Difference between revisions of "Minkowski question mark"
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If $x \in \mathbb{R}$ is rational with continued fraction expansion $x=[a_0;a_1,a_2,\ldots,a_m]$ then define | If $x \in \mathbb{R}$ is rational with continued fraction expansion $x=[a_0;a_1,a_2,\ldots,a_m]$ then define | ||
$$?(x) = a_0 + 2\displaystyle\sum_{n=1}^m \dfrac{(-1)^{n+1}}{2^{a_1+\ldots+a_m}}.$$ | $$?(x) = a_0 + 2\displaystyle\sum_{n=1}^m \dfrac{(-1)^{n+1}}{2^{a_1+\ldots+a_m}}.$$ | ||
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Latest revision as of 18:29, 24 May 2016
If $x \in \mathbb{R}$ is irrational with continued fraction expansion $x=[a_0;a_1,a_2,\ldots]$ then define $$?(x) = a_0 + 2 \displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}}{2^{a_1+\ldots+a_n}}.$$ If $x \in \mathbb{R}$ is rational with continued fraction expansion $x=[a_0;a_1,a_2,\ldots,a_m]$ then define $$?(x) = a_0 + 2\displaystyle\sum_{n=1}^m \dfrac{(-1)^{n+1}}{2^{a_1+\ldots+a_m}}.$$
