- This article aims to be an accessible introduction.
For the mathematical definition, see Decimal representation.
The
decimal numeral
system (also called
base ten or occasionally
denary) has
ten as its
base. It is the numerical base
most widely used by modern civilizations.
.
Decimal notation often refers to the base-10
positional notation such as the
Hindu-Arabic numeral system,
however it can also be used more generally to refer to
non-positional systems such as
Roman
or
Chinese numerals which are still
based on powers of ten.
Decimal notation
Decimal notation is the writing of
numbers in
a base-10
numeral system. Examples
are
Roman numerals,
Brahmi numerals, and
Chinese numerals, as well as the
Hindu-Arabic numerals used by speakers
of English. Roman numerals have symbols for the decimal powers (1,
10, 100, 1000) and secondary symbols for half these values (5, 50,
500). Brahmi numerals had symbols for the nine numbers 1–9, the
nine decades 10–90, plus a symbol for 100 and another for 1000.
Chinese has symbols for 1–9, and fourteen additional symbols for
higher powers of 10, which in modern usage reach
10
^{44}.
However, when people who use
Hindu-Arabic numerals speak of decimal
notation, they often mean not just decimal numeration, as above,
but also decimal fractions, all conveyed as part of a
positional system. Positional decimal
systems include a zero and use symbols (called
digits) for the ten values (0, 1, 2, 3, 4,
5, 6, 7, 8, and 9) to represent any number, no matter how large or
how small. These digits are often used with a
decimal separator which indicates the
start of a fractional part, and with a symbol such as the plus sign
+ (for positive) or minus sign − (for negative) adjacent to the
numeral to indicate its polarity.
Positional notation uses positions for each power of ten: units,
tens, hundreds, thousands,
etc. The position of each digit
within a number denotes the multiplier (power of ten) multiplied
with that digit—each position has a value ten times that of the
position to its right. There were two independent sources of
positional decimal systems in ancient civilization: the
Chinese counting rod system and the
Hindu-Arabic numeral system,
which descended from Brahmi numerals.
Ten is the number which is the count of
fingers and thumbs on both hands (or toes on the feet). In many
languages the word
digit or its translation is
also the anatomical term referring to fingers and toes. In English,
decimal (decimus
Lat.) means
tenth,
decimate means
reduce by a tenth, and denary (denarius
Lat.) means
the unit of
ten. The symbols for the digits in common use around the
globe today are called
Arabic numerals by Europeans and
Indian numerals by Arabs, the two
groups' terms both referring to the culture from which they learned
the system. However, the symbols used in different areas are not
identical; for instance, Western Arabic numerals (from which the
European numerals are derived) differ from the forms used by other
Arab cultures.
Decimal fractions
A
decimal fraction is a
fraction where the
denominator is a
power of ten.
Decimal fractions are commonly expressed without a denominator, the
decimal separator being inserted
into the numerator (with
leading zeros
added if needed), at the position from the right corresponding to
the power of ten of the denominator. e.g., 8/10, 83/100, 83/1000,
and 8/10000 are expressed as: 0
.8,
0
.83, 0
.083, and
0
.0008. In English-speaking and many Asian
countries, a period (
.) or raised period
(
•) is used as the decimal separator; in many
other countries, a comma is used.
The
integer part or
integral part
of a decimal number is the part to the left of the decimal
separator (see also
floor function).
The part from the decimal separator to the right is the fractional
part; if considered as a separate number, a zero is often written
in front. Especially for negative numbers, we have to distinguish
between the fractional part of the notation and the fractional part
of the number itself, because the latter gets its own minus sign.
It is usual for a decimal number whose
absolute value is less than one to have a
leading zero.
Trailing zeros after the decimal point
are not necessary, although in science, engineering and
statistics they can be retained to indicate a
required precision or to show a level of confidence in the accuracy
of the number: Whereas 0
.080 and
0
.08 are numerically equal, in engineering
0
.080 suggests a measurement with an error of up
to 1 part in two thousand (±0
.0005), while
0
.08 suggests a measurement with an error of up to
1 in two hundred (see
Significant figures).
Other rational numbers
Any
rational number which cannot be
expressed as a finite decimal fraction has a unique infinite
decimal expansion ending with
recurring decimals.
The decimal fractions are those with denominator divisible by only
2 and or 5.
- 1/2 = 0.5
- 1/20 = 0.05
- 1/5 = 0.2
- 1/50 = 0.02
- 1/4 = 0.25
- 1/40 = 0.025
- 1/25 = 0.04
- 1/8 = 0.125
- 1/125= 0.008
- 1/10 = 0.1
- 1/3 = 0.333333… (with 3 repeating)
- 1/9 = 0.111111… (with 1 repeating)
100-1=99=9×11
- 1/11 = 0.090909… (with 09 or 90 repeating)
1000-1=9×111=27×37
- 1/27 = 0.037037037…
- 1/37 = 0.027027027…
- 1/111 = 0 .009009009…
also:
- 1/81= 0.012345679012… (with 012345679 repeating)
Other prime factors in the denominator will give longer recurring
sequences, see for instance
7,
13.
That a rational number must have a
finite
or recurring decimal expansion can be seen to be a consequence of
the
long division algorithm, in that there are only q-1 possible
nonzero
remainders on division by q, so
that the recurring pattern will have a period less than q. For
instance to find 3/7 by long division:
0.4 2 8 5 7 1 4 ...
7 ) 3.0 0 0 0 0 0 0 0
2 8 30/7 = 4 r 2
2 0
1 4 20/7 = 2 r 6
6 0
5 6 60/7 = 8 r 4
4 0
3 5 40/7 = 5 r 5
5 0
4 9 50/7 = 7 r 1
1 0
7 10/7 = 1 r 3
3 0
2 8 30/7 = 4 r 2
2 0
etc
The converse to this observation is that every
recurring decimal represents a rational
number
p/
q. This is a consequence of the fact the
recurring part of a decimal representation is, in fact, an infinite
geometric series which will sum to
a rational number. For instance,
- 0.0123123123\cdots = \frac{123}{10000} \sum_{k=0}^\infty
0.001^k = \frac{123}{10000}\ \frac{1}{1-0.001} = \frac{123}{9990} =
\frac{41}{3330}
Real numbers
Every
real number has a (possibly
infinite) decimal representation, i.e., it can be written as
- x = \mathop{\rm sign}(x) \sum_{i\in\mathbb Z} a_i\,10^i
where
- sign() is the sign function,
- a_{i} ∈ { 0,1,…,9 } for all i ∈
Z, are its decimal digits, equal
to zero for all i greater than some number (that number
being the common logarithm of
|x|).
Such a sum converges as
i decreases, even if there are
infinitely many nonzero
a_{i}.
Rational numbers (e.g. p/q) with
prime factors in the denominator other
than 2 and 5 (when reduced to simplest terms) have a unique
recurring decimal
representation.
Non-uniqueness of decimal representation
Consider those rational numbers which have only the factors 2 and 5
in the denominator, i.e. which can be written as
p/(2
^{a}5
^{b}). In this case there is a terminating
decimal representation. For instance 1/1=1, 1/2=0.5, 3/5=0.6,
3/25=0.12 and 1306/1250=1.0448. Such numbers are the only real
numbers which do not have a unique decimal representation, as they
can also be written as a representation that has a recurring 9, for
instance 1=
0.99999…, 1/2=0.499999…,
etc.
The number
0=0/1 is special in that
it has no representation with recurring 9.
This leaves the
irrational
numbers. They also have unique infinite decimal representation,
and can be characterised as the numbers whose decimal
representations neither terminate nor recur.
So in general the decimal representation is unique, if one excludes
representations that end in a recurring 9.
The same
trichotomy holds for other
base-n
positional numeral
systems:
- Terminating representation: rational where the denominator
divides some n^{k}
- Recurring representation: other rational
- Non-terminating, non-recurring representation: irrational
and a version of this even holds for irrational-base numeration
systems, such as
golden mean base
representation.
History
History of decimal numbers
According
to the Cambridge
University scholars, decimal numbers originated in China. The
earliest evidence of use dates back to the
14th century BC, although it was almost
certainly in use long before that.
History of the Hindu-Arabic numeral system
The modern
numeral system format, known as the
Hindu-Arabic numeral
system, originated in India by the
9th century. Its ideas were
transmitted to China and the Islamic world during and after that
time. It was notably introduced to the west through
Muhammad ibn
Mūsā al-Khwārizmī's
On the Calculation with Hindu
Numerals.
History of decimal fractions
While the mathematician
Jamshīd al-Kāshī claimed to
have discovered decimal fractions himself in the 15th century, J.
Lennart Berggrenn notes that decimal fractions were used five
centuries before him by Arab mathematician
Abu'l-Hasan al-Uqlidisi as early as
the 10th century. Furthermore, according to the Cambridge
University scholars, decimal fractions were first developed and
used by the Chinese in the
1st century
BC, and then spread to the Middle East and from there to
Europe.
Natural languages
A straightforward decimal system, in which 11 is expressed as
ten-one and 23 as
two-ten-three, is found in
Chinese languages, and in
Vietnamese with a few irregularities.
Japanese,
Korean, and
Thai have imported the Chinese decimal system.
Many other languages with a decimal system have special words for
the numbers between 10 and 20, and decades.
Incan languages such as
Quechua and
Aymara have an almost straightforward
decimal system, in which 11 is expressed as
ten with one
and 23 as
two-ten with three.
Some psychologists suggest irregularities of the English names of
numerals may hinder children's counting ability.
Alternative bases
Some cultures do, or did, use other numeral systems, most notably
In addition, it has been suggested that many other cultures
developed alternative numeral systems (although the extent is
debated):
- Many or all of the Chumashan
languages originally used a base 4 counting system, in which
the names for numbers were structured according to multiples of 4
and 16.
- Many languages use quinary number systems, including Gumatj, Nunggubuyu,, Kuurn Kopan Noot and Saraveca. Of these, Gumatj is the only true "5-25"
language known, in which 25 is the higher group of 5.
- Some
Nigerians use base 12
systems
- The
Huli language of Papua New
Guinea is reported to have base 15 numerals.
Ngui means 15, ngui ki means 15×2 = 30, and
ngui ngui means 15×15 = 225.
- Umbu-Ungu, also known as Kakoli, is reported to have base-24 numerals.
Tokapu means 24, tokapu talu means 24×2 = 48, and
tokapu tokapu means 24×24 = 576.
- Base 27 is used in two natural
languages, the Telefol language
and the Oksapmin language of Papua
New Guinea.
- Ngiti is reported to have a base 32
numeral system with base 4 cycles.
Computer hardware and software systems commonly use
a binary representation,
internally (although a few of the earliest computers, such as
ENIAC, did use decimal representation internally).
For external use by computer specialists, this binary
representation is sometimes presented in the related
octal or
hexadecimal
systems.For most purposes, however, binary values are converted to
the equivalent decimal values for presentation to and manipulation
by humans.
Both computer hardware and software also use internal
representations which are effectively decimal for storing decimal
values and doing arithmetic. Often this arithmetic is done on data
which are encoded using some variant of
binary-coded decimal, especially in
database implementations, but there are other decimal
representations in use (such as in the new
IEEE
754 Standard for Floating-Point Arithmetic). Decimal arithmetic
is used in computers so that decimal fractional results can be
computed exactly, which is not possible using a binary fractional
representation.This is often important for financial and other
calculations.
See also
References
- The History of Arithmetic, Louis
Charles Karpinski, 200pp, Rand McNally & Company,
1925.
- Fingers or Fists? (The Choice of Decimal or Binary
Representation), Werner Buchholz, Communications of the ACM,
Vol. 2 #12, pp3–11, ACM Press, December 1959.
- Decimal Computation, Hermann Schmid, John Wiley & Sons
1974 (ISBN 047176180X); reprinted in 1983 by Robert E. Krieger
Publishing Company (ISBN 0898743184)
- Histoire universelle des chiffres, Georges Ifrah, Robert
Laffont, 1994 (Also: The Universal History of Numbers: From
prehistory to the invention of the computer, Georges Ifrah, ISBN
0471393401, John Wiley and Sons Inc., New York, 2000. Translated
from the French by David Bellos, E.F. Harding, Sophie Wood and Ian
Monk)
- Decimal Floating-Point: Algorism for Computers,
Cowlishaw, M.
F., Proceedings 16th IEEE Symposium on Computer Arithmetic,
ISBN 0-7695-1894-X, pp104-111, IEEE Comp. Soc., June 2003
- Science and Civilisation in China Vol 3 (See under
heading Decimal System)
- The Genius of China by Robert Temple (See under
heading Decimal System)
- Ifrah, page 346
- Britannica Concise Encyclopedia (2007). algebra
- Science and Civilisation in China Vol 3 (See under
heading Decimal Fractions)
- The Genius of China by Robert Temple
- .
- There is a surviving list of Ventureño language number words
up to 32 written down by a Spanish priest ca. 1819. "Chumashan
Numerals" by Madison S. Beeler, in Native American
Mathematics, edited by Michael P. Closs (1986), ISBN
0-292-75531-7.
- Harald Hammarström, Rarities in Numeral Systems: "Bases 5, 10, and 20 are
omnipresent."
- Dawson, J. " Australian Aborigines: The Languages and Customs of
Several Tribes of Aborigines in the Western District of
Victoria (1881), p. xcviii.
- Decimal Arithmetic - FAQ
External links