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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
is said to be simply normal in an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
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 In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the ...
 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 ( binary) or rolls of a die (
base 6 A senary () numeral system (also known as base-6, heximal, or seximal) has six as its base. It has been adopted independently by a small number of cultures. Like decimal, it is a semiprime, though it is unique as the product of the only two con ...
). 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 In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathema ...
real numbers are normal (meaning that the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of non-normal numbers has
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wi ...
zero), this proof is not
constructive Although the general English usage of the adjective constructive is "helping to develop or improve something; helpful to someone, instead of upsetting and negative," as in the phrase "constructive criticism," in legal writing ''constructive'' has ...
, and only a few specific numbers have been shown to be normal. For example,
Chaitin's constant In the computer science subfield of algorithmic information theory, a Chaitin constant (Chaitin omega number) or halting probability is a real number that, informally speaking, represents the probability that a randomly constructed program will ...
is normal (and
uncomputable Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can do ...
). It is widely believed that the (computable) numbers , , and '' e'' are normal, but a proof remains elusive.


Definitions

Let be a finite
alphabet An alphabet is a standardized set of basic written graphemes (called letters) that represent the phonemes of certain spoken languages. Not all writing systems represent language in this way; in a syllabary, each character represents a syllab ...
of -digits, the set of all infinite
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
s that may be drawn from that alphabet, and the set of finite sequences, or
strings String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * ''Strings'' (1991 film), a Canadian anim ...
. 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 Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
\lim_ \frac = \frac for each . Now let be any finite string in and let be the number of times the string appears as a
substring In formal language theory and computer science, a substring is a contiguous sequence of characters within a string. For instance, "''the best of''" is a substring of "''It was the best of times''". In contrast, "''Itwastimes''" is a subsequenc ...
in the first digits of the sequence . (For instance, if , then .) is normal if, for all finite strings , \lim_ \frac = \frac 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 (a sequence over the alphabet ), and each occur with frequency ; , , , and each occur with frequency ; , , , , , , , and each occur with frequency ; etc. Roughly speaking, the
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
of finding the string in any given position in is precisely that expected if the sequence had been produced at
random In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual ra ...
. Suppose now that is an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
greater than 1 and is a
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
. Consider the infinite digit sequence expansion of in the base
positional number system Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
(we ignore the decimal point). 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 (or sometimes an absolutely 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 (in which case it is not a normal number). For bases and with
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
(so that and ) 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 A disjunctive sequence is an infinite sequence (over a finite alphabet of characters) in which every finite string appears as a substring. For instance, the binary Champernowne sequence :0\ 1\ 00\ 01\ 10\ 11\ 000\ 001 \ldots formed by concatenat ...
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 Rich may refer to: Common uses * Rich, an entity possessing wealth * Rich, an intense flavor, color, sound, texture, or feeling ** Rich (wine), a descriptor in wine tasting Places United States * Rich, Mississippi, an unincorporated comm ...
'' 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 Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
in the
unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
. denotes the fractional part of . 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 (but not normal or -dense,) -dense (but not simply normal or normal), normal (and thus simply normal and -dense), 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 In probability theory, the Borel–Cantelli lemma is a theorem about sequences of events. In general, it is a result in measure theory. It is named after Émile Borel and Francesco Paolo Cantelli, who gave statement to the lemma in the first de ...
, he proved that
almost all In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathema ...
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 Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is close ...
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 all the digits of a particular normal number. The set of non-normal numbers, despite being "large" in the sense of being
uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
, 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 and even
comeagre In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...
). 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 number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s 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 The Copeland–Erdős constant is the concatenation of "0." with the base 10 representations of the prime numbers in order. Its value, using the modern definition of prime, is approximately :0.235711131719232931374143… . The constant is irration ...
obtained by concatenating the
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s 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''(''n'') 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''(''n'') = ''n''2, obtained by concatenating the
square number In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
s 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''(''x'') is any non-constant
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
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 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 int ...
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 where the αs and βs are real numbers with β > β1 > β2 > ... > β''d'' ≥ 0, and ''f''(''x'') > 0 for all ''x'' > 0. show an explicit
uncountably infinite In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
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 , π,
ln(2) The decimal value of the natural logarithm of 2 is approximately :\ln 2 \approx 0.693\,147\,180\,559\,945\,309\,417\,232\,121\,458. The logarithm of 2 in other bases is obtained with the formula :\log_b 2 = \frac. The common logarithm in particula ...
, 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). It has also been conjectured that every
irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
algebraic number is absolutely normal (which would imply that is 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 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 ...
is normal in any base, since the digit sequences of rational numbers are eventually periodic. However, a rational number can be ''simply'' normal in a particular base. For example, \frac=\frac=0.\overline is simply normal in base 10. gives an example of an irrational number that is absolutely abnormal. Let f\left(n\right) = \begin n^\frac, & n\in\mathbb\cap\left[3,\infty\right) \\ 4, & n = 2 \end \alpha = \prod_^\infty \left(\right) = \left(1-\frac\right)\left(1-\frac\right)\left(1-\frac\right)\left(1-\frac\right)\left(1-\frac 1\right)\ldots=0.6562499999956991\underbrace_8528404201690728\ldots 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 X\subseteq\R^+ if the complement of ''X'' has measure 0. * If ''x'' is normal in base ''b'' and ''a'' ≠ 0 is a rational number, then x \cdot a is also normal in base ''b''. * If A\subseteq\N is ''dense'' (for every \alpha<1 and for all sufficiently large ''n'', , A \cap \, \geq n^\alpha) and a_1,a_2,a_3,\ldots are the base-''b'' expansions of the elements of ''A'', then the number 0.a_1a_2a_3\ldots, formed by concatenating the elements of ''A'', is normal in base ''b'' (Copeland and Erdős 1946). From this it follows that Champernowne's number is normal in base 10 (since the set of all positive integers is obviously dense) and that the Copeland–Erdős constant is normal in base 10 (since the prime number theorem implies that the set of primes is dense). * A sequence is normal
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...
every ''block'' of equal length appears with equal frequency. (A block of length ''k'' is a substring of length ''k'' appearing at a position in the sequence that is a multiple of ''k'': e.g. the first length-''k'' block in ''S'' is ''S'' ..''k'' the second length-''k'' block is ''S'' 'k''+1..2''k'' etc.) 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 k\in\mathbb^. 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 m_1 where the number is simply normal in bases ''b''''m'' for all m\in\. 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 Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
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 machine A finite-state machine (FSM) or finite-state automaton (FSA, plural: ''automata''), finite automaton, or simply a state machine, is a mathematical model of computation. It is an abstract machine that can be in exactly one of a finite number o ...
s and normal sequences: every infinite subsequence selected from a normal sequence by a
regular language In theoretical computer science and formal language theory, a regular language (also called a rational language) is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science (as opposed to ...
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 (FSGs) and information lossless finite-state compressors (ILFSCs). * A finite-state gambler (a.k.a. finite-state martingale) is a finite-state machine over a finite alphabet \Sigma, each of whose states is labelled with percentages of money to bet on each digit in \Sigma. For instance, for an FSG over the binary alphabet \Sigma = \, the current state ''q'' bets some percentage q_0 \in ,1/math> of the gambler's money on the bit 0, and the remaining q_1 = 1-q_0 fraction of the gambler's money on the bit 1. The money bet on the digit that comes next in the input (total money times percent bet) is multiplied by , \Sigma, , 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., if\limsup_ d(S \upharpoonright n) = \infty,where d(S \upharpoonright n) is the amount of money the gambler ''d'' has after reading the first ''n'' digits of ''S'' (see
limit superior In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a ...
). * A finite-state compressor is a finite-state machine with output strings labelling its state transitions, including possibly the empty string. (Since one digit is read from the input sequence for each state transition, it is necessary to be able to output the empty string in order to achieve any compression at all). 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 f: \Sigma^* \to \Sigma^* \times Q, mapping the input string of ''C'' to the output string and final state of ''C'', is 1–1. Compression techniques such as
Huffman coding In computer science and information theory, a Huffman code is a particular type of optimal prefix code that is commonly used for lossless data compression. The process of finding or using such a code proceeds by means of Huffman coding, an algo ...
or
Shannon–Fano coding In the field of data compression, Shannon–Fano coding, named after Claude Shannon and Robert Fano, is a name given to two different but related techniques for constructing a prefix code based on a set of symbols and their probabilities (estimat ...
can be implemented with ILFSCs. An ILFSC ''C'' compresses an infinite sequence ''S'' if\liminf_ \frac < 1,where , C(S \upharpoonright n), is the number of digits output by ''C'' after reading the first ''n'' digits of ''S''. The compression ratio (the
limit inferior In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For ...
above) 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 Converse may refer to: Mathematics and logic * Converse (logic), the result of reversing the two parts of a definite or implicational statement ** Converse implication, the converse of a material implication ** Converse nonimplication, a logical c ...
. Therefore: Ziv and Lempel showed: (they actually showed that the sequence's optimal compression ratio over all ILFSCs is exactly its ''
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
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. 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 sequence Intuitively, an algorithmically random sequence (or random sequence) is a Sequence#Infinite sequences in theoretical computer science, sequence of binary digits that appears random to any algorithm running on a (prefix-free or not) universal Turi ...
s, which are those infinite sequences that appear random to any algorithm (and in fact have similar gambling and compression characterizations with
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
s replacing finite-state machines).


Connection to equidistributed sequences

A number ''x'' is normal in base ''b''
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...
the sequence _^\infty is
equidistributed In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be equidistributed, or uniformly distributed, if the proportion of terms falling in a subinterval is proportional to the length of that subinterval. Such sequences ...
modulo 1, or equivalently, using
Weyl's criterion In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be equidistributed, or uniformly distributed, if the proportion of terms falling in a subinterval is proportional to the length of that subinterval. Such sequences ...
, if and only if \lim_\frac\sum_^e^=0 \quad\text m\geq 1. This connection leads to the terminology that ''x'' is normal in base β for any real number β if and only if the sequence \left(\right)_^\infty is equidistributed modulo 1.


Notes


See also

*
Champernowne constant In mathematics, the Champernowne constant is a transcendental real constant whose decimal expansion has important properties. It is named after economist and mathematician D. G. Champernowne, who published it as an undergraduate in 1933. For ...
*
De Bruijn sequence In combinatorial mathematics, a de Bruijn sequence of order ''n'' on a size-''k'' alphabet ''A'' is a cyclic sequence in which every possible length-''n'' string on ''A'' occurs exactly once as a substring (i.e., as a ''contiguous'' subseq ...
*
Infinite monkey theorem The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, such as the complete works of William Shakespeare. In fact, the monkey would ...
*
The Library of Babel "The Library of Babel" ( es, La biblioteca de Babel) is a short story by Argentine author and librarian Jorge Luis Borges (1899–1986), conceiving of a universe in the form of a vast library containing all possible 410-page books of a certain ...


References

* * * * * * * * * * * * * * * * * * * * * * * * *


Further reading

* * * * * * * *


External links


We are in Digits of Pi and Live Forever
by Clifford A. Pickover *{{MathWorld, title=Normal number, id=NormalNumber Number theory Sets of real numbers Irrational numbers