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# Disjunctive numbers

A rich number or disjunctive number (in french: nombre univers; this might suggest the english translation: universe number, although this term is not used in the literature) is a real number whose expansion, in a given base ${\displaystyle b}$ is a disjunctive sequence over the alphabet ${\displaystyle \{0,...,b-1\}}$, i.e. it contains any finite sequence of digits, at least once, expressible in that base (in other words, it contains the whole "universe" of finite integers, at least once, expressible in base ${\displaystyle b}$). The simply normal numbers (in base ${\displaystyle b}$) constitute a subset of the disjonctive numbers (in same base ${\displaystyle b}$). (For simply normal numbers, in a given base ${\displaystyle b}$, every finite sequence of digits, expressible in that base, must appear infinitely often and with equiprobability between sequences with the same number of digits, while for disjunctive numbers, the only constraint is "at least once" with no say on the probalities.)

Thus, disjunctive numbers (in base ${\displaystyle b}$) contain, at least once, all ${\displaystyle n}$-digits approximations (ignoring the fractional point) for all real numbers (including at least a second occurrence of itself?), which means that disjunctive numbers are transcendental numbers and constitute an uncountable subset of the real numbers. Also, all ${\displaystyle n}$-digits approximations for all digital encodings of any past/present/future speech/music/book/image/photo/video/etc., whether finite or infinite (such as any picture of the Mandelbrot set with infinite resolution), is to be found at least once in a disjunctive number.

A number that is disjunctive to every base is called absolutely disjunctive or is said to be a lexicon. A set is called "residual" if it contains the intersection of a countable family of open dense sets. The set of absolutely disjunctive reals is a residual.[1] It is conjectured that every real irrational algebraic number is absolutely disjunctive.[2]

## Examples

The Champernowne constant (0,123456789101112...) is a disjunctive number, and normal, in base 10. The number defined by ${\displaystyle n}$ times the 0 digit inserted between ${\displaystyle n}$ and ${\displaystyle n+1}$ for all positive integers (0,10200300000040000000000005...) is a disjunctive number, but is not normal, in base 10.

## Is pi a disjunctive number?

It is not known whether ${\displaystyle \pi }$ = 3.1415926535897932384626433832795... is a disjunctive number.[3]

### Pi approximations within the decimal expansion of pi

Pi approximations within the decimal expansion of pi
${\displaystyle n}$ Digits Starting position
1 3 9
2 31 137
3 314 2120
4 3141 3496
5 31415 88008
6 314159 176451
7 3141592 25198140
8 31415926 50366472

A081876 ${\displaystyle a(n)}$ is the starting position of the second occurrence of a string of the initial ${\displaystyle n}$ decimal digits of Pi in the decimal expansion of Pi.

{9, 137, 2120, 3496, 88008, 176451, 25198140, 50366472, ...}

### e approximations within the decimal expansion of pi

${\displaystyle e}$ approximations within the decimal expansion of ${\displaystyle \pi }$
${\displaystyle n}$ Digits Starting position
1 2 6
2 27 28
3 271 241
4 2718 11706
5 27182 28024
6 271828 33789
7 2718281 1526800
8 27182818 73154827

A090898 Index of first occurrence of the first ${\displaystyle n}$ digits of ${\displaystyle e}$ in the decimal expansion of Pi.

{6, 28, 241, 11706, 28024, 33789, 1526800, 73154827, ...}

(...)

## Notes

1. Calude & Zamfirescu (1999).
2. Adamczewski & Bugeaud (2010) p.414.
3. André Boileau, Les nombres univers, Mathématiques ludiques et technologie, AMQ, octobre 2012 (Cégep de Ste-Foy). (Contains a tool to search for a base 10 digit sequence within the decimal expansion of ${\displaystyle \pi }$.)

## References

• Adamczewski, Boris; Bugeaud, Yann (2010). "8. Transcendence and diophantine approximation". in Berthé, Valérie; Rigo, Michael. Combinatorics, automata, and number theory. Encyclopedia of Mathematics and its Applications. 135. Cambridge: Cambridge University Press. p. 410–451. Zbl pre05879512. ISBN 978-0-521-51597-9.
• Calude, C.S.; Zamfirescu, T. (1999). “Most numbers obey no probability laws”. Publicationes Mathematicae Debrecen 54 (Supplement): pp. 619–623.