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As shown above, the “binary” algorithm (Fig. 1.6) “generates” the “binary” system (1.12). But then it is clear that the optimal (n, k, 0)-algorithm in general case also “generates” some positional numeral system with the base R = k + 1, that is, the following representation of the natural number N in the form:

where ai {0, 1, 2, …, k} is the numeral of the ith digit.

In particular, for the case of k = 9, the formula (1.23) reduces to the classical decimal system:

For the case k = 59, the formula (1.23) reduces to the Babylonian positional system with the base of 60. From these examples and arguments, it follows that the optimal (n, k, 0)-algorithms “generate” all well-known positional numeral systems (including the Babylonian system with the base 60, decimal, binary, and other positional systems).