count tzolk'in overlaps vigesimal named days with tridecimal numbered days. A haab' consists of vigesimal days, octodecimal months, and base-52 years forming a round. In addition, a long count of vigesimal days, octodecimal winal, then vigesimal tun, k'atun, b'ak'tun, etc. tracks historical dates. A second example of a mixed radix numeral system in current use is in the design and use of currency, where a limited set of denominations are printed or minted with the objective of being able to represent any monetary quantity; the amount of money is then represented by the number of coins or banknotes of each denomination. When deciding which denominations to create (and hence which radices to mix), a compromise is aimed for between a minimal number of different denominations, and a minimal number of individual pieces of coinage required to represent typical quantities. So, for example, in the UK, banknotes are printed for £50, £20, £10 and £5, and coins are minted for £2, £1, 50p, 20p, 10p, 5p, 2p and 1p—these follow the 1-2-5 series of preferred values. Prior to decimalisation, monetary amounts in the UK were described in terms of pounds, shillings, and pence, with 12 pence per shilling and 20 shillings per pound, so that "£1 7s 6d", for example, corresponded to the mixed-radix numeral 1∞720612. United States customary units are generally mixed-radix systems, with multipliers varying from one size unit to the next in the same manner that units of time do. Mixed-radix representation is also relevant to mixed-radix versions of the Cooley–Tukey FFT algorithm, in which the indices of the input values are expanded in a mixed-radix representation, the indices of the output values are expanded in a corresponding mixed-radix representation with the order of the bases and digits reversed, and each subtransform can be regarded as a Fourier transform in one digit for all values of the remaining digits. Manipulation Mixed-radix numbers of the same base can be manipulated using a generalization of manual arithmetic algorithms. Conversion of values from one mixed base to another is easily accomplished by first converting the place values of the one system into the other, and then applying the digits from the one system against these. APL and J include operators to convert to and from mixed-radix systems. Factorial number system Another proposal is the so-called factorial number system: For example, the biggest number that could be represented with six digits would