Is there a named set of numbers (containing, as a subset, the Normal Numbers) comprised of numbers that contain every finite sequence of digits at least once? (The Normal Numbers have all finite sequences of the same length present uniformly, which seems a stronger requirement.)

My question has relevance to the ending of the science fiction book (not film) "Contact" by Carl Sagan, where the heroine finds a "planted" message within the digits of pi. This is objectionable 1) philosophically: in euclidean space pi is a priori and thus can not be "synthesized" by any intelligence, and perhaps 2) mathematically: if pi is Normal (as it has been shown to be at least in base 8) then pi will contain all messages. Even if pi is only a member of the broader set I specify above, all messages must occur.

There exist similar questions on this site, e.g.: Patterns in pi in “Contact”

Steve Lord
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1 Answers1


The term "disjunctive in base $10$" has been used --- see Wikipedia for Disjunctive sequence

The term "lexicon" has been used --- see Calude/Zamfirescu’s 1998 paper The typical number is a lexicon

The following is an abstract for a talk given at the 32nd meeting of the Iowa MAA Section meeting, held on 15 April 1944 (American Mathematical Monthly 51 #7, August-September 1944, pp. 431-432).

  1. Repetitious numbers, by Professor E. S. Allen, Iowa State College, introduced by the Secretary. Professor Allen defined a repetitious number as one which, written decimally, has at some point every possible finite succession of digits. He pointed out that every such number has then each such succession an infinite number of times. An example of a repetitious number was given, and methods for obtaining others were outlined. For instance, he explained a method for obtaining from a single one a set which can be put into a one-to-one correspondence with all real numbers

In internet postings I’ve seen other names given to this property. Two such names that I know of off-hand are Omni-transcental numbers --- see this 18 February 2003 sci.math post and my 19 February 2003 reply --- and Alexandrian numbers --- see the 9 October 2013 Mathematics Stack Exchange question Which Well-Known Numbers Are Alexandrian.

I’ve written a fair amount about the messages in pi stuff in Carl Sagan’s 1985 book Contact. See my 23 October 2000 sci.math post and my 24 April 2001 math-teach post. Incidentally, much of what I’ve written about normal numbers up to September 2009 is summarized/collected together in this 18 September 2009 sci.math post.

There are also MANY related Stack Exchange questions that are worth looking at:

Prove there are no hidden messages in Pi (5 February 2011)

For any irrational number such as pi, would any sequence of length n appear in its decimal places? (27 June 2011)

Do the digits of $\pi$ contain every possible finite-length digit sequence? (5 January 2012)

Does Pi contain all possible number combinations? (18 October 2012)

Irrational Numbers Containing Other Irrational Numbers (8 January 2013)

Difference between irrational numbers with and without a pattern. (24 January 2013)

Do irrational number contain infinite/every patterns of sequences? (21 March 2013)

Infinite irrational number sequences? (7 May 2013)

Does the decimal representation of $\pi$ contain almost all numbers? (15 May 2013)

$\pi$, disjunctive numbers, and finite sequences of given length (8 September 2013)

Pi might contain all finite sets, can it also contain infinite sets? (11 September 2013)

Which Well-Known Numbers Are Alexandrian (9 October 2013)

Using decimals of $\pi$ to store data (14 July 2014)

Patterns in pi in “Contact” (14 January 2015)

Decimal Expansion of Pi (5 June 2015)

Randomness in pi and other irrational numbers (27 June 2015)

PI as an infinite set of integers (1 October 2015)

Where, if ever, does the decimal representation of $\pi$ repeat its initial segment? (28 October 2015)

Pi's Recursiveness (3 March 2016)

Are there infinite self-locating strings in the decimal expansion of $\pi$? (2 May 2016)

How to find sequence of digits in pi? (4 May 2016)

Can any number sequence be found in $\pi$? (21 July 2016)

Sequence in 10-base decimal representation of $\pi$ (4 March 2017)

Is the problem of answering the “Does [a random, but computable irrational number] contain all finite sequences of digits?" question computable? (10 March 2017)

Does digits of pi contain all possible substrings? (15 March 2017)

The set of normal numbers is uncountable (10 July 2017)

Dave L. Renfro
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