Normal Schmormal: My occasionally helpful guide to parenting kids with special needs (Down syndrome, autism, ADHD, neurodivergence)

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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. 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.

BaileyandCrandall( 2002) show an explicit uncountably infinite class of b-normal numbers by perturbing Stoneham numbers. No rational number is normal in any base, since the digit sequences of rational numbers are eventually periodic. 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.displaystyle \alpha =\prod _{m=2} 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).

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. For bases r and s with log r / log s irrational, there are uncountably many numbers normal in each base but not the other. 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.Intuitively, a number being simply normal means that no digit occurs more frequently than any other. Using the Borel–Cantelli lemma, he proved that almost all real numbers are normal, establishing the existence of normal numbers. It is widely believed that the (computable) numbers √ 2, π, and e are normal, but a proof remains elusive.

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). 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.We defined a number to be simply normal in base b if each individual digit appears with frequency 1⁄ b. m = 2 ∞ ( 1 − 1 f ( m ) ) = ( 1 − 1 4 ) ( 1 − 1 9 ) ( 1 − 1 64 ) ( 1 − 1 152587890625 ) ( 1 − 1 6 ( 5 15 ) ) … = 0.

It has been an elusive goal to prove the normality of numbers that are not artificially constructed. A number is said to be normal in base b if, for every positive integer n, all possible strings n digits long have density b − n.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. Consider the infinite digit sequence expansion S x, b of x in the base b positional number system (we ignore the decimal point).