Normal number

As a normal number a real number is called in mathematics occur under their Nachkommaziffern for each all possible k- digit number blocks of the same asymptotic relative frequencies.

Definition

Is an integer and x is an arbitrary real number. Now the number will x to the base b, respectively (see number system ). For each k- digit number block of digits of base b denote that number by which the numeric keypad among the first n digits of x occurs.

Normal number

The number x is called "normal " in base b if and only if

For all and all k- digit number blocks apply.

It can be shown that a number, x is normal to base b if the sequence

Uniformly distributed modulo 1.

Absolutely normal number

The number x is " absolutely normal " if it is normal to each base.

Just normal number

The number x is called " just normal " in base b if it satisfies the condition of a normal number for k = 1, ie for single-digit numeric keypads ( = digits).

For example, the number of (periodic block of 01 in base 2 ) is simply normal in base 2, since the digits 0 and 1 occur with equal frequency.

There is the following equivalence: the number x is normal to the base b if and only if they simply normal to each of the bases b, b2, b3, ... is.

Number of normal numbers

The term " normal number " was introduced in 1909 by Émile Borel. He also proved equal using the lemma of Borel - Cantelli that almost all ( in the Lebesgue sense) real numbers are normal or even absolutely normal.

The amount of non-normal numbers is uncountable, however, as it is easy to show by way of a construction corresponding to the Cantor'schen discontinuum.

Construction of normal numbers

Waclaw Sierpinski delivered in 1917 the first construction of a normal number. Verónica cup and Santiago Figueira 2002 gave an algorithm for computing the constructed of Sierpinski number. Chaitinsche the constant is an example of a non- predictable normal number.

David Gawen Champernowne gave in 1933 to the first explicit construction of a normal number, which is known as Champernowne number. In the decimal are the first places:

It is the result A033307 in OEIS and is formed by " chaining " of the natural numbers in base 10, the Champernowne number is not normal with respect to some other bases.

The Copeland - Erdős number, named after Arthur Herbert Copeland and Paul Erdős, is another example of a base 10 normal number sequence A33308 in OEIS. The first decimal as follows:

It is formed by the juxtaposition of all prime numbers in base 10.

Wolfgang Schmidt examined 1960, under which conditions, and numbers that are normal to the base, and the base are normally and showed: When a rational number ( equivalent to: if there are positive natural numbers m and n is ), then every to base normal number also to the base normal. The converse is also true, and even, if irrational, then, the set of numbers that are not normal to the base and normal to the base, the thickness of the continuum.

Abnormal numbers

A rational number can be normal to any base, since their representation is always periodic. There are also constructions of irrational numbers which are normal to any base ( we call such numbers " absolutely abnormal" ).

Kreiszahl

It is not known if irrational numbers in the decimal system as the number of circle, the Euler constant E or the natural logarithm of the number 2 are normal or not.

Of the mathematicians David H. Bailey and Richard E. Crandall to this day unproven conjecture was erected in 2001 that every irrational algebraic number could be normal.