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

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We defined a number to be simply normal in base b if each individual digit appears with frequency 1⁄ b. 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. 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.

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.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).

Consider the infinite digit sequence expansion S x, b of x in the base b positional number system (we ignore the decimal point).

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. Using the Borel–Cantelli lemma, he proved that almost all real numbers are normal, establishing the existence of normal numbers.

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. m = 2 ∞ ( 1 − 1 f ( m ) ) = ( 1 − 1 4 ) ( 1 − 1 9 ) ( 1 − 1 64 ) ( 1 − 1 152587890625 ) ( 1 − 1 6 ( 5 15 ) ) … = 0. 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. 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).

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. It has been an elusive goal to prove the normality of numbers that are not artificially constructed.