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thomae
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There are 6 letters in THOMAE ( A1E1H4M3O1T1 )
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Thomae's function, named after Carl Johannes Thomae, has many names: the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function, the Riemann function, or the Stars over Babylon (John Horton Conway's name). This real-valued function f(x) of the real variable x is defined as: * * * * f * ( * x * ) * = * * * { * * * * * * 1 * q * * * * * * if * * x * = * * * * p * q * * * * * ( * x * * is rational), with * * p * ∈ * * Z * * * and * * q * ∈ * * N * * * coprime * * * * * * 0 * * * * if * * x * * is irrational. * * * * * * * * * * {\displaystyle f(x)={\begin{cases}{\frac {1}{q}}&{\text{if }}x={\tfrac {p}{q}}\quad (x{\text{ is rational), with }}p\in \mathbb {Z} {\text{ and }}q\in \mathbb {N} {\text{ coprime}}\\0&{\text{if }}x{\text{ is irrational.}}\end{cases}}} * Since every rational number has a unique representation with relatively prime * * * * p * ∈ * * Z * * * * * {\displaystyle p\in \mathbb {Z} \;} * and * * * * q * ∈ * * N * * * * {\displaystyle q\in \mathbb {N} } * , the function is well-defined. Note that * * * * + * 1 * * * {\displaystyle +1} * is the only number in * * * * * N * * * * {\displaystyle \mathbb {N} } * which is coprime to * * * * 0. * * * {\displaystyle 0.} * * It is a modification of the Dirichlet function, which is 1 at rational numbers and 0 elsewhere. |