An automatic real number is a real number for which the base-b expansion is an automatic sequence. All such numbers are either rational or transcendental, but not a U-number. Rational numbers are k-automatic in base b for all k and b.
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Famous quotes containing the words automatic, real and/or number:
“What we learn for the sake of knowing, we hold; what we learn for the sake of accomplishing some ulterior end, we forget as soon as that end has been gained. This, too, is automatic action in the constitution of the mind itself, and it is fortunate and merciful that it is so, for otherwise our minds would be soon only rubbish-rooms.”
—Anna C. Brackett (1836–1911)
“Sometimes among our more sophisticated, self-styled intellectuals—and I say self-styled advisedly; the real intellectual I am not sure would ever feel this way—some of them are more concerned with appearance than they are with achievement. They are more concerned with style then they are with mortar, brick and concrete. They are more concerned with trivia and the superficial than they are with the things that have really built America.”
—Lyndon Baines Johnson (1908–1973)
“In the multitude of middle-aged men who go about their vocations in a daily course determined for them much in the same way as the tie of their cravats, there is always a good number who once meant to shape their own deeds and alter the world a little.”
—George Eliot [Mary Ann (or Marian)