Editing Normal number (section)

===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.