In
mathematics and
computer science, a
digit
is a symbol (a number symbol, e.g. "3" or "7") used in
numeral (combinations of symbols, e.g. "37"),
to represent
numbers, (
integers or
real numbers)
in
positional numeral systems. The name "digit" comes from
the fact that the 10 digits (ancient
Latin
digita meaning fingers) of the hands correspond to the 10
symbols of the common base 10 number system, i.e. the decimal
(ancient
Latin adjective
dec. meaning
ten) digits.
In a given number system, if the
base is an
integer, the number of digits required is always equal to the
absolute value of the base.
Overview
In a basic digital system, a
numeral
is a sequence of digits, which may be of arbitrary length. Each
position in the sequence has a
place
value, and each digit has a value. The total value of the
numeral is computed by multiplying each digit in the sequence by
its place value, and summing the results.
Digital values
Each digit in a number system represents an integer. For example,
in
decimal the digit "1" represents the
integer
one, and in the
hexadecimal system, the letter "A" represents
the number
ten. A
positional number system must have
a digit representing the integers from
zero up
to, but not including, the
radix of the number
system.
Computation of place values
The
HinduArabic numeral
system uses a decimal
separator, commonly a period in the United Kingdom and United
States or a comma in Europe, to denote the "ones place," which has a place
value one. Each successive place to the left of this has a
place value equal to the place value of the previous digit times
the
base. Similarly, each
successive place to the right of the separator has a place value
equal to the place value of the previous digit divided by the base.
For example, in the numeral
10.34 (written in
base ten),
 the 0 is immediately to the left of the
separator, so it is in the ones place;
 the 1 to the left of the zero has a place
value of one, and is in the tens place;
 the 3 is to the right of the ones place, so it
is in the tenths place; and
 the 4 to the right of the tenths place is in
the hundredths place.
The total value of the number is 1 ten, 0 ones, 3 tenths, and 4
hundredths. Note that the zero, which contributes no value to the
number, indicates that the 1 is in the tens place rather than the
ones place.
The place value of any given digit in a numeral can be given by a
simple calculation, which in itself is a compliment to the logic
behind numeral systems. The calculation involves the multiplication
of the given digit by the base raised by the exponent n1, where
'n' represents the position of the digit from the separator; the
value of n is positive (+), but this is only if the digit is to the
left of the separator. And to the right, the digit is multiplied by
the base raised by a negative () n. For example, in the number
10.34 (written in
base
ten),
 the 1 is second to the left of the separator,
so based on calculation, its value is,
 n  1 = 2  1 = 1
 1 × 10^{1} = 10
 the 4 is second to the right of the separator,
so based on calculation its value is,
 n = 2
 4 × 10^{2} =
History
Glyphs used to represent digits of the
HinduArabic numeral system.
The first true written
positional numeral system is
considered to be the
HinduArabic numeral system.
This system was established by the 7th centuryO'Connor, J. J. and
Robertson, E. F.
Arabic Numerals. January 2001. Retrieved on
20070220., but was not yet in its modern form because the use of
the digit
zero had not yet been widely
accepted. Instead of a zero, a space was left in the numeral as a
placeholder. The first widely acknowledged use of zero was in 876.
Although the original HinduArabic system was very similar to the
modern one, even down to the
glyphs used to
represent digits, the direction of writing was reversed, so that
place values increased to the right rather than to the left.
The digits of the Maya numeral system,
with HinduArabic equivalents
By the 13th century,
HinduArabic
numerals were accepted in European mathematical circles
(
Fibonacci used them in his
Liber Abaci). They began to enter common
use in the 15th century. By the end of the 20th century virtually
all noncomputerized calculations in the world were done with
Arabic numerals, which have replaced native numeral systems in most
cultures.
Other historical numeral systems using digits
The exact age of the
Maya numerals is
unclear, but it is possible that it is older than the HinduArabic
system. The system was
vigesimal (base
twenty), so it has twenty digits. The Mayas
used a shell symbol to represent zero. Numerals were written
vertically, with the ones place at the bottom. The
Mayas had no equivalent of the modern
decimal separator, so their system could
not represent fractions.
The
Thai numeral system is identical
to the
HinduArabic numeral
system except for the symbols used to represent digits.
The use of
these digits is less common in Thailand than it once
was, but they are still used alongside HinduArabic
numerals.
The rod
numerals, the written forms of counting
rods once used by Chinese and Japanese
mathematicians, are a decimal positional system able to represent
not only zero but also negative numbers. Counting rods
themselves predate HinduArabic numeral system. The
Suzhou numerals are
variants of rod numerals.
Rod numerals (vertical)
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 










0 
1 
2 
3 
4 
5 
6 
7 
8 
9 










Modern digital systems
In computer science
The
binary (base 2),
octal (base 8), and
hexadecimal (base 16) systems, extensively used
in
computer science, all follow the
conventions of the
HinduArabic numeral system. The
binary system uses only the digits "0" and "1", while the octal
system uses the digits from "0" through "7". The hexadecimal system
uses all the digits from the decimal system, plus the letters "A"
through "F", which represent the numbers 10 to 15
respectively.
Unusual systems
The
ternary and
balanced ternary systems have sometimes
been used. They are both basethree systems.
Balanced ternary is unusual in having the digit values 1, 0 and 1.
Balanced ternary turns out to have some useful properties and the
system has been used in the experimental Russian
Setun computers.
Digits in mathematics
Despite the essential role of digits in describing numbers, they
are relatively unimportant to modern
mathematics. Nevertheless, there are a few
important mathematical concepts that make use of the representation
of a number as a sequence of digits.
Digital roots
The digital root is the singledigit number obtained by summing the
digits of a given number, then summing the digits of the result,
and so on until a singledigit number is obtained.
Casting out nines
Casting out nines is a procedure
for checking arithmetic done by hand. To describe it, let f(x)\,
represent the
digital root of x\,, as
described above. Casting out nines makes use of the fact that if A
+ B = C\,, then f(f(A) + f(B)) = f(C)\,. In the process of casting
out nines, both sides of the latter
equation are computed, and if they are not equal
the original addition must have been faulty.
Repunits and repdigits
Repunits are integers that are represented with only the digit 1.
For example, 1111 (one thousand, one hundred eleven) is a repunit.
Repdigits are a generalization of repunits;
they are integers represented by repeated instances of the same
digit. For example, 333 is a repdigit. The
primacy of repunits is of interest to
mathematicians
Palindromic numbers and Lychrel numbers
Palindromic numbers are numbers that read the same when their
digits are reversed. A
Lychrel number
is a positive integer that never yields a palindromic number when
subjected to the iterative process of being added to itself with
digits reversed. The question of whether there are any
Lychrel numbers in base 10 is an open problem
in
recreational
mathematics; the smallest candidate is
196.
See also
References