IntegerDigits[n, 2] . … Flatten[IntegerDigits[ Append[2 - With[{w = Floor[Log[3, 2n]]}, IntegerDigits[n - (3 w + 1 - 1)/2, 3, w]], 3], 2, 2]] (e) Fibonacci encoding. … Apply[Take, RealDigits[(N[#, N[Log[10, #] + 3]] &)[ n √ 5 /GoldenRatio 2 + 1/2], GoldenRatio]] The representations of all the first Fibonacci[n] - 1 numbers can be obtained from (the version in the main text has Rest[RotateLeft[Join[#, {0, 1}]]] & applied) Apply[Join, Map[Last, NestList[{# 〚 2 〛 ], Join[Map[Join[{1, 0}, Rest[#]] & , # 〚 2 〛 ], Map[Join[{1, 0}, #] &, # 〚 1 〛 ]]} &, {{}, {{1}}}, n-3]]]

Lyapunov exponents If one thinks of cells to the right of a point in a 1D cellular automaton as being like digits in a real number, then linear growth in the region of differences associated with a change further to the right is analogous to the exponentially sensitive dependence on initial conditions shown on page 155 .

But insofar as such a rule is itself more complicated it may in the end achieve no real reduction in computational effort. … The way things are set up the initial conditions for these cellular automata consist of digit sequences of numbers that give positions. The color of a particular cell is then found by evolving for a number of steps equal to the length of these input digit sequences.

(Related sequences can be generated from RealDigits[1/p, 2] as discussed on page 912 .)

The idea of arithmetic coding is to represent each such bin by the digit sequence of the shortest number within the bin—after trailing zeros have been dropped. For any sequence s this can be done using Module[{c, m = 0}, Map[c[#] = {m, m += Count[s, #]/Length[s]} &, Union[s]]; Function[x, (First[RealDigits[2 # Ceiling[2 -# Min[x]], 2, -#, -1]] &)[Floor[Log[2, Max[x] - Min[x]]]]][ Fold[(Max[#1] - Min[#1]) c[#2] + Min[#1] &, {0, 1}, s]]] Huffman coding of a sequence containing a single 0 block together with n 1 blocks will yield output of length about n ; arithmetic coding will yield length about Log[n] .

Constructible reals Instead of finding successive digits using systems like Turing machines, one can imagine constructing complete real numbers using idealizations of mechanical processes.

And part (d) finally shows the base 2 digits of the successive values attained by the second register when the first register has just decreased to zero. … Yet particularly when such extended instruction sets are used, register machines can provide fairly accurate idealizations of the low-level operations of real computers.

Hashing Given data in the form of sequences of numbers between 0 and k - 1 , a very simple hashing scheme is just to compute FromDigits[Take[list, n], k] . … The slightly modified but still very simple scheme Mod[FromDigits[list, k], m] , where m is usually chosen to be a prime, is what is most often used in practice. … There are several subtleties associated with setting up hash codes that appropriately handle approximate real numbers and Mathematica patterns.

But this is analogous to a digit sequence of a real number.

In connection with early work on chaos theory, it was noted that there are some systems that act like "full shifts", in the sense that the set of sequences they generate includes all possibilities—and corresponds to what one would get by starting with any possible number, then successively shifting digits to the left, and at each step picking off the leading digit. … The locality of cellular automaton rules was thought of as making them the analog for symbol sequences of continuous functions for real numbers (compare page 869 ).