= Solution
In the absolute-value description, a <place of a number field> is an equivalence class of nontrivial absolute values, where equivalent absolute values give the same topology, or differ by a positive power. More generally this defines the rank-one absolute-value places used in weak approximation for a field. For a <number field>, finite places correspond to nonzero <prime ideals> of its <ring of integers>, while infinite places correspond to real embeddings and pairs of conjugate complex embeddings. The completion at $v$ is denoted $K_v$.
A <valuation> place can also be described by a <valuation> ring $\mathcal O_v\subset K$ and its residue map into $k_v$, extended by sending elements outside that ring to infinity. The finite places above are the discrete-valuation instances of this description.
The <weak approximation theorem> states that for finitely many pairwise inequivalent nontrivial absolute values $v_1,\ldots,v_r$, the diagonal image of $K$ is dense in $\prod_iK_{v_i}$. Equivalently, for arbitrary $a_i\in K_{v_i}$ and $\epsilon_i>0$, there is $a\in K$ such that
$$
\boxed{|a-a_i|_{v_i}<\epsilon_i\quad(1\leq i\leq r).}
$$
The places must be distinct: two requirements at the same completion need not be compatible. In the number-field setting this permits simultaneous prescribed finite congruences and archimedean approximations.
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