic (more details in this discussion http://mathoverflow.net/questions/25630/major-mathematical-advances-past-age-fifty ).

But moreover, we know that we can only discover a measure zero subset of the transcendental numbers in this way. So while we can approximate real numbers by rationals all day long, the amount of information we get out of a rational approximation can vary wildly.