# Radix

In digital numeral systems, the radix or base is the number of unique digits, including the digit zero, used to represent numbers in a positional numeral system. For example, for the decimal/denary system (the most common system in use today) the radix (base number) 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

In the system with radix 13, for example, a string of digits such as 398 denotes the (decimal) number 3 × 132 + 9 × 131 + 8 × 130 = 632.

More generally, in a system with radix b (b > 1), a string of digits d1dn denotes the number d1bn−1 + d2bn−2 + … + dnb0, where 0 ≤ di < 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]

Commonly used numeral systems include:

Base/radix Name Description
2 Binary numeral system Used internally by nearly all computers, is base 2. 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 The most used system of numbers in the world, is used in arithmetic. 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.
60 Sexagesimal system Originated 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

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

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

0 < rm < b and 0 ≤ ri < 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. ^ 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.
2. ^ "Binary: How Do Computers Talk? | Experimonkey". experimonkey.com. Retrieved 2018-12-02.
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.