Radix

Last updated July 17, 2024

In a positional numeral system, the radix (pl.: radices) or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal system (the most common system in use today) the radix is ten, because it uses the ten digits from 0 through 9.

In any standard positional numeral system, a number is conventionally written as (x)_y with x as the string of digits and y as its base, although for base ten the subscript is usually assumed (and omitted, together with the pair of parentheses), as it is the most common way to express value. For example, (100)₁₀ is equivalent to 100 (the decimal system is implied in the latter) and represents the number one hundred, while (100)₂ (in the binary system with base 2) represents the number four.^[1]

Etymology

Radix is a Latin word for "root". Root can be considered a synonym for base, in the arithmetical sense.

In numeral systems

Generally, in a system with radix b (b > 1), a string of digits d₁ ... d_n denotes the number d₁bⁿ⁻¹ + d₂bⁿ⁻² + … + d_nb⁰, where 0 ≤ d_i < b.^[1] In contrast to decimal, or radix 10, which has a ones' place, tens' place, hundreds' place, and so on, radix b would have a ones' place, then a b¹s' place, a b²s' place, etc.^[2]

For example, if b = 12, a string of digits such as 59A (where the letter "A" represents the value of ten) would represent the value 5 × 12² + 9 × 12¹ + 10 × 12⁰ = 838 in base 10.

Commonly used numeral systems include:

Base/radix	Name	Description
2	Binary numeral system	Used internally by nearly all computers. The two digits are "0" and "1", expressed from switches displaying OFF and ON, respectively. Used in most electric counters.
8	Octal system	Used occasionally in computing. The eight digits are "0"–"7" and represent 3 bits (2³).
10	Decimal system	Used by humans in the vast majority of cultures. Its ten digits are "0"–"9". Used in most mechanical counters.
12	Duodecimal (dozenal) system	Sometimes advocated due to divisibility by 2, 3, 4, and 6. It was traditionally used as part of quantities expressed in dozens and grosses.
16	Hexadecimal system	Often used in computing as a more compact representation of binary (1 hex digit per 4 bits). The sixteen digits are "0"–"9" followed by "A"–"F" or "a"–"f".
20	Vigesimal system	Traditional numeral system in several cultures, still used by some for counting. Historically also known as the score system in English, now most famous in the phrase "four score and seven years ago" in the Gettysburg Address.
36	Base36	Base36 is a binary-to-text encoding scheme that represents binary data in an ASCII string format by translating it into a radix-36 representation. The choice of 36 is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z (the ISO basic Latin alphabet). Each base36 digit needs less than 6 bits of information to be represented.
60	Sexagesimal system	Originally used in modified form in ancient Sumer and passed to the Babylonians.^[3] Used today as the basis of modern circular coordinate system (degrees, minutes, and seconds) and time measuring (minutes, and seconds) by analogy to the rotation of the Earth.

The octal and hexadecimal systems are often used in computing because of their ease as shorthand for binary. Every hexadecimal digit corresponds to a sequence of four binary digits, since sixteen is the fourth power of two; for example, hexadecimal 78₁₆ is binary 1111000₂. Similarly, every octal digit corresponds to a unique sequence of three binary digits, since eight is the cube of two.

This representation is unique. Let b be a positive integer greater than 1. Then every positive integer a can be expressed uniquely in the form

a=r_{m}b^{m}+r_{m-1}b^{m-1}+\dotsb +r_{1}b+r_{0},

where m is a nonnegative integer and the r's are integers such that

0 < r_m < b and 0 ≤ r_i < b for i = 0, 1, ... , m − 1.^[4]

Radices are usually natural numbers. However, other positional systems are possible, for example, golden ratio base (whose radix is a non-integer algebraic number),^[5] and negative base (whose radix is negative).^[6] A negative base allows the representation of negative numbers without the use of a minus sign. For example, let b = −10. Then a string of digits such as 19 denotes the (decimal) number 1 × (−10)¹ + 9 × (−10)⁰ = −1.

Notes

1 2 Mano, M. Morris; Kime, Charles (2014). Logic and Computer Design Fundamentals (4th ed.). Harlow: Pearson. pp. 13–14. ISBN 978-1-292-02468-4.
↑ "Binary". experimonkey.com. Retrieved 2023-05-14.
↑ Bertman, Stephen (2005). Handbook to Life in Ancient Mesopotamia (Paperback ed.). Oxford [u.a.]: Oxford Univ. Press. p. 257. ISBN 978-019-518364-1.
↑ McCoy (1968 , p. 75)
↑ Bergman, George (1957). "A Number System with an Irrational Base". Mathematics Magazine. 31 (2): 98–110. doi:10.2307/3029218. JSTOR 3029218.
↑ William J. Gilbert (September 1979). "Negative Based Number Systems" (PDF). Mathematics Magazine. 52 (4): 240–244. doi:10.1080/0025570X.1979.11976792 . Retrieved 7 February 2015.

Related Research Articles

The decimal numeral system is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as decimal notation.

In mathematics and computing, the hexadecimal numeral system is a positional numeral system that represents numbers using a radix (base) of sixteen. Unlike the decimal system representing numbers using ten symbols, hexadecimal uses sixteen distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9 and "A"–"F" to represent values from ten to fifteen.

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 computer number format is the internal representation of numeric values in digital device hardware and software, such as in programmable computers and calculators. Numerical values are stored as groupings of bits, such as bytes and words. The encoding between numerical values and bit patterns is chosen for convenience of the operation of the computer; the encoding used by the computer's instruction set generally requires conversion for external use, such as for printing and display. Different types of processors may have different internal representations of numerical values and different conventions are used for integer and real numbers. Most calculations are carried out with number formats that fit into a processor register, but some software systems allow representation of arbitrarily large numbers using multiple words of memory.

A ternary numeral system has three as its base. Analogous to a bit, a ternary digit is a trit. One trit is equivalent to log₂ 3 bits of information.

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 2i 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.

Mixed radix numeral 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:

... 32, 5, 07, 45; 15, 500 ... ∞, 7, 24, 60; 60, 1000

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.

Quaternary is a numeral system with four as its base. It uses the digits 0, 1, 2, and 3 to represent any real number. Conversion from binary is straightforward.

Positional notation 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 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.

Base36 is a binary-to-text encoding scheme that represents binary data in an ASCII string format by translating it into a radix-36 representation. The choice of 36 is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z.

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:

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 $r$ .

References

McCoy, Neal H. (1968), Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN 68015225

External links

MathWorld entry on base

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[morris_mano_p13_14-1] 1 2 Mano, M. Morris; Kime, Charles (2014). Logic and Computer Design Fundamentals (4th ed.). Harlow: Pearson. pp. 13–14. ISBN 978-1-292-02468-4.

[2] "Binary". experimonkey.com. Retrieved 2023-05-14.

[3] Bertman, Stephen (2005). Handbook to Life in Ancient Mesopotamia (Paperback ed.). Oxford [u.a.]: Oxford Univ. Press. p. 257. ISBN 978-019-518364-1.

[4] McCoy (1968 , p. 75)

[5] Bergman, George (1957). "A Number System with an Irrational Base". Mathematics Magazine. 31 (2): 98–110. doi:10.2307/3029218. JSTOR 3029218.

[6] William J. Gilbert (September 1979). "Negative Based Number Systems" (PDF). Mathematics Magazine. 52 (4): 240–244. doi:10.1080/0025570X.1979.11976792 . Retrieved 7 February 2015.

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