Normal Number - Properties and Examples

Properties and Examples

The concept of a normal number was introduced by Émile Borel in 1909. Using the Borel-Cantelli lemma, he proved the normal number theorem: almost all real numbers are normal, in the sense that the set of non-normal numbers has Lebesgue measure zero (Borel 1909). This theorem established the existence of normal numbers. In 1917, Waclaw Sierpinski showed that it is possible to specify a particular such number. Becher and Figueira proved in 2002 that there is a computable normal number.

The set of non-normal numbers, though "small" in the sense of being a null set, is "large" in the sense of being uncountable. For instance, there are uncountably many numbers whose decimal expansion does not contain the digit 5, and none of these are normal.

Champernowne's number

0.1234567891011121314151617...,

obtained by concatenating the decimal representations of the natural numbers in order, is normal in base 10, but it might not be normal in some other bases.

The Copeland–Erdős constant

0.235711131719232931374143...,

obtained by concatenating the prime numbers in base 10, is normal in base 10, as proved by Copeland and Erdős (1946). More generally, the latter authors proved that the real number represented in base b by the concatenation

0.f(1)f(2)f(3)...,

where f(n) is the nth prime expressed in base b, is normal in base b. Besicovitch (1935) proved that the number represented by the same expression, with f(n) = n2,

0.149162536496481100121144...,

obtained by concatenating the square numbers in base 10, is normal in base 10. Davenport & Erdős (1952) proved that the number represented by the same expression, with f being any polynomial whose values on the positive integers are positive integers, expressed in base 10, is normal in base 10.

Nakai & Shiokawa (1992) proved that if f(x) is any non-constant polynomial with real coefficients such that f(x) > 0 for all x > 0, then the real number represented by the concatenation

0....,

where is the integer part of f(n) expressed in base b, is normal in base b. (This result includes as special cases all of the above-mentioned results of Champernowne, Besicovitch, and Davenport & Erdős.) The authors also show that the same result holds even more generally when f is any function of the form

f(x) = α·xβ + α₁·xβ₁ + ... + α_d·xβ_d,

where the α's and β's are real numbers with β > β₁ > β₂ > ... > β_d ≥ 0, and f(x) > 0 for all x > 0.

Every Chaitin's constant is a normal number (Calude, 1994). A computable normal number was constructed in (Becher 2002). Although these constructions do not directly give the digits of the numbers constructed, the second shows that it is possible in principle to enumerate all the digits of a particular normal number.

Bailey and Crandall show an explicit uncountably infinite class of b-normal numbers by perturbing Stoneham numbers.

It has been an elusive goal to prove the normality of numbers which were not explicitly constructed for the purpose. It is for instance unknown whether √2, π, ln(2) or e is normal (but all of them are strongly conjectured to be normal, because of some empirical evidence). It is not even known whether all digits occur infinitely often in the decimal expansions of those constants. It has been conjectured that every irrational algebraic number is normal; while no counterexamples are known, there also exists no algebraic number that has been proven to be normal in any base.