Normal number


In mathematics, a real number is said to be simply normal in an integer base b 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 b−n.
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 or rolls of a die. Even though there will be sequences such as 10, 100, or more consecutive tails or fives or even 10, 100, or more repetitions of a sequence such as tail-head or 6-1, 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 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, 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. It is widely believed that the numbers [square root of 2|], [pi|], and e are normal, but a proof remains elusive.

Definitions

Let be a finite alphabet of -digits, the set of all infinite sequences that may be drawn from that alphabet, and the set of finite sequences, or strings. Let be such a sequence. For each in let denote the number of times the digit appears in the first digits of the sequence. We say that is simply normal if the limit
for each. Now let be any finite string in and let be the number of times the string appears as a substring in the first digits of the sequence. is normal if, for all finite strings,
where denotes the length of the string. In other words, is normal if all strings of equal length occur with equal asymptotic frequency. For example, in a normal binary sequence, and each occur with frequency ;,,, and each occur with frequency ;,,,,,,, and each occur with frequency ; etc. Roughly speaking, the probability of finding the string in any given position in is precisely that expected if the sequence had been produced at random.
Suppose now that is an integer greater than 1 and is a real number. Consider the infinite digit sequence expansion of in the base positional number system. We say that is simply normal in base if the sequence is simply normal and that is normal in base if the sequence is normal. The number is called a normal number if it is normal in base for every integer greater than 1.
A given infinite sequence is either normal or not normal, whereas a real number, having a different base- expansion for each integer, may be normal in one base but not in another. For bases and with rational every number normal in base is normal in base. For bases and with irrational, there are uncountably many numbers normal in each base but not the other.
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 is one whose expansion in base is disjunctive: one that is disjunctive to every base is called absolutely disjunctive or is said to be a lexicon. A number normal in base is rich in base, but not necessarily conversely. The real number is rich in base if and only if the set is dense in the unit interval.
We defined a number to be simply normal in base if each individual digit appears with frequency. For a given base, a number can be simply normal, rich, normal, or none of these. A number is absolutely non-normal or absolutely abnormal if it is not simply normal in any base.

Properties and examples

The concept of a normal number was introduced by. Using the Borel–Cantelli lemma, he proved that almost all real numbers are normal, establishing the existence of normal numbers. showed that it is possible to specify a particular such number. 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. Also, the non-normal numbers are dense in the reals: the set of non-normal numbers between two distinct real numbers is non-empty since it contains [#Non-normal numbers|every rational number]. For instance, there are uncountably many numbers whose decimal expansions do not contain the digit 1, and none of those numbers are 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 are normal in their respective bases, 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. More generally, the latter authors proved that the real number represented in base b by the concatenation
where f is the n-th prime expressed in base b, is normal in base b. proved that the number represented by the same expression, with f = n2,
obtained by concatenating the square numbers in base 10, is normal in base 10. 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.
proved that if f is any non-constant polynomial with real coefficients such that f > 0 for all x > 0, then the real number represented by the concatenation
where is the integer part of f expressed in base b, is normal in base b. The authors also show that the same result holds even more generally when f is any function of the form
where the αs and βs are real numbers with β > β1 > β2 >... > βd ≥ 0, and f > 0 for all x > 0.
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 [square root of 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. It has also been conjectured that every irrational algebraic number is absolutely normal, and no counterexamples are known in any base. However, no irrational algebraic number has been proven to be normal in any base.

Non-normal numbers

No rational number is normal in any base, since the digit sequence of a rational number is eventually periodic, and thus most of the strings that are longer than the period do not appear in the digit sequence.
gives an example of an irrational number that is absolutely abnormal. Let
Then α is a Liouville number and is absolutely abnormal.

Properties

Additional properties of normal numbers include:
  • Every non-zero real number is the product of two normal numbers. This follows from the general fact that every number is the product of two numbers from a set if the complement of X has measure 0.
  • If x is normal in base b, and a ≠ 0 is a rational number, then is also normal in base b.
  • If is dense, and are the base-b expansions of the elements of A, then the number, formed by concatenating the elements of A, is normal in base b. From this it follows that Champernowne's number is normal in base 10 and that the Copeland–Erdős constant is normal in base 10.
  • A sequence is normal if and only if every block of equal length appears with equal frequency. This was implicit in the work of and made explicit in the work of.
  • A number is normal in base b if and only if it is simply normal in base bk for all. This follows from the previous block characterization of normality: Since the n-th block of length k in its base b expansion corresponds to the n-th digit in its base bk expansion, a number is simply normal in base bk if and only if blocks of length k appear in its base b expansion with equal frequency.
  • A number is normal if and only if it is simply normal in every base. This follows from the previous characterization of base b normality.
  • A number is b-normal if and only if there exists a set of positive integers where the number is simply normal in bases bm for all No finite set suffices to show that the number is b-normal.
  • All normal sequences are closed under finite variations: adding, removing, or changing a finite number of digits in any normal sequence leaves it normal. Similarly, if a finite number of digits are added to, removed from, or changed in any simply normal sequence, the new sequence is still simply normal.

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.
A deeper connection exists with finite-state gamblers and information lossless finite-state compressors.
  • A finite-state gambler is a finite-state machine over a finite alphabet, each of whose states is labelled with percentages of money to bet on each digit in. For instance, for an FSG over the binary alphabet, the current state q bets some percentage of the gambler's money on the bit 0, and the remaining fraction of the gambler's money on the bit 1. The money bet on the digit that comes next in the input is multiplied by, and the rest of the money is lost. After the bit is read, the FSG transitions to the next state according to the input it received. A FSG d succeeds on an infinite sequence S if, starting from $1, it makes unbounded money betting on the sequence; i.e., ifwhere is the amount of money the gambler d has after reading the first n digits of S.
  • A finite-state compressor is a finite-state machine with output strings labelling its state transitions, including possibly the empty string.. An information lossless finite-state compressor is a finite-state compressor whose input can be uniquely recovered from its output and final state. In other words, for a finite-state compressor C with state set Q, C is information lossless if the function, mapping the input string of C to the output string and final state of C, is 1–1. Compression techniques such as Huffman coding or Shannon–Fano coding can be implemented with ILFSCs. An ILFSC C compresses an infinite sequence S ifwhere is the number of digits output by C after reading the first n digits of S. The compression ratio can always be made to equal 1 by the 1-state ILFSC that simply copies its input to the output.
Schnorr and Stimm showed that no FSG can succeed on any normal sequence, and Bourke, Hitchcock and Vinodchandran showed the converse. Therefore:
Ziv and Lempel showed:
. 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.
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.

Connection to equidistributed sequences

A number x is normal in base b if and only if the sequence is equidistributed modulo 1, 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.