# Notes

## Section 5: Data Compression

Completeness [of number representations]

If one successively reads 0's and 1's from an infinite sequence then the representations (c), (d) and (e) have the property that eventually one will always accumulate a valid representation for some number or another. The pictures below show which sequences of 0's and 1's correspond to complete numbers in these representations. Every vertical column is a possible sequence of 0's and 1's, and the column is shown to terminate when a complete number is obtained.

With an infinite random sequence of 0's and 1's, different number representations yield different distributions of sizes of numbers. Representation (b), for example, is more weighted towards large numbers, while (c) is more weighted towards small numbers. Maximal compression for a sequence of numbers with a particular distribution of sizes is obtained by choosing a representation that yields a matching such distribution. (See also page 949.)

From Stephen Wolfram: A New Kind of Science [citation]