In an unpublished manuscript Turing anticipated by nearly thirty years the basic ideas behind the theory of algorithmic randomness, using a computationally constrained version of ‘measure theory’ to answer a question posed by Émile Borel in number theory: this question concerned constructing what are called ‘absolutely normal’ numbers. In this chapter we explain what these mysterious terms mean, and what Turing did. Mathematicians have always been fascinated with patterns in numbers. At an early stage in our education we learn about the special nature of decimal expansions of ‘rational numbers’, fractions that we can write in the form m/n, for some whole numbers m and n with n ≠ 0. The Greeks proved that some numbers, such as √2, 3√7 and √2 + √3 are not rational—indeed, it can be shown that ‘most’ numbers (in a precise mathematical sense) are irrational. It can be shown that a real number is rational if and only if it has a finite decimal expansion, or a decimal expansion that repeats from some point onwards; for example, 1/4 = 0.25 and 3/7 = 0.428571 428571 428571... . Note that we can also think of 1/4 as a repeating decimal, 0.25000000. . . ; we can also write it as 0.24999999 ... , but for simplicity we ignore such ambiguities. We can also count using bases different from 10. The binary system uses base 2, where each place in the representation corresponds to a power of 2; for example, just as 2301 in the decimal system refers to (2 × 103) + (3 × 102) + (0 × 101) + (1 × 100), so in base 2 the decimal number 13 = (1 × 23) + (1 × 22) + (0 × 21) + (1 × 20) is represented by 1101. In base 3 we use only the numbers 0, 1, 2 and express numbers using powers of 3, so the decimal number 25 = (2 × 32) + (2 × 31) + (1 × 30) is represented by 221. Note that when we use bases larger than 10 we have to invent extra symbols to represent the larger ‘digits’; for example, in base 12 we might use the digits 0, 1, 2, . . . , 9, T, E, with T and E representing ‘ten’ and ‘eleven’.
