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

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

BaileyandCrandall( 2002) show an explicit uncountably infinite class of b-normal numbers by perturbing Stoneham numbers. 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. It has been an elusive goal to prove the normality of numbers that are not artificially constructed.

Consider the infinite digit sequence expansion S x, b of x in the base b positional number system (we ignore the decimal point). We defined a number to be simply normal in base b if each individual digit appears with frequency 1⁄ b. 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. 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.

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

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. 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).

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 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).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 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. It is widely believed that the (computable) numbers √ 2, π, and e are normal, but a proof remains elusive. 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). 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.