A schizophrenic number (also known as mock rational number) is an

irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...

that displays certain characteristics of

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...

Definition

The Universal Book of Mathematics ''The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes'' (2004) is a bestselling book by British author David Darling (astronomer), David Darling. Summary The book is presented in a dictionary format. The book is divided into ...

'' defines "schizophrenic number" as: The sequence of numbers generated by the recurrence relation ''f''(''n'') = 10 ''f''(''n'' − 1) + ''n'' described above is: :0, 1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 1234567900, ... . :''f''(49) = 1234567901234567901234567901234567901234567901229 The

integer part In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least inte ...

s 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 The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part can ...

of each square root.

Characteristics

The ''schizophrenic number'' shown above is the special case of a more general phenomenon that appears in the

b

-ary expansions of square roots of the solutions of the recurrence

f_b(n)=b f_b(n-1)+n

, for all

b\geq2

, with initial value

f(0) = 0

taken at odd positive integers

n

. The case

b=10

and

n=49

corresponds to the example above. Indeed, Tóth showed that these irrational numbers present ''schizophrenic patterns'' within their

b

-ary expansion, composed of blocks that begin with a non-repeating digit block followed by a repeating digit block. When put together in base

b

, these blocks form the ''schizophrenic'' pattern. For instance, in base 8, the number

\sqrt

begins:

1111111111111111111111111.1111111111111111111111 0600
444444444444444444444444444444444444444444444 02144
333333333333333333333333333333333333333333 175124422
666666666666666666666666666666666666666 ....

The pattern is due to the

Taylor expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...

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.

Other properties

In some cases, instead of repeating digit sequences we find repeating ''digit patterns''. For instance, the number

\sqrt

1111111111111111111111111.1111111111111111111111111111111 01200 
202020202020202020202020202020202020202020 11010102 
00120012000012001200120012001200120012 0010
21120020211210002112100021121000211210 ...

shows repeating digit patterns in base

3

. Numbers that are ''schizophrenic'' in base

b

are also ''schizophrenic'' in base

b^m

(up to a certain limit, see Tóth). An example is

\sqrt

above, which is still schizophrenic in base

9

1444444444444.4444444444 350
666666666666666666666 4112
0505050505050505050 337506
75307530753075307 40552382 ...

History

Clifford A. Pickover Clifford Alan Pickover (born August 15, 1957) is an American author, editor, and columnist in the fields of science, mathematics, science fiction, innovation, and creativity. For many years, he was employed at the IBM Thomas J. Watson Research ...

has said that the schizophrenic numbers were discovered by Kevin Brown. In his book '' Wonders of Numbers'' he has so described the history of schizophrenic numbers:

References

External links