The decimal numeral system (also called base-ten positional numeral
system, and occasionally called denary) 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.[1] The way
of denoting numbers in the decimal system is often referred to as
Contents 1 Origin
2
5.1 Rational numbers 6
7.1 History of decimal fractions 7.2 Natural languages 7.3 Other bases 8 See also 9 References 10 External links Origin[edit] Ten fingers on two hands, the possible starting point of the decimal counting. Many numeral systems of ancient civilisations use ten and its powers
for representing numbers, probably because there are ten fingers on
two hands and people started counting by using their fingers. Examples
are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals,
and Chinese numerals. Very large numbers were difficult to represent
in these old numeral systems, and, only the best mathematicians were
able to multiply or divide large numbers. These difficulties were
completely solved with the introduction of the Hindu–Arabic numeral
system for representing integers. This system has been extended to
represent some non-integer numbers, called decimal fractions or
decimal numbers for forming the decimal numeral system.
either a (finite) sequence of digits such as 2017, or in full generality, a m a m − 1 … a 0 displaystyle a_ m a_ m-1 ldots a_ 0 (in this case, the (entire) decimal represents an integer) or two sequence of digits separated by a decimal mark such as 3.14159, 15.00, or in full generality a m a m − 1 … a 0 . b 1 b 2 … b n displaystyle a_ m a_ m-1 ldots a_ 0 .b_ 1 b_ 2 ldots b_ n It is generally assumed that, if m > 0, the first digit am is not zero, but, in some circumstances, it may be useful to have one or more 0's on the left. This does not change the value represented by the decimal. For example, 3.14 = 03.14 = 003.14. Similarly, if bn =0, it may be removed, and conversely, trailing zeros may be added without changing the represented number. For example, 15 = 15.0 = 15.00 and 5.2 = 5.20 = 5.200. sometimes, the unnecessary zeros are used for indicating the accuracy of a measurement. For example, 15.00m may indicate that the measurement error is less than one centimeter, while 15m may mean that the length is roughly fifteen meters, and that the error may exceed 10 cm. For representing a negative number, a minus sign is placed before am. The numeral a m a m − 1 … a 0 . b 1 b 2 … b n displaystyle a_ m a_ m-1 ldots a_ 0 .b_ 1 b_ 2 ldots b_ n represents the number a m 10 m + a m − 1 10 m − 1 + ⋯ + a 0 10 0 + b 1 10 1 + b 2 10 2 + ⋯ + b n 10 n displaystyle a_ m 10^ m +a_ m-1 10^ m-1 +cdots +a_ 0 10^ 0 + frac b_ 1 10^ 1 + frac b_ 2 10^ 2 +cdots + frac b_ n 10^ n Therefore, the contribution of each digit to the value of a number
depends on its position in the numeral. That is, the decimal system is
a positional numeral system
0.8 , 14.89 , 0.00024 displaystyle 0.8,14.89,0.00024 represent the fractions 8/10, 1489/100, 24/100000. More generally, a decimal with n digits after the separator represents the fraction with denominator 10n, whose numerator is the integer obtained by removing the separator. Expressed as a fully reduced fraction, the decimal numbers are those, whose denominator is a product of a power of 2 by a power of 5. Thus the smallest denominators of decimal numbers are 1 = 2 0 ⋅ 5 0 , 2 = 2 1 ⋅ 5 0 , 4 = 2 2 ⋅ 5 0 , 5 = 2 0 ⋅ 5 1 , 8 = 2 3 ⋅ 5 0 , 10 = 2 1 ⋅ 5 1 , 16 = 2 4 ⋅ 5 0 , 25 = 2 0 ⋅ 5 2 , … displaystyle 1=2^ 0 cdot 5^ 0 ,2=2^ 1 cdot 5^ 0 ,4=2^ 2 cdot 5^ 0 ,5=2^ 0 cdot 5^ 1 ,8=2^ 3 cdot 5^ 0 ,10=2^ 1 cdot 5^ 1 ,16=2^ 4 cdot 5^ 0 ,25=2^ 0 cdot 5^ 2 ,ldots The integer part, or integral part of a decimal is the integer written
to the left of the decimal separator (see also truncation). For a
nonnegative decimal, it is the largest integer that is not greater
than the decimal. The part from the decimal separator to the right is
the fractional part, which equals the difference between the numeral
and its integer part.
When the integral part of a numeral is zero, it may occur, typically
in computing, that the integer part is not written (for example .1234,
instead of 0.1234). In normal writing, this is generally avoided
because of the risk of confusion between the decimal mark and other
punctuation.
[x]n = [x]0.d1d2...dn-1dn, and the difference of [x]n–1 and [x]n amounts to [x]n - [x]n–1 = dn ⋅ 10-n < 10-n+1, which is either 0, if dn = 0, or gets arbitrarily small, when n tends to infinity. According to the definition of a limit, x is the limit of [x]n when n tends to infinity. This is written as x = lim n → ∞ [ x ] n textstyle ;x=lim _ nrightarrow infty [x]_ n ; or x = [x]0.d1d2...dn..., which is called an infinite decimal expansion of x. Conversely, for any integer [x]0 and any sequence of digits ( d n ) n = 1 ∞ textstyle ;(d_ n )_ n=1 ^ infty the (infinite) expression [x]0.d1d2...dn... is an infinite decimal expansion of a real number x. This expansion is unique if neither all dn are equal to 9 nor all dn are equal to 0 for n large enough (for all n greater than some natural number N). If all dn for n > N equal to 9 and [x]n = [x]0.d1d2...dn, the limit of the sequence ( [ x ] n ) n = 1 ∞ textstyle ;([x]_ n )_ n=1 ^ infty is the decimal fraction obtained by replacing the last digit that is not a 9, i.e.: dN, by dN + 1, and replacing all subsequent 9s by 0s (see 0.999...). Any such decimal fraction, i.e., dn = 0 for n > N, may be converted to its equivalent infinite decimal expansion by replacing dN by dN - 1, and replacing all subsequent 0s by 9s (see 0.999...). In summary, every real number that is not a decimal fraction has a unique infinite decimal expansion. Each decimal fraction has exactly two infinite decimal expansions, one containing only 0s after some place, which is obtained by the above definition of [x]n, and the other containing only 9s after some place, which is obtained by defining [x]n as the greatest number that is less than x, having exactly n digits after the decimal mark. Rational numbers[edit] Main article: Repeating decimal The long division allows computing the infinite decimal expansion of a rational number. If the rational number is a decimal fraction, the division stops eventually, producing a decimal numeral, which may be prolongated into an infinite expansion by adding infinitely many 0. If the rational number is not a decimal fraction, the division may continue indefinitely. However, as all successive remainder are less than the divisor, there are only a finite number of possible remainders, and after some place, the same sequence of digits must be repeated indefinitely in the quotient. That is, one has a repeating decimal. For example, 1/81 = 0.012345679012... (with 012345679 repeating). Conversely, every eventually repeating sequence of digit is the infinite decimal expansion of a rational number. This is a consequence of the fact that the recurring part of a decimal representation is, in fact, an infinite geometric series which will sum to a rational number. For example, 0.0123123123 … = 123 10000 ∑ k = 0 ∞ 0.001 k = 123 10000
1 1 − 0.001 = 123 9990 = 41 3330 displaystyle 0.0123123123ldots = frac 123 10000 sum _ k=0 ^ infty 0.001^ k = frac 123 10000 frac 1 1-0.001 = frac 123 9990 = frac 41 3330
Diagram of the world's earliest multiplication table (c. 305 BC) from
the
Modern computer hardware and software systems commonly use a binary
representation internally (although many early computers, such as the
10 displaystyle 10 have no finite binary fractional representation; and is generally
impossible for multiplication (or division).[10][11] See
The world's earliest decimal multiplication table was made from bamboo
slips, dating from 305 BC, during the
Many ancient cultures calculated with numerals based on ten, sometimes
argued due to human hands typically having ten digits.[12]
Standardized weights used in
The world's earliest positional decimal system Upper row vertical form Lower row horizontal form History of decimal fractions[edit] counting rod decimal fraction 1/7
寸 , meaning 寸 096644 [27]
J. Lennart Berggren notes that positional decimal fractions appear for
the first time in a book by the Arab mathematician Abu'l-Hasan
al-Uqlidisi written in the 10th century.[28] The Jewish mathematician
A forerunner of modern European decimal notation was introduced by
Units of information shannon or bit (base 2) nat (base e) hartley, ban or dit (base 10) qubit (quantum) v t e Some cultures do, or did, use other bases of numbers.
See also[edit] Algorism
Binary-coded decimal
References[edit] ^ The History of Arithmetic, Louis Charles Karpinski, 200pp, Rand
McNally & Company, 1925.
^
External links[edit]
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