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Let $p$ be a prime number and $K$ be a field with embeddings into$\mathbb{R}$ and $\mathbb{Q}_p$. We propose an algorithm that generatescontinued fraction expansions converging in $\mathbb{Q}_p$ and is expected tosimultaneously converge in both $\mathbb{R}$ and $\mathbb{Q}_p$. This algorithmproduces finite continued fraction expansions for rational numbers. In the caseof $p=2$ and if $K$ is a quadratic field, the continued fraction expansionsgenerated by this algorithm converge in $\mathbb{R}$, and they are eventuallyperiodic or finite. For an element $\alpha$ in $K$, let $p_n/q_n$ denote the$n$-th convergent. There exist constants $u_1$ and $u_2$ in ${\mathbb R}_{>0}$with $u_1 + u_2 = 2$, and constants $C_1$ and $C_2$ in ${\mathbb R}_{>0}$ suchthat $|\alpha - p_n/q_n| < C_1/|q_n|^{u_1}$ and $|\alpha - p_n/q_n|_2 <C_2/|q_n|^{u_2}$. Here, $|\cdot|_2$ represents the $2$-adic distance. For primenumbers $p > 2$, we present numerical experiences.

Simultaneous Convergent Continued Fraction Algorithm for Real and $p$-adic Fields with Applications to Quadratic Fields