The Universal Book of Mathematics defines "schizophrenic number" 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.
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.
However, by increasing n we can forestall the disappearance of the repeating strings as long as we like.
[1]The sequence of numbers generated by the recurrence relation f (n) = 10 f (n − 1) + n described above is: The integer parts of their square roots, alternate between numbers with irregular digits and numbers with repeating digits, in a similar way to the alternations appearing within the decimal part of each square root.
The schizophrenic number shown above is the special case of a more general phenomenon that appears in the
-ary expansions of square roots of the solutions of the recurrence
Indeed, Tóth showed that these irrational numbers present schizophrenic patterns within their
, these blocks form the schizophrenic pattern.
begins: The pattern is due to the Taylor expansion of the square root of the recurrence's solution taken at odd positive integers.
The various digit contributions of the Taylor expansion yield the non-repeating and repeating digit blocks that form the schizophrenic pattern.
In some cases, instead of repeating digit sequences, we find repeating digit patterns.
: shows repeating digit patterns in base
: Clifford A. Pickover has said that the schizophrenic numbers were discovered by Kevin Brown.
In Wonders of Numbers Pickover described the history of schizophrenic numbers thus: The construction and discovery of schizophrenic numbers was prompted by a claim (posted in the Usenet newsgroup sci.math) that the digits of an irrational number chosen at random would not be expected to display obvious patterns in the first 100 digits.
It was said that if such a pattern were found, it would be irrefutable proof of the existence of either God or extraterrestrial intelligence.
Transcendental numbers like e and π, and noninteger surds such as square root of 2 are irrational.