Radix

In apositional numeral system,theradix(pl.:radices) orbaseis the number of uniquedigits,including the digit zero, used to represent numbers. For example, for thedecimal 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)_ywithxas thestringof digits andyas its base, although for base ten the subscript is usually assumed (and omitted, together with the pair ofparentheses), as it is the most common way to expressvalue.For example,(100)₁₀is equivalent to 100(the decimal system is implied in the latter) and represents the number one hundred, while (100)₂(in thebinary systemwith base 2) represents the number four.^[1]

Etymology

Radixis a Latin word for "root".Rootcan be considered a synonym forbase,in the arithmetical sense.

In numeral systems

Generally, in a system with radixb(b> 1), a string of digitsd₁...d_ndenotes the numberd₁bⁿ⁻¹+d₂bⁿ⁻²+… +d_nb⁰,where0 ≤d_i<b.^[1]In contrast to decimal, or radix 10, which has a ones' place, tens' place, hundreds' place, and so on, radixbwould have a ones' place, then ab¹s' place, ab²s' place, etc.^[2]

For example, ifb= 12, a string of digits such as 59A (where the letter "A" represents the value of ten) would represent the value5×12²+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 allcomputers.The two digits are "0" and "1", expressed from switches displaying OFF and ON, respectively. Used in most electriccounters.
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 mostmechanical 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 indozensandgrosses.
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 thescoresystemin English, now most famous in the phrase "four score and seven years ago" in theGettysburg Address.
36	Base36	Base36is abinary-to-text encodingscheme that representsbinary datain anASCIIstring format by translating it into aradix-36 representation. The choice of 36 is convenient in that the digits can be represented using theArabic numerals0–9 and theLatin lettersA–Z (theISO basic Latin Alpha bet). Each base36 digit needs less than 6 bits of information to be represented.
60	Sexagesimal system	Originally used in modified form in ancientSumerand passed to theBabylonians.^[3]Used today as the basis of moderncircular coordinate system(degrees, minutes, and seconds) andtimemeasuring (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 78₁₆is binary1111000₂.Similarly, every octal digit corresponds to a unique sequence of three binary digits, since eight is the cube of two.

This representation is unique. Letbbe a positive integer greater than 1. Then every positive integeracan be expressed uniquely in the form

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

wheremis a nonnegative integer and ther's are integers such that

0 <r_m<band 0 ≤r_i<bfori= 0, 1,...,m− 1.^[4]

Radices are usuallynatural numbers.However, other positional systems are possible, for example,golden ratio base(whose radix is a non-integeralgebraic number),^[5]andnegative base(whose radix is negative).^[6] A negative base allows the representation of negative numbers without the use of a minus sign. For example, letb= −10. Then a string of digits such as 19 denotes the (decimal) number1 × (−10)¹+ 9 × (−10)⁰= −1.

Notes

^^a ^b 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.Retrieved2023-05-14.
^ 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.Retrieved7 February2015.

References

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

External links

MathWorld entry on base

[morris_mano_p13_14-1] 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.Retrieved2023-05-14.

[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.Retrieved7 February2015.

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