In mathematics, a real number is said to be simply normal in an integer base b[1] if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b. A number is said to be normal in base b if, for every positive integer n, all possible strings n digits long have density bn.

Intuitively, a number being simply normal means that no digit occurs more frequently than any other. If a number is normal, no finite combination of digits of a given length occurs more frequently than any other combination of the same length. A normal number can be thought of as an infinite sequence of coin flips (binary) or rolls of a die (base 6). Even though there will be sequences such as 10, 100, or more consecutive tails (binary) or fives (base 6) or even 10, 100, or more repetitions of a sequence such as tail-head (two consecutive coin flips) or 6-1 (two consecutive rolls of a die), there will also be equally many of any other sequence of equal length. No digit or sequence is "favored".

A number is said to be normal (sometimes called absolutely normal) if it is normal in all integer bases greater than or equal to 2.

While a general proof can be given that almost all real numbers are normal (meaning that the set of non-normal numbers has Lebesgue measure zero),[2] this proof is not constructive, and only a few specific numbers have been shown to be normal. For example, any Chaitin's constant is normal (and uncomputable). It is widely believed that the (computable) numbers 2, π, and e are normal, but a proof remains elusive.[3]


Let Σ be a finite alphabet of b-digits, Σω the set of all infinite sequences that may be drawn from that alphabet, and Σ the set of finite sequences, or strings.[4] Let SΣω be such a sequence. For each a in Σ let NS(a, n) denote the number of times the digit a appears in the first n digits of the sequence S. We say that S is simply normal if the limit

for each a. Now let w be any finite string in Σ and let NS(w, n) be the number of times the string w appears as a substring in the first n digits of the sequence S. (For instance, if S = 01010101..., then NS(010, 8) = 3.) S is normal if, for all finite strings wΣ,

where |w| denotes the length of the string w. In other words, S is normal if all strings of equal length occur with equal asymptotic frequency. For example, in a normal binary sequence (a sequence over the alphabet {0,1}), 0 and 1 each occur with frequency 12; 00, 01, 10, and 11 each occur with frequency 14; 000, 001, 010, 011, 100, 101, 110, and 111 each occur with frequency 18; etc. Roughly speaking, the probability of finding the string w in any given position in S is precisely that expected if the sequence had been produced at random.

Suppose now that b is an integer greater than 1 and x is a real number. Consider the infinite digit sequence expansion Sx, b of x in the base b positional number system (we ignore the decimal point). We say that x is simply normal in base b if the sequence Sx, b is simply normal[5] and that x is normal in base b if the sequence Sx, b is normal.[6] The number x is called a normal number (or sometimes an absolutely normal number) if it is normal in base b for every integer b greater than 1.[7][8]

A given infinite sequence is either normal or not normal, whereas a real number, having a different base-b expansion for each integer b ≥ 2, may be normal in one base but not in another[9][10] (in which case it is not a normal number). For bases r and s with log r / log s rational (so that r = bm and s = bn) every number normal in base r is normal in base s. For bases r and s with log r / log s irrational, there are uncountably many numbers normal in each base but not the other.[10]

A disjunctive sequence is a sequence in which every finite string appears. A normal sequence is disjunctive, but a disjunctive sequence need not be normal. A rich number in base b is one whose expansion in base b is disjunctive:[11] one that is disjunctive to every base is called absolutely disjunctive or is said to be a lexicon. A number normal in base b is rich in base b, but not necessarily conversely. The real number x is rich in base b if and only if the set {x bn mod 1 : nN} is dense in the unit interval.[11][12]

We defined a number to be simply normal in base b if each individual digit appears with frequency 1b. For a given base b, a number can be simply normal (but not normal or rich), rich (but not simply normal or normal), normal (and thus simply normal and rich), or none of these. A number is absolutely non-normal or absolutely abnormal if it is not simply normal in any base.[7][13]

Properties and examples

The concept of a normal number was introduced by Émile Borel (1909). Using the Borel–Cantelli lemma, he proved that almost all real numbers are normal, establishing the existence of normal numbers. Wacław Sierpiński (1917) showed that it is possible to specify a particular such number. Becher and Figueira (2002) proved that there is a computable absolutely normal number. Although this construction does not directly give the digits of the numbers constructed, it shows that it is possible in principle to enumerate each digit of a particular normal number.

The set of non-normal numbers, despite being "large" in the sense of being uncountable, is also a null set (as its Lebesgue measure as a subset of the real numbers is zero, so it essentially takes up no space within the real numbers). Also, the non-normal numbers (as well as the normal numbers) are dense in the reals: the set of non-normal numbers between two distinct real numbers is non-empty since it contains every rational number (in fact, it is uncountably infinite[14] and even comeagre). For instance, there are uncountably many numbers whose decimal expansions (in base 3 or higher) do not contain the digit 1, and none of these numbers is normal.

Champernowne's constant


obtained by concatenating the decimal representations of the natural numbers in order, is normal in base 10. Likewise, the different variants of Champernowne's constant (done by performing the same concatenation in other bases) are normal in their respective bases (for example, the base-2 Champernowne constant is normal in base 2), but they have not been proven to be normal in other bases.

The Copeland–Erdős constant


obtained by concatenating the prime numbers in base 10, is normal in base 10, as proved by A. H. Copeland and Paul Erdős (1946). More generally, the latter authors proved that the real number represented in base b by the concatenation


where f(n) is the nth prime expressed in base b, is normal in base b. Besicovitch (1935) proved that the number represented by the same expression, with f(n) = n2,


obtained by concatenating the square numbers in base 10, is normal in base 10. Harold Davenport and Erdős (1952) proved that the number represented by the same expression, with f being any non-constant polynomial whose values on the positive integers are positive integers, expressed in base 10, is normal in base 10.

Nakai and Shiokawa (1992) proved that if f(x) is any non-constant polynomial with real coefficients such that f(x) > 0 for all x > 0, then the real number represented by the concatenation


where [f(n)] is the integer part of f(n) expressed in base b, is normal in base b. (This result includes as special cases all of the above-mentioned results of Champernowne, Besicovitch, and Davenport & Erdős.) The authors also show that the same result holds even more generally when f is any function of the form

f(x) = α·xβ + α1·xβ1 + ... + αd·xβd,

where the αs and βs are real numbers with β > β1 > β2 > ... > βd ≥ 0, and f(x) > 0 for all x > 0.

Bailey and Crandall (2002) show an explicit uncountably infinite class of b-normal numbers by perturbing Stoneham numbers.

It has been an elusive goal to prove the normality of numbers that are not artificially constructed. While 2, π, ln(2), and e are strongly conjectured to be normal, it is still not known whether they are normal or not. It has not even been proven that all digits actually occur infinitely many times in the decimal expansions of those constants (for example, in the case of π, the popular claim "every string of numbers eventually occurs in π" is not known to be true).[15] It has also been conjectured that every irrational algebraic number is absolutely normal (which would imply that 2 is normal), and no counterexamples are known in any base. However, no irrational algebraic number has been proven to be normal in any base.

Martin (2001) gives an example of an irrational number that is absolutely abnormal.[16] Let

Then α is a Liouville number and is absolutely abnormal.

Non-normal numbers

No rational number is normal in any base, since the digit sequences of rational numbers are eventually periodic.


Additional properties of normal numbers include:

Connection to finite-state machines

Agafonov showed an early connection between finite-state machines and normal sequences: every infinite subsequence selected from a normal sequence by a regular language is also normal. In other words, if one runs a finite-state machine on a normal sequence, where each of the finite-state machine's states are labeled either "output" or "no output", and the machine outputs the digit it reads next after entering an "output" state, but does not output the next digit after entering a "no output state", then the sequence it outputs will be normal.[19]

A deeper connection exists with finite-state gamblers (FSGs) and information lossless finite-state compressors (ILFSCs).

Schnorr and Stimm showed that no FSG can succeed on any normal sequence, and Bourke, Hitchcock and Vinodchandran showed the converse. Therefore:

A sequence is normal if and only if there is no finite-state gambler that succeeds on it.

Ziv and Lempel showed:

A sequence is normal if and only if it is incompressible by any information lossless finite-state compressor

(they actually showed that the sequence's optimal compression ratio over all ILFSCs is exactly its entropy rate, a quantitative measure of its deviation from normality, which is 1 exactly when the sequence is normal). Since the LZ compression algorithm compresses asymptotically as well as any ILFSC, this means that the LZ compression algorithm can compress any non-normal sequence.[20]

These characterizations of normal sequences can be interpreted to mean that "normal" = "finite-state random"; i.e., the normal sequences are precisely those that appear random to any finite-state machine. Compare this with the algorithmically random sequences, which are those infinite sequences that appear random to any algorithm (and in fact have similar gambling and compression characterizations with Turing machines replacing finite-state machines).

Connection to equidistributed sequences

A number x is normal in base b if and only if the sequence is equidistributed modulo 1,[21][22] or equivalently, using Weyl's criterion, if and only if

This connection leads to the terminology that x is normal in base β for any real number β if and only if the sequence is equidistributed modulo 1.[22]


  1. ^ The only bases considered here are natural numbers greater than 1
  2. ^ Beck 2009.
  3. ^ Bailey & Crandall 2002.
  4. ^ ω is the smallest infinite ordinal number; is the Kleene star.
  5. ^ Bugeaud 2012, p. 78.
  6. ^ Bugeaud 2012, p. 79.
  7. ^ a b Bugeaud 2012, p. 102.
  8. ^ Adamczewski & Bugeaud 2010, p. 413.
  9. ^ Cassels 1959.
  10. ^ a b Schmidt 1960.
  11. ^ a b Bugeaud 2012, p. 92.
  12. ^ x bn mod 1 denotes the fractional part of x bn.
  13. ^ Martin 2001.
  14. ^ Billingsley 2012.
  15. ^ Bailey et al. 2012.
  16. ^ Bugeaud 2012, p. 113.
  17. ^ Wall 1949.
  18. ^ Long 1957.
  19. ^ Agafonov 1968.
  20. ^ Ziv & Lempel 1978.
  21. ^ Bugeaud 2012, p. 89.
  22. ^ a b Everest et al. 2003, p. 127.

See also


Further reading