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A schizophrenic number (also known as mock rational number) is an irrational number that displays certain characteristics of rational numbers. ==Definition== The definition of schizophrenic numbers is given in ''The Universal Book of Mathematics'' as:〔.〕 :''An informal name for an irrational number that displays such persistent patterns in its decimal expansion, that it has the appearance of a rational number. A schizophrenic number can be obtained as follows. For any positive integer n let f(n) denote the integer given by the recurrence f(n) = 10 f(n − 1) + n with the initial value f(0) = 0. Thus, f(1) = 1, f(2) = 12, f(3) = 123, and so on. The square roots of f(n) for odd integers n give rise to a curious mixture appearing to be rational for periods, and then disintegrating into irrationality. This is illustrated by the first 500 digits of :'' :''The repeating strings become progressively shorter and the scrabbled strings become larger until eventually the repeating strings disappear. However, by increasing n we can forestall the disappearance of the repeating strings as long as we like. The repeating digits are always 1, 5, 6, 2, 4, 9, 6, 3, 9, 2, . . . .'' The sequence of numbers generated by the recurrence relation described above is: :0, 1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 1234567900, ... . The integer parts of their square roots, :1, 3, 11, 35, 111, 351, 1111, 3513, 11111, 35136, 111111, 351364, 1111111, ... , alternate between numbers with irregular digits and numbers with repeating digits, in a similar way to the alternations appearing within the fractional part of each square root. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Schizophrenic number」の詳細全文を読む スポンサード リンク
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