E.g., the sexagesimal Babylonian notation and the Chinese rod numerals, which can be classified as standard systems of base 60 and 10, respectively, counting the space representing zero as a numeral, can also be classified as non-standard systems, more specifically, mixed-base systems with unary components, considering the primitive repeated glyphs making up the numerals.
However, most of the non-standard systems listed below have never been intended for general use, but were devised by mathematicians or engineers for special academic or technical use.
The value of the digit string pqrs given by the polynomial form can be simplified into p + q + r + s since bn = 1 for all n. Non-standard features of this system include: In some systems, while the base is a positive integer, negative digits are allowed.
Cistercian numerals are a decimal positional numeral system, but the positions are not aligned as in common decimal notation; instead, they are attached to the top-right, top-left, bottom-right and bottom-left of a vertical stem, respectively, and thus limited to four in number (so only integers from 0 to 9999 can be represented).
The system has close similarities to standard positional numeral systems, but may also be compared to e.g. Greek numerals, where different sets of symbols (in fact, Greek letters) are used for the ones, tens, hundreds and thousands, likewise giving an upper limit on the numbers that can be represented.
Similarly, in computers, e.g. the long integer format is a standard binary system (apart from the sign bit), but it has a limited number of positions, and the physical locations for the representations of the digits may not be aligned.
A mixed base system one may already be familiar with is the measurement of time in mix of base-12 (for hours) & base-60 (for minutes & seconds), although the measurement is often reported in terms of base-10, e.g. 20:00:00 commonly represents twenty hours (past midnight).
In a mixed-radix system such as the factorial number system, the weights form a sequence where each weight is an integer multiple of the previous one, and the number of permitted digit values varies accordingly from position to position.
In these, not only are the bases of a given digit different, they can be also nonuniform and altered in an asymmetric way to encode information more efficiently.