Editing Normal number (section)

== Properties and examples ==
The concept of a normal number was introduced by {{harvs|first=Émile|last=Borel|authorlink=Émile Borel|year=1909|txt}}. Using the [[Borel–Cantelli lemma]], he proved that [[almost all]] real numbers are normal, establishing the existence of normal numbers. {{harvs|first=Wacław|last=Sierpiński|authorlink=Wacław Sierpiński|year=1917|txt}} showed that it is possible to specify a particular such number. {{harvs|last1=Becher|last2=Figueira|year=2002|txt}} proved that there is a [[computable number|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 [[#Non-normal numbers|every rational number]] (in fact, it is uncountably infinite{{sfn|Billingsley|2012}} 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 those numbers are normal.

[[Champernowne constant|Champernowne's constant]]

{{block indent|0.1234567891011121314151617181920212223242526272829...,}}

obtained by concatenating the decimal representations of the [[natural number]]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]]

{{block indent|0.23571113171923293137414347535961677173798389...,}}

obtained by concatenating the [[prime number]]s in base 10, is normal in base 10, as proved by {{harvs|first1=A. H.|last1=Copeland|author1-link=Arthur Herbert Copeland|first2=Paul|last2=Erdős|author2-link=Paul Erdős|year=1946|txt}}. More generally, the latter authors proved that the real number represented in base ''b'' by the concatenation 

{{block indent|0.''f''(1)''f''(2)''f''(3)...,}}

where ''f''(''n'') is the ''n''<sup>th</sup> prime expressed in base ''b'', is normal in base ''b''. {{harvs|last=Besicovitch|authorlink=Abram Samoilovitch Besicovitch|year=1935|txt}} proved that the number represented by the same expression, with ''f''(''n'') = ''n''<sup>2</sup>,

{{block indent|0.149162536496481100121144...,}}

obtained by concatenating the [[square number]]s in base 10, is normal in base 10. {{harvs|first1=Harold|last1=Davenport|author1-link=Harold Davenport|last2=Erdős|year=1952|txt}} 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.

{{harvs|last1=Nakai|last2=Shiokawa|year=1992|txt}} 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  

{{block indent|0.[''f''(1)][''f''(2)][''f''(3)]...,}}

where [''f''(''n'')] is the [[Floor and ceiling functions|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

{{block indent|1=''f''(''x'') = α·''x''<sup>β</sup> + α<sub>1</sub>·''x''<sup>β<sub>1</sub></sup> + ... + α<sub>''d''</sub>·''x''<sup>β<sub>''d''</sub></sup>,}}

where the αs and βs are real numbers with β > β<sub>1</sub> > β<sub>2</sub> > ... > β<sub>''d''</sub> ≥ 0, and ''f''(''x'') > 0 for all ''x'' > 0.

{{harvs|last1=Bailey|last2=Crandall|year=2002|txt}} show an explicit [[uncountably infinite]] class of ''b''-normal numbers by perturbing [[Stoneham number]]s.

{{anchor|Conjecture}}It has been an elusive goal to prove the normality of numbers that are not artificially constructed. While [[square root of 2|{{sqrt|2}}]], [[pi|π]], [[natural logarithm of 2|ln(2)]], and [[e (mathematical constant)|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).{{sfn|Bailey|Borwein|Calude|Dinneen|2012}} It has also been conjectured that every [[irrational number|irrational]] [[algebraic number]] is absolutely normal (which would imply that {{sqrt|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.

===Non-normal numbers===
No [[rational number]] is normal in any base, since the digit sequences of rational numbers are [[Repeating decimal|eventually periodic]].

{{harvs|last=Martin|year=2001|txt}} gives an example of an irrational number that is absolutely abnormal.{{sfn|Bugeaud|2012|page=113}}  Let

<math display="block">f\left(n\right) = \begin{cases}
n^\frac{f\left(n-1\right)}{n-1}, & n\in\mathbb{Z}\cap\left[3,\infty\right) \\
4, & n = 2
\end{cases}
</math>

<math display="block">\begin{align} & \alpha = \prod_{m=2}^\infty \left({1 - \frac{1}{f\left(m\right)}}\right) = \left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)\left(1-\frac{1}{64}\right)\left(1-\frac{1}{152587890625}\right)\left(1-\frac 1{6^{\left(5^{15}\right)}}\right)\ldots = \\ &=0.6562499999956991\underbrace{99999\ldots99999}_{23,747,291,559}8528404201690728\ldots\end{align}</math>

Then &alpha; 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 <math>X\subseteq\R^+</math> if the complement of ''X'' has measure 0.
* If ''x'' is normal in base ''b'' and ''a'' ≠ 0 is a rational number, then <math>x \cdot a</math> is also normal in base ''b''.{{sfn|Wall|1949}}
* If <math>A\subseteq\N</math> is ''dense'' (for every <math>\alpha<1</math> and for all sufficiently large ''n'', <math>|A \cap \{1,\ldots,n\}| \geq n^\alpha</math>) and <math>a_1,a_2,a_3,\ldots</math> are the base-''b'' expansions of the elements of ''A'', then the number <math>0.a_1a_2a_3\ldots</math>, 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]] 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''[1..''k''], the second length-''k'' block is ''S''[''k''+1..2''k''], etc.) This was implicit in the work of {{harvs|last1=Ziv|last2=Lempel|year=1978|txt}} and made explicit in the work of {{harvs|last1=Bourke|last2=Hitchcock|last3=Vinodchandran|year=2005|txt}}.
* A number is normal in base ''b'' if and only if it is simply normal in base ''b<sup>k</sup>'' for all <math>k\in\mathbb{Z}^{+}</math>. This follows from the previous block characterization of normality: Since the ''n''<sup>th</sup> block of length ''k'' in its base ''b'' expansion corresponds to the ''n''<sup>th</sup> digit in its base ''b<sup>k</sup>'' expansion, a number is simply normal in base ''b<sup>k</sup>'' 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 <math>m_1<m_2<m_3<\cdots</math> where the number is simply normal in bases ''b''<sup>''m''</sup> for all <math>m\in\{m_1,m_2,\ldots\}.</math>{{sfn|Long|1957}} 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 set|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.