Examples
A mixed radix numeral system can often benefit from a tabular summary. The familiar system for describing the 604800 seconds of a week starting from midnight on Sunday runs as follows:
| Radix: | 7 | 2 | 12 | 60 | 60 |
| Denomination: | day | half-day | hour | minute | second |
| Place value (seconds): | 86400 | 43200 | 3600 | 60 | 1 |
| Digit translations … | |||||
| day: | 0=Sunday, 1=Monday, 2=Tuesday, 3=Wednesday, 4=Thursday, 5=Friday, 6=Saturday | ||||
| half-day: | 0=AM, 1=PM | ||||
| hour: | 0 is written as "12" (!) | ||||
In this numeral system, the mixed radix numeral 371251251605760 seconds would be interpreted as 05:51:57 p.m. on Wednesday, and 070201202602460 would be 12:02:24 a.m. on Sunday. Ad hoc notations for mixed radix numeral systems are commonplace.
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. In theory, balanced ternary minimizes the number of pieces of coinage required to represent any quantity.
A historical example of a mixed radix numeral system is the system of Mayan numerals, which generally used base-20, except for the second place (the "10s" in decimal) which was base-18, so that the first two places counted up to 360 (an approximation to the number of days in the year).
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.
Read more about this topic: Mixed Radix
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