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

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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. A normal number can be thought of as an infinite sequence of coin flips ( binary) or rolls of a die ( base 6). Using the Borel–Cantelli lemma, he proved that almost all real numbers are normal, establishing the existence of normal numbers. 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.

It is widely believed that the (computable) numbers √ 2, π, and e are normal, but a proof remains elusive. 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 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. 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. 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 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 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. 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. 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 been an elusive goal to prove the normality of numbers that are not artificially constructed. 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. 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 irrational, there are uncountably many numbers normal in each base but not the other.

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. displaystyle \alpha =\prod _{m=2} 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. 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. 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.

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