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

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No rational number is normal in any base, since the digit sequences of rational numbers are eventually periodic. 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). 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. We say that x is simply normal in base b if the sequence S x, b is simply normal [5] and that x is normal in base b if the sequence S x, b is normal.

It is widely believed that the (computable) numbers √ 2, π, and e are normal, but a proof remains elusive. 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. We defined a number to be simply normal in base b if each individual digit appears with frequency 1⁄ b. For each a in Σ let N S( a, n) denote the number of times the digit a appears in the first n digits of the sequence S.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.

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).

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. Even though there will be sequences such as 10, 100, or more consecutive tails (binary) or fives (base 6) or even 10, 100, or more repetitions of a sequence such as tail-head (two consecutive coin flips) or 6-1 (two consecutive rolls of a die), there will also be equally many of any other sequence of equal length. 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. 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. Using the Borel–Cantelli lemma, he proved that almost all real numbers are normal, establishing the existence of normal numbers.

Consider the infinite digit sequence expansion S x, b of x in the base b positional number system (we ignore the decimal point). 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. For bases r and s with log r / log s irrational, there are uncountably many numbers normal in each base but not the other. 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. 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.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 example, in a normal binary sequence (a sequence over the alphabet { 0, 1}), 0 and 1 each occur with frequency 1⁄ 2; 00, 01, 10, and 11 each occur with frequency 1⁄ 4; 000, 001, 010, 011, 100, 101, 110, and 111 each occur with frequency 1⁄ 8; etc. 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. Intuitively, a number being simply normal means that no digit occurs more frequently than any other.

BaileyandCrandall( 2002) show an explicit uncountably infinite class of b-normal numbers by perturbing Stoneham numbers.If a number is normal, no finite combination of digits of a given length occurs more frequently than any other combination of the same length. While √ 2, π, ln(2), and e are strongly conjectured to be normal, it is still not known whether they are normal or not. It has been an elusive goal to prove the normality of numbers that are not artificially constructed. In mathematics, a real number is said to be simply normal in an integer base b [1] if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density1/ b.