In any standard positional numeral system, a number is conventionally written as(x)y withx as thestring of digits andy as its base. For base ten, the subscript is usually assumed and omitted (together with the enclosingparentheses), as it is the most common way to expressvalue. 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 thebinary system with base 2) represents the number four.[1]
Generally, in a system with radixb (b > 1), a string of digitsd1 ...dn denotes the numberd1bn−1 +d2bn−2 + ... +dnb0, where0 ≤di <b.[1] In contrast to decimal, or radix 10, which has a ones' place, tens' place, hundreds' place, and so on, radixb would 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.
Used internally by nearly allcomputers. The two digits are "0" and "1", expressed from switches displaying OFF and ON, respectively. Used in most electriccounters.
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".
Traditional numeral system in several cultures, still used by some for counting. Historically also known as thescore system in English, now most famous in the phrase "four score and seven years ago" in theGettysburg Address.
Base36 is abinary-to-text encoding scheme that representsbinary data in anASCII string format by translating it into a radix-36 representation. The choice of 36 is convenient in that the digits can be represented using theArabic numerals 0–9 and theLatin letters A–Z (theISO basic Latin alphabet). Each base36 digit needs less than 6 bits of information to be represented.
Originally used in modified form in ancientSumer and passed to theBabylonians.[3] Used today as the basis of moderncircular coordinate system (degrees, minutes, and seconds) andtime 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 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. Letb be a positive integer greater than 1. Then every positive integera can be expressed uniquely in the form
wherem is a nonnegative integer and ther's are integers such that
Radices are usuallynatural numbers. However, other positional systems are possible, for example,golden ratio base (whoseradix 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.
Different bases are especially used in connection with computers.The commonly used bases are 10 (decimal), 2 (binary), 8 (octal), and 16 (hexadecimal).Abyte with 8bits can represent values from 0 to 255, often expressed withleading zeros in base 2, 8 or 16 to give the same length.[7]
The first row in the tables is the base written in decimal.
