Editing Positional notation (section)

=== Digits and numerals ===
A ''digit'' is a symbol that is used for positional notation, and a ''numeral'' consists of one or more digits used for representing a [[number]] with positional notation. Today's most common digits are the [[Arabic numerals|decimal digits]] "0", "1", "2", "3", "4", "5", "6", "7", "8", and "9". The distinction between a digit and a numeral is most pronounced in the context of a number base.

A non-zero ''numeral'' with more than one digit position will mean a different number in a different number base, but in general, the ''digits'' will mean the same.<ref>The digit will retain its meaning in other number bases, in general, because a higher number base would normally be a notational extension of the lower number base in any systematic organization. In the [[mathematical science]]s there is virtually only one positional-notation numeral system for each base below 10, and this extends with few, if insignificant, variations on the choice of alphabetic digits for those bases above 10.</ref> For example, the base-8 numeral 23<sub>8</sub> contains two digits, "2" and "3", and with a base number (subscripted) "8". When converted to base-10, the 23<sub>8</sub> is equivalent to 19<sub>10</sub>, i.e. 23<sub>8</sub> = 19<sub>10</sub>. In our notation here, the subscript "<sub>8</sub>" of the numeral 23<sub>8</sub> is part of the numeral, but this may not always be the case.

Imagine the numeral "23" as having [[#Non-standard positional numeral systems|an ambiguous base]] number. Then "23" could likely be any base, from base-4 up. In base-4, the "23" means 11<sub>10</sub>, i.e. 23<sub>4</sub> = 11<sub>10</sub>. In base-60, the "23" means the number 123<sub>10</sub>, i.e. 23<sub>60</sub> = 123<sub>10</sub>. The numeral "23" then, in this case, corresponds to the set of base-10 numbers {11, 13, 15, 17, 19, 21, '''23''', ..., 121, 123} while its digits "2" and "3" always retain their original meaning: the "2" means "two of", and the "3" means "three of".

In certain applications when a numeral with a fixed number of positions needs to represent a greater number, a higher number-base with more digits per position can be used. A three-digit, decimal numeral can represent only up to '''999'''. But if the number-base is increased to 11, say, by adding the digit "A", then the same three positions, maximized to "AAA", can represent a number as great as '''1330'''. We could increase the number base again and assign "B" to 11, and so on (but there is also a possible encryption between number and digit in the number-digit-numeral hierarchy). A three-digit numeral "ZZZ" in base-60 could mean '''{{val|215999}}'''. If we use the entire collection of our [[alphanumerics]] we could ultimately serve a base-''62'' numeral system, but we remove two digits, uppercase "I" and uppercase "O", to reduce confusion with digits "1" and "0".<ref>We do ''not'' usually remove the ''lowercase'' digits "l" and lowercase "o", for in most fonts they are discernible from the digits "1" and "0".</ref>
We are left with a base-60, or sexagesimal numeral system utilizing 60 of the 62 standard alphanumerics. (But see ''[[#Sexagesimal system|Sexagesimal system]]'' below.) In general, the number of possible values that can be represented by a <math>d</math> digit number in base <math>r</math> is <math>r^d</math>.

The common numeral systems in computer science are binary (radix 2), octal (radix 8), and hexadecimal (radix 16). In [[Binary numeral system|binary]] only digits "0" and "1" are in the numerals. In the [[octal]] numerals, are the eight digits 0–7. [[Hexadecimal|Hex]] is 0–9 A–F, where the ten numerics retain their usual meaning, and the alphabetics correspond to values 10–15, for a total of sixteen digits. The numeral "10" is binary numeral "2", octal numeral "8", or hexadecimal numeral "16".