Consider the infinite digit sequence expansion S x, b of x in the base b positional number system (we ignore the decimal point). While √ 2, π, ln(2), and e are strongly conjectured to be normal, it is still not known whether they are normal or not. For a given base b, a number can be simply normal (but not normal or b-dense, [ clarification needed]) b-dense (but not simply normal or normal), normal (and thus simply normal and b-dense), or none of these. 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).

While a general proof can be given that almost all real numbers are normal (meaning that the set of non-normal numbers has Lebesgue measure zero), [2] this proof is not constructive, and only a few specific numbers have been shown to be normal.Now let w be any finite string in Σ ∗ and let N S( w, n) be the number of times the string w appears as a substring in the first n digits of the sequence S.

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. 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 [14] and even comeagre).Roughly speaking, the probability of finding the string w in any given position in S is precisely that expected if the sequence had been produced at random. The real number x is rich in base b if and only if the set { x b n mod 1: n ∈ N} is dense in the unit interval.

HaroldDavenportandErdős( 1952) 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. Let Σ be a finite alphabet of b-digits, Σ ω the set of all infinite sequences that may be drawn from that alphabet, and Σ ∗ the set of finite sequences, or strings. It has also been conjectured that every irrational algebraic number is absolutely normal (which would imply that √ 2 is normal), and no counterexamples are known in any base. BaileyandCrandall( 2002) show an explicit uncountably infinite class of b-normal numbers by perturbing Stoneham numbers. A given infinite sequence is either normal or not normal, whereas a real number, having a different base- b expansion for each integer b ≥ 2, may be normal in one base but not in another [9] [10] (in which case it is not a normal number).For bases r and s with log r / log s rational (so that r = b m and s = b n) every number normal in base r is normal in base s. displaystyle \alpha =\prod _{m=2} We defined a number to be simply normal in base b if each individual digit appears with frequency 1⁄ b.