In Section 4.2, we have constructed explicitly real numbers which are normal to a given base. In the first section of this chapter, we describe another class of explicit real numbers with the same property. Then, we discuss the existence of explicit examples of absolutely normal numbers.

Definition 5.1. A real number is called absolutely normal if it is normal to every integer base b ≥ 2. A real number is called absolutely non-normal if it is normal to no integer base b ≥ 2.

We briefly and partially mention in Section 5.2 the point of view of complexity and calculability theory. Then, in Section 5.3, we give an explicit example of an absolutely non-normal irrational number. We end this chapter with some words on a method proposed by Bailey and Crandall to investigate the random character of arithmetical constants.

Korobov's and Stoneham's normal numbers

In 1946 Good [324] introduced the so-called ‘normal recurring decimals’. Integers b ≥ 2and k ≥ 1 being given, he constructed rational numbers ξ whose b-ary expansion has period bk and is such that every sequence of k digits from {0, 1,…,b − 1} occurs in the b-ary expansion of ξ with the same frequency b−k. An example with b = 2 and k = 3 is given by the rational 23/255 with purely periodic binary expansion of period 00010111. A similar result was independently proved by de Bruijn [143] also in 1946.