In general there are many different expressions that correspond to a given number, though there is always a unique Cantor normal form—essentially a finite sequence of digits giving coefficients of descending powers of ω . … Yet as discussed on page 1127 , one can also consider larger cardinal numbers, such as ℵ 1 , considered in connection with the number of real numbers, and so on.

There were occasional comments about complicated behavior (notably by Arthur Cayley in 1879) but no real investigation seems to have been made. … (Already in the late 1940s John von Neumann had suggested using x  4x (1 - x) as a random number generator, commenting on its extraction of initial condition digits, as mentioned on page 921 .)

In 1909 Emile Borel had formulated the notion of normal numbers (see page 912 ) whose infinite digit sequences contain all blocks with equal frequency. … Starting in the late 1940s the development of information theory began to suggest connections between randomness and inability to compress data, but emphasis on p Log[p] measures of information content (see page 1071 ) reinforced the idea that block frequencies are the only real criterion for randomness.

But after an observation is made, it is in effect assumed that a system can be described by ordinary real-number probabilities—so that for example no interference is possible. … But this does not mean that in more complicated systems more characteristic of real measuring devices there may not be other sources of randomness that end up dominating. … For certainly in the standard formalism, quantum probability amplitudes are taken to be continuous quantities in which an arbitrary number of digits can be specified.