A number is said to be simply normal to base if its base- expansion has each digit appearing with average frequency tending to .A normal number is an irrational number for which any finite pattern of numbers occurs with the expected limiting frequency in the expansion in a given base (or all bases). For example, for a normal decimal number, each digit 0-9 would be expected to occur 1/10 of the time, each pair of digits 00-99 would be expected to occur 1/100 of the time, etc. A number that is normal in base- is often called -normal.A number that is -normal for every , 3, ... is said to be absolutely normal (Bailey and Crandall 2003).As stated by Kac (1959), "As is often the case, it is much easier to prove that an overwhelming majority of objects possess a certain property than to exhibit even one such object....It is quite difficult to exhibit a 'normal' number!" (Stoneham 1970).If a real number is -normal, then it is also -normal for and integers (Kuipers..
A sequence of real numbers is equidistributed on an interval if the probability of finding in any subinterval is proportional to the subinterval length. The points of an equidistributed sequence form a dense set on the interval .However, dense sets need not necessarily be equidistributed. For example, , where is the fractional part, is dense in but not equidistributed, as illustrated above for to 5000 (left) and to (right)Hardy and Littlewood (1914) proved that the sequence , of power fractional parts is equidistributed for almost all real numbers (i.e., the exceptional set has Lebesgue measure zero). Exceptional numbers include the positive integers, the silver ratio (Finch 2003), and the golden ratio .The top set of above plots show the values of for equal to e, the Euler-Mascheroni constant , the golden ratio , and pi. Similarly, the bottom set of above plots show a histogram of the distribution of for these constants. Note that while most..
Champernowne's constant(1)(OEIS A033307) is the number obtained by concatenating the positive integers and interpreting them as decimal digits to the right of a decimal point. It is normal in base 10 (Champernowne 1933, Bailey and Crandall 2002). Mahler (1961) showed it to also be transcendental. The constant has been computed to digits by E. W. Weisstein (Jul. 3, 2013) using the Wolfram Language.The infinite sequence of digits in Champernowne's constant is sometimes known as Barbier's infinite word (Allouche and Shallit 2003, pp. 114, 299 and 334).The number of digits after concatenation of the first, second, ... primes are givenby 1, 2, 3, 4, 6, 8, 10, 12, 14, 16, ... (OEIS A068670).The Champernowne constant continued fraction contains sporadic very large terms, making the continued fraction difficult to calculate. However, the size of the continued fraction high-water marks display apparent patterns (Sikora..
A real number that is -normal for every base 2, 3, 4, ... is said to be absolutely normal. As proved by Borel (1922, p. 198), almost all real numbers in are absolutely normal (Niven 1956, p. 103; Stoneham 1970; Kuipers and Niederreiter 1974, p. 71; Bailey and Crandall 2002).The first specific construction of an absolutely normal number was by Sierpiński (1917), with another method presented by Schmidt (1962). These results were both obtained by complex constructive devices (Stoneham 1970), and are by no means easy to construct (Stoneham 1970, Sierpiński and Schinzel 1988).