Decimal

A decimal system (also called base-ten, denary or decenary) is a numeral system that uses ten as its radix (base). Decimal systems are the global standard for denoting integer and non-integer numbers. The way of denoting numbers in a decimal system is often referred to as decimal notation.^[1] Presently, the most common decimal system is the Hindu–Arabic numeral system, which is a positional numeral system. However, there are also non-positional base-ten systems, such as Roman or Chinese numerals.

A decimal numeral (also often just decimal or, less correctly, decimal number), generally refers to the notation of a number in a decimal numeral system. Non-integral decimals may be identified by a decimal separator (usually "." or "," as in 25.9703 or 3.1415).^[2] In English, the word decimal often refers to the digits after the decimal separator, for example, that "3.14 is the approximation of π to two decimals" or "two decimal places."

The numbers that may be represented exactly by a decimal of finite length are the decimal fractions. That is, fractions of the form a/10ⁿ, where a is an integer, and n is a non-negative integer. Decimal fractions also result from the addition of an integer and a fractional part; the resulting sum sometimes is called a fractional number.

Decimals are commonly used to approximate real numbers. By increasing the number of digits after the decimal separator, one can make the approximation errors as small as one wants, when one has a method for computing the new digits. In the sciences, the number of decimal places given generally gives an indication of the precision to which a quantity is known; for example, if a mass is given as 1.32 milligrams, it usually means there is reasonable confidence that the true mass is somewhere between 1.315 milligrams and 1.325 milligrams, whereas if it is given as 1.320 milligrams, then it is likely between 1.3195 and 1.3205 milligrams. The same holds in pure mathematics; for example, if one computes the square root of 22 to two digits past the decimal point, the answer is 4.69, whereas computing it to three digits, the answer is 4.690. The extra 0 at the end is meaningful, in spite of the fact that 4.69 and 4.690 are the same real number.

In principle, the decimal expansion of any real number can be carried out as far as desired past the decimal point. If the expansion reaches a point where all remaining digits are zero, then the remainder can be omitted, and such an expansion is called a terminating decimal. A repeating decimal is an infinite decimal that, after some place, repeats indefinitely the same sequence of digits (e.g., 5.123144144144144... = 5.123144).^[3] An infinite decimal represents a rational number, the quotient of two integers, if and only if it is a repeating decimal or has a finite number of non-zero digits.

Many numeral systems of ancient civilizations use ten and its powers for representing numbers, possibly because there are ten fingers on two hands and people started counting by using their fingers. Examples are firstly the Egyptian numerals, then the Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, and Chinese numerals.^[4] 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.^[4]

For writing numbers, a decimal system typically uses ten decimal digits, a decimal mark, and, for negative numbers, a minus sign "−". The decimal digits in Arabic numerals are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9;^[5] the decimal separator is the dot "." in many countries (mostly English-speaking),^[6] and a comma "," in other countries.^[2]

For representing a non-negative number, a decimal numeral consists of

• either a (finite) sequence of digits (such as "2017"), where the entire sequence represents an integer:

  ${\displaystyle a_{m}a_{m-1}\ldots a_{0}}$

• or a decimal mark separating two sequences of digits (such as "20.70828")

${\displaystyle a_{m}a_{m-1}\ldots a_{0}.b_{1}b_{2}\ldots b_{n}}$.

If m > 0, that is, if the first sequence contains at least two digits, it is generally assumed that the first digit a_m is not zero. 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 the final digit on the right of the decimal mark is zero—that is, if b_n = 0—it may be removed; conversely, trailing zeros may be added after the decimal mark without changing the represented number; ^{[note 1]} for example, 15 = 15.0 = 15.00 and 5.2 = 5.20 = 5.200.

For representing a negative number, a minus sign is placed before a_m.

The numeral ${\displaystyle a_{m}a_{m-1}\ldots a_{0}.b_{1}b_{2}\ldots b_{n}}$ represents the number

${\displaystyle a_{m}10^{m}+a_{m-1}10^{m-1}+\cdots +a_{0}10^{0}+{\frac {b_{1}}{10^{1}}}+{\frac {b_{2}}{10^{2}}}+\cdots +{\frac {b_{n}}{10^{n}}}}$.

The integer part or integral part of a decimal numeral is the integer written to the left of the decimal separator (see also truncation). For a non-negative decimal numeral, 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.

In brief, the contribution of each digit to the value of a number depends on its position in the numeral.

Decimal fractions (sometimes called decimal numbers, especially in contexts involving explicit fractions) are the rational numbers that may be expressed as a fraction whose denominator is a power of ten.^[7] For example, the decimal expressions ${\displaystyle 0.8,14.89,0.00079,1.618,3.14159}$representan las fracciones ⁠ 8 / 10 ⁠ , ⁠ 1489 / 100 ⁠ , ⁠ 79 / 100000 ⁠ , ⁠ + 1618 / 1000 ⁠ y ⁠ + 314159 / 100000 ⁠ , y por lo tanto denotan fracciones decimales. Un ejemplo de una fracción que no puede representarse mediante una expresión decimal (con un número finito de dígitos) es ⁠ 1 / 3 ⁠ , ya que 3 no es una potencia de 10.

En términos más generales, un decimal con n dígitos después del separador (un punto o una coma) representa la fracción con denominador 10 ⁿ , cuyo numerador es el entero obtenido al eliminar el separador.

De ello se deduce que un número es una fracción decimal si y solo si tiene una representación decimal finita.

Expresados ​​como fracciones totalmente simplificadas , los números decimales son aquellos cuyo denominador es producto de una potencia de 2 y una potencia de 5. Por lo tanto, los denominadores más pequeños de los números decimales son:

${\displaystyle 1=2^{0}\cdot 5^{0},2=2^{1}\cdot 5^{0},4=2^{2}\cdot 5^{0},5=2^{0}\cdot 5^{1},8=2^{3}\cdot 5^{0},10=2^{1}\cdot 5^{1},16=2^{4}\cdot 5^{0},20=2^{2}\cdot 5^{1},25=2^{0}\cdot 5^{2},\ldots }$

For a real numberx and an integer n ≥ 0, let [x]_n denote the (finite) decimal expansion of the greatest number that is not greater than x that has exactly n digits after the decimal mark. Let d_i denote the last digit of [x]_i. It is straightforward to see that [x]_n may be obtained by appending d_n to the right of [x]_n−1. This way one has

[x]_n = [x]₀.d₁d₂...d_n−1d_n,

and the difference of [x]_n−1 and [x]_n amounts to

${\displaystyle \left\vert \left[x\right]_{n}-\left[x\right]_{n-1}\right\vert =d_{n}\cdot 10^{-n}<10^{-n+1}}$,

which is either 0, if d_n = 0, or gets arbitrarily small as 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${\textstyle \;x=\lim _{n\rightarrow \infty }[x]_{n}\;}$or

x = [x]₀.d₁d₂...d_n...,

which is called an infinite decimal expansion of x.

Por el contrario, para cualquier entero [ x ] ₀ y cualquier secuencia de dígitos${\textstyle \;(d_{n})_{n=1}^{\infty }}$La expresión (infinita) [ x ] ₀ . d ₁ d ₂ ... d _n ... es una expansión decimal infinita de un número real x . Esta expansión es única si ni todos los d _n son iguales a 9 ni todos los d _n son iguales a 0 para n suficientemente grande (para todo n mayor que algún número natural N ).

Si todos los d _n para n > N son iguales a 9 y [ x ] _n = [ x ] ₀ . d ₁ d ₂ ... d _n , el límite de la secuencia${\textstyle \;([x]_{n})_{n=1}^{\infty }}$es la fracción decimal que se obtiene al reemplazar el último dígito que no es un 9, es decir: d _N , por d _N + 1 , y reemplazar todos los 9 subsiguientes por 0 (ver 0,999... ).

Cualquier fracción decimal de este tipo, es decir: d _n = 0 para n > N , puede convertirse a su expansión decimal infinita equivalente reemplazando d _N por d _N − 1 y reemplazando todos los 0 subsiguientes por 9 (ver 0,999... ).

En resumen, cada número real que no es una fracción decimal tiene una única expansión decimal infinita. Cada fracción decimal tiene exactamente dos expansiones decimales infinitas: una que contiene solo ceros después de alguna posición, que se obtiene mediante la definición anterior de [ x ] _n , y otra que contiene solo nueves después de alguna posición, que se obtiene definiendo [ x ] _n como el mayor número menor que x , que tiene exactamente n dígitos después de la coma decimal.

Most modern computer hardware and software systems commonly use a binary representation internally (although many early computers, such as the ENIAC or the IBM 650, used decimal representation internally).^[8] 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 or from the equivalent decimal values for presentation to or input from humans; computer programs express literals in decimal by default. (123.1, for example, is written as such in a computer program, even though many computer languages are unable to encode that number precisely.)

Tanto el hardware como el software informático utilizan representaciones internas que son efectivamente decimales para almacenar valores decimales y realizar operaciones aritméticas. A menudo, estas operaciones se realizan sobre datos codificados mediante alguna variante del sistema decimal codificado en binario (BCD ) , ^{[ 9 ] [ 10 ]} especialmente en implementaciones de bases de datos, pero existen otras representaciones decimales en uso (incluida la coma flotante decimal, como en las revisiones más recientes del estándar IEEE 754 para aritmética de coma flotante ). ^{[ 11 ]}

La aritmética decimal se utiliza en las computadoras para que los resultados fraccionarios decimales de la suma (o resta) de valores con una longitud fija de su parte fraccionaria siempre se calculen con esa misma longitud de precisión. Esto es especialmente importante para los cálculos financieros, por ejemplo, que requieren en sus resultados múltiplos enteros de la unidad monetaria más pequeña para fines contables. Esto no es posible en binario, porque las potencias negativas de${\displaystyle 10}$no tienen representación fraccionaria binaria finita; y generalmente es imposible multiplicarlas (o dividirlas). ^{[ 12 ] [ 13 ]} Véase Aritmética de precisión arbitraria para cálculos exactos.

Many ancient cultures calculated with numerals based on ten, perhaps because two human hands have ten fingers.^[14] Standardized weights used in the Indus Valley Civilisation (c.3300–1300 BCE) were based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, while their standardized ruler – the Mohenjo-daro ruler – was divided into ten equal parts.^[15][16][17]Egyptian hieroglyphs, in evidence since around 3000 BCE, used a purely decimal system,^[18] as did the Linear A script (c.1800–1450 BCE) of the Minoans^[19][20] and the Linear B script (c. 1400–1200 BCE) of the Mycenaeans. The Únětice culture in central Europe (2300-1600 BC) used standardised weights and a decimal system in trade.^[21] The number system of classical Greece also used powers of ten, including an intermediate base of 5, as did Roman numerals.^[22] Notably, the polymath Archimedes (c. 287–212 BCE) invented a decimal positional system in his Sand Reckoner which was based on 10⁸.^[22][23]Hittite hieroglyphs (since 15th century BCE) were also strictly decimal.^[24]

The Egyptian hieratic numerals, the Greek alphabet numerals, the Hebrew alphabet numerals, the Roman numerals, the Chinese numerals and early Indian Brahmi numerals are all non-positional decimal systems, and required large numbers of symbols. For instance, Egyptian numerals used different symbols for 10, 20 to 90, 100, 200 to 900, 1,000, 2,000, 3,000, 4,000, to 10,000.^[25] The world's earliest positional decimal system was the Chinese rod calculus.^[26]

History of decimal fractions

Starting from the 2nd century BCE, some Chinese units for length were based on divisions into ten; by the 3rd century CE these metrological units were used to express decimal fractions of lengths, non-positionally.^[27] Calculations with decimal fractions of lengths were performed using positional counting rods, as described in the 3rd–5th century CE Sunzi Suanjing. The 5th century CE mathematician Zu Chongzhi calculated a 7-digit approximation of π. Qin Jiushao's book Mathematical Treatise in Nine Sections (1247) explicitly writes a decimal fraction representing a number rather than a measurement, using counting rods.^[28] The number 0.96644 is denoted

寸

.

Historians of Chinese science have speculated that the idea of decimal fractions may have been transmitted from China to the Middle East.^[26]

Al-Khwarizmi introduced fractions to Islamic countries in the early 9th century CE, written with a numerator above and denominator below, without a horizontal bar. This form of fraction remained in use for centuries.^[26][29]

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.^[30] The Jewish mathematician Immanuel Bonfils used decimal fractions around 1350 but did not develop any notation to represent them.^[31] The Persian mathematician Jamshid al-Kashi significantly advanced the theory in the 15th century. In his work, The Key to Arithmetic (Miftah al-Hisab), al-Kashi provided the first systematic and comprehensive treatment of decimal fractions as a complete system, predating similar European developments by nearly 175 years.^[32]

A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century. Stevin's influential booklet De Thiende ("the art of tenths") was first published in Dutch in 1585 and translated into French as La Disme.^[33]

John Napier introduced using the period (.) to separate the integer part of a decimal number from the fractional part in his book on constructing tables of logarithms, published posthumously in 1620.^{[34]:p. 8,archive p. 32}