Mixed radixnumeral systems are non-standard positional numeral systems in which the numerical base varies from position to position. Such numerical representation applies when a quantity is expressed using a sequence of units that are each a multiple of the next smaller one, but not by the same factor. Such units are common for instance in measuring time; a time of 32 weeks, 5 days, 7 hours, 45 minutes, 15 seconds, and 500 milliseconds might be expressed as a number of minutes in mixed-radix notation as:
In the tabular format, the digits are written above their base, and a semicolon indicates the radix point. In numeral format, each digit has its associated base attached as a subscript, and the radix point is marked by a full stop or period. The base for each digit is the number of corresponding units that make up the next larger unit. As a consequence there is no base (written as ∞) for the first (most significant) digit, since here the "next larger unit" does not exist (and one could not add a larger unit of "month" or "year" to the sequence of units, as they are not integer multiples of "week").
Examples
The most familiar example of mixed-radix systems is in timekeeping and calendars. Western time radices include, both cardinally and ordinally, decimal years, decades, and centuries, septenary for days in a week, duodecimal months in a year, bases 28–31 for days within a month, as well as base 52 for weeks in a year. Time is further divided into hours counted in base 24 hours, sexagesimal minutes within an hour and seconds within a minute, with decimal fractions of the latter.
A standard form for dates is 2021-04-10 16:31:15, which would be a mixed radix number by this definition, with the consideration that the quantities of days vary both per month, and with leap years. One proposed calendar instead uses base 13 months, quaternary weeks, and septenary days.
A mixed radix numeral system is often best expressed with a table. A table describing what can be understood as the 604800 seconds of a week is as follows, with the week beginning on hour 0 of day 0 (midnight on Sunday):
Radix
7
24
60
60
Denomination
day
hour
minute
second
Place value (seconds)
86400
3600
60
1
In this numeral system, the mixed radix numeral 37172451605760 seconds would be interpreted as 17:51:57 on Wednesday, and 0702402602460 would be 00:02:24 on Sunday. Ad hoc notations for mixed radix numeral systems are commonplace.
The Maya calendar consists of several overlapping cycles of different radices. A short count tzolk'in overlaps base 20 named days with tridecimal numbered days. A haab' consists of vigesimal days, octodecimalmonths, and base-52 years forming a round. In addition, a long count of vigesimal days, octodecimal winal, then base 24 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.
Another proposal is the so-called factorial number system:
Radix
8
7
6
5
4
3
2
1
Place value
7!
6!
5!
4!
3!
2!
1!
0!
Place value in decimal
5040
720
120
24
6
2
1
1
Highest digit allowed
7
6
5
4
3
2
1
0
For example, the biggest number that could be represented with six digits would be 543210 which equals 719 in decimal: 5×5! + 4×4! + 3×3! + 2×2! + 1×1! It might not be clear at first sight but the factorial based numbering system is unambiguous and complete. Every number can be represented in one and only one way because the sum of respective factorials multiplied by the index is always the next factorial minus one:
There is a natural mapping between the integers 0, ..., n!−1 and permutations of n elements in lexicographic order, which uses the factorial representation of the integer, followed by an interpretation as a Lehmer code.
The above equation is a particular case of the following general rule for any radix (either standard or mixed) base representation which expresses the fact that any radix (either standard or mixed) base representation is unambiguous and complete. Every number can be represented in one and only one way because the sum of respective weights multiplied by the index is always the next weight minus one:
In mathematics, the factorial of a non-negative integer , denoted by , is the product of all positive integers less than or equal to . The factorial of also equals the product of with the next smaller factorial: For example, The value of 0! is 1, according to the convention for an empty product.
A numeral system is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.
Octal is a numeral system with eight as the base.
A senary numeral system has six as its base. It has been adopted independently by a small number of cultures. Like the decimal base 10, the base is a semiprime, though it is unique as the product of the only two consecutive numbers that are both prime. As six is a superior highly composite number, many of the arguments made in favor of the duodecimal system also apply to the senary system.
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically "0" (zero) and "1" (one). A binary number may also refer to a rational number that has a finite representation in the binary numeral system, that is, the quotient of an integer by a power of two.
The quater-imaginary numeral system is a numeral system, first proposed by Donald Knuth in 1960. Unlike standard numeral systems, which use an integer as their bases, it uses the imaginary number as its base. It is able to uniquely represent every complex number using only the digits 0, 1, 2, and 3. Numbers less than zero, which are ordinarily represented with a minus sign, are representable as digit strings in quater-imaginary; for example, the number −1 is represented as "103" in quater-imaginary notation.
A numerical digit or numeral is a single symbol used alone or in combinations, to represent numbers in a positional numeral system. The name "digit" comes from the fact that the ten digits of the hands correspond to the ten symbols of the common base 10 numeral system, i.e. the decimal digits.
The Cooley–Tukey algorithm, named after J. W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size in terms of N1 smaller DFTs of sizes N2, recursively, to reduce the computation time to O(N log N) for highly composite N (smooth numbers). Because of the algorithm's importance, specific variants and implementation styles have become known by their own names, as described below.
Balanced ternary is a ternary numeral system that uses a balanced signed-digit representation of the integers in which the digits have the values −1, 0, and 1. This stands in contrast to the standard (unbalanced) ternary system, in which digits have values 0, 1 and 2. The balanced ternary system can represent all integers without using a separate minus sign; the value of the leading non-zero digit of a number has the sign of the number itself. The balanced ternary system is an example of a non-standard positional numeral system. It was used in some early computers and has also been used to solve balance puzzles.
Positional notation, also known as place-value notation, positional numeral system, or simply place value, usually denotes the extension to any base of the Hindu–Arabic numeral system. More generally, a positional system is a numeral system in which the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit. In early numeral systems, such as Roman numerals, a digit has only one value: I means one, X means ten and C a hundred. In modern positional systems, such as the decimal system, the position of the digit means that its value must be multiplied by some value: in 555, the three identical symbols represent five hundreds, five tens, and five units, respectively, due to their different positions in the digit string.
In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.
In combinatorics, the factorial number system, also called factoradic, is a mixed radix numeral system adapted to numbering permutations. It is also called factorial base, although factorials do not function as base, but as place value of digits. By converting a number less than n! to factorial representation, one obtains a sequence of n digits that can be converted to a permutation of n elements in a straightforward way, either using them as Lehmer code or as inversion table representation; in the former case the resulting map from integers to permutations of n elements lists them in lexicographical order. General mixed radix systems were studied by Georg Cantor.
In mathematical notation for numbers, a signed-digit representation is a positional numeral system with a set of signed digits used to encode the integers.
Bijective numeration is any numeral system in which every non-negative integer can be represented in exactly one way using a finite string of digits. The name refers to the bijection that exists in this case between the set of non-negative integers and the set of finite strings using a finite set of symbols.
Non-standard positional numeral systems here designates numeral systems that may loosely be described as positional systems, but that do not entirely comply with the following description of standard positional systems:
In mathematics and computer science, optimal radix choice is the problem of choosing the base, or radix, that is best suited for representing numbers. Various proposals have been made to quantify the relative costs of using different radices in representing numbers, especially in computer systems. One formula is the number of digits needed to express it in that base, multiplied by the base. This expression also arises in questions regarding organizational structure, networking, and other fields.
A non-integer representation uses non-integer numbers as the radix, or base, of a positional numeral system. For a non-integer radix β > 1, the value of
A negative base may be used to construct a non-standard positional numeral system. Like other place-value systems, each position holds multiples of the appropriate power of the system's base; but that base is negative—that is to say, the base b is equal to −r for some natural number r.
In arithmetic, a complex-base system is a positional numeral system whose radix is an imaginary or complex number.
↑ S. S. Pillai, "An arithmetical function concerning primes", Annamalai University Journal (1930), pp. 159–167.
↑ József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, pp. 384-386.
↑ E. E. Kummer, Neuer elementarer Beweis des Satzes, dass die Anzahl aller Primzahlen eine unendliche ist, Monatsberichte der Königlichen Preussische Akademie des Wissenschaften zu Berlin 1878/9, pp. 777–778.
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