Last updated

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/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]


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:

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. 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.
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

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.

See also


  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: How Do Computers Talk? | Experimonkey". experimonkey.com. Retrieved 2018-12-02.[ dead link ]
  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.

Related Research Articles

Arithmetic Elementary branch of mathematics

Arithmetic is a branch of mathematics that consists of the study of numbers, especially concerning the properties of the traditional operations on them—addition, subtraction, multiplication, division, exponentiation and extraction of roots. Arithmetic is an elementary part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry, and analysis. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory, and are sometimes still used to refer to a wider part of number theory.

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 16. Unlike the common way of representing numbers using 10 symbols, hexadecimal uses 16 distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9, and "A"–"F" to represent values 10 to 15.

Numeral system Notation for expressing numbers

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.

The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7, that is to say 10 represents 8 in decimal and 100 represents 64 in decimal. However, English uses a base-10 number language system and so a true octal system might use different language to avoid confusion with the decimal system.

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.

Babylonian cuneiform numerals Numeral system

Assyro-Chaldean Babylonian cuneiform numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record.

A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" (one).

Numerical digit Symbols used in combinations to write numbers

A numerical digit 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.

A quaternary numeral system is base-4. It uses the digits 0, 1, 2 and 3 to represent any real number. Conversion from binary is straightforward.

Positional notation Method for representing or encoding numbers

Positional notation denotes usually 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.

Many letters of the Latin alphabet, both capital and small, are used in mathematics, science, and engineering to denote by convention specific or abstracted constants, variables of a certain type, units, multipliers, or physical entities. Certain letters, when combined with special formatting, take on special meaning.

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 derives from this bijection 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.

In computer science, an integer literal is a kind of literal for an integer whose value is directly represented in source code. For example, in the assignment statement x = 1, the string 1 is an integer literal indicating the value 1, while in the statement x = 0x10 the string 0x10 is an integer literal indicating the value 16, which is represented by 10 in hexadecimal.