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52-437: To calculate the five trillionth digit (and the following seventy-six digits) took 13,500 CPU hours, using 25 computers from 6 countries. The forty trillionth digit required 84,500 CPU hours and 126 computers from 18 countries. The highest calculation, the one quadrillionth digit, took 1.2 million CPU hours and 1,734 computers from 56 countries. Total resources: 1,885 computers donated 1.3 million CPU hours. The average computer that

104-601: A binary representation internally (although many early computers, such as the ENIAC or the IBM 650 , used 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 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,

156-423: A decimal mark , and, for negative numbers , a minus sign "−". The decimal digits are 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 ; the decimal separator is the dot " . " in many countries (mostly English-speaking), and a comma " , " in other countries. For representing a non-negative number , a decimal numeral consists of If m > 0 , that is, if the first sequence contains at least two digits, it

208-677: 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 . Very large numbers were difficult to represent in these old numeral systems, and only

260-414: A real number x 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 and

312-605: 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. The number 0.96644 is denoted Historians of Chinese science have speculated that the idea of decimal fractions may have been transmitted from China to the Middle East. Al-Khwarizmi introduced fractions to Islamic countries in

364-428: A certain number of digits after the decimal point, which indicate the error bounds. For example, although 0.080 and 0.08 denote the same number, the decimal numeral 0.080 suggests a measurement with an error less than 0.001, while the numeral 0.08 indicates an absolute error bounded by 0.01. In both cases, the true value of the measured quantity could be, for example, 0.0803 or 0.0796 (see also significant figures ). For

416-617: A decimal system have special words for the numbers between 10 and 20, and decades. For example, in English 11 is "eleven" not "ten-one" or "one-teen". 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. Some cultures do, or did, use other bases of numbers. Egyptian numerals The system of ancient Egyptian numerals

468-412: A decimal with n digits after the separator (a point or comma) represents the fraction with denominator 10 , whose numerator is the integer obtained by removing the separator. It follows that a number is a decimal fraction if and only if it has a finite decimal representation. Expressed as fully reduced fractions , the decimal numbers are those whose denominator is a product of a power of 2 and

520-413: A different numeral system, using individual signs for the numbers 1 to 9, multiples of 10 from 10 to 90, the hundreds from 100 to 900, and the thousands from 1000 to 9000. A large number like 9999 could thus be written with only four signs—combining the signs for 9000, 900, 90, and 9—as opposed to 36 hieroglyphs. Boyer saw the new hieratic numerals as ciphered, mapping one number onto one Egyptian letter for

572-400: A fraction looked like a mouth, which meant "part": Fractions were written with this fractional solidus , i.e. , the numerator 1, and the positive denominator below. Thus, 1 ⁄ 3 was written as: Special symbols were used for 1 ⁄ 2 and for the non-unit fractions 2 ⁄ 3 and, less frequently, 3 ⁄ 4 : If the denominator became too large, the "mouth"

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624-574: A placeholder in Egyptian numeration and zero was not used in calculations. However, the symbol nefer ( nfr 𓄤, "good", "complete", "beautiful") was apparently also used for two numeric purposes: According to Carl Boyer , a deed from Edfu contained a "zero concept" replacing the magnitude in geometry. Rational numbers could also be expressed, but only as sums of unit fractions , i.e. , sums of reciprocals of positive integers, except for 2 ⁄ 3 and 3 ⁄ 4 . The hieroglyph indicating

676-522: A power of 5. Thus the smallest denominators of decimal numbers are Decimal numerals do not allow an exact representation for all real numbers . Nevertheless, they allow approximating every real number with any desired accuracy, e.g., the decimal 3.14159 approximates π , being less than 10 off; so decimals are widely used in science , engineering and everyday life. More precisely, for every real number x and every positive integer n , there are two decimals L and u with at most n digits after

728-559: A set of ten symbols emerged in India. Several Indian languages show a straightforward decimal system. Dravidian languages have numbers between 10 and 20 expressed in a regular pattern of addition to 10. The Hungarian language also uses a straightforward decimal system. All numbers between 10 and 20 are formed regularly (e.g. 11 is expressed as "tizenegy" literally "one on ten"), as with those between 20 and 100 (23 as "huszonhárom" = "three on twenty"). A straightforward decimal rank system with

780-433: A value. The numbers that may be represented in the decimal system are the decimal fractions . That is, fractions of the form a /10 , 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

832-559: A word for each order (10 十 , 100 百 , 1000 千 , 10,000 万 ), and in which 11 is expressed as ten-one and 23 as two-ten-three , and 89,345 is expressed as 8 (ten thousands) 万 9 (thousand) 千 3 (hundred) 百 4 (tens) 十 5 is found in Chinese , and in Vietnamese with a few irregularities. Japanese , Korean , and Thai have imported the Chinese decimal system. Many other languages with

884-679: 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 . d 1 d 2 ... d n ... is an infinite decimal expansion of a real number x . This expansion is unique if neither all d n are equal to 9 nor all d n are equal to 0 for n large enough (for all n greater than some natural number N ). If all d n for n > N equal to 9 and [ x ] n = [ x ] 0 . d 1 d 2 ... d n ,

936-405: 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

988-413: 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

1040-422: Is not possible in binary, because the negative powers of 10 {\displaystyle 10} have no finite binary fractional representation; and is generally impossible for multiplication (or division). See Arbitrary-precision arithmetic for exact calculations. Many ancient cultures calculated with numerals based on ten, perhaps because two human hands have ten fingers. Standardized weights used in

1092-534: Is written as such in a computer program, even though many computer languages are unable to encode that number precisely.) 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 (including decimal floating point such as in newer revisions of

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1144-482: The IEEE 754 Standard for Floating-Point Arithmetic ). Decimal arithmetic is used in computers so that decimal fractional results of adding (or subtracting) values with a fixed length of their fractional part always are computed to this same length of precision. This is especially important for financial calculations, e.g., requiring in their results integer multiples of the smallest currency unit for book keeping purposes. This

1196-569: 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. Egyptian hieroglyphs , in evidence since around 3000 BCE, used a purely decimal system, as did the Linear A script ( c.  1800–1450 BCE ) of

1248-541: The Minoans 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. The number system of classical Greece also used powers of ten, including an intermediate base of 5, as did Roman numerals . Notably, the polymath Archimedes (c. 287–212 BCE) invented a decimal positional system in his Sand Reckoner which

1300-533: The base-ten positional numeral system and denary / ˈ d iː n ər i / or decanary ) is the standard system for denoting integer and non-integer numbers . It is the extension to non-integer numbers ( decimal fractions ) of the Hindu–Arabic numeral system . The way of denoting numbers in the decimal system is often referred to as decimal notation . A decimal numeral (also often just decimal or, less correctly, decimal number ), refers generally to

1352-540: The 13.3 trillionth digit in base 10. Unlike most computations of π , which compute results in base 10 , PiHex computed in base 2 (bits), because Bellard's formula and the BBP formula could only be used to compute π in base 2 at the time. The final bit strings for each of the three calculations resulted as such: This software-engineering -related article is a stub . You can help Misplaced Pages by expanding it . Base 10 The decimal numeral system (also called

1404-587: The 15th century. 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 . 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. A method of expressing every possible natural number using

1456-432: 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 . For writing numbers, the decimal system uses ten decimal digits ,

1508-451: The decimal mark such that L ≤ x ≤ u and ( u − L ) = 10 . Numbers are very often obtained as the result of measurement . As measurements are subject to measurement uncertainty with a known upper bound , the result of a measurement is well-represented by a decimal with n digits after the decimal mark, as soon as the absolute measurement error is bounded from above by 10 . In practice, measurement results are often given with

1560-471: The decimal mark without changing the represented number; 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 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

1612-442: The decimal mark. 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 zeros. If the rational number is not a decimal fraction, the division may continue indefinitely. However, as all successive remainders are less than

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1664-401: The decimal separator (see decimal representation ). In this context, the usual decimals, with a finite number of non-zero digits after the decimal separator, are sometimes called terminating decimals . A repeating decimal is an infinite decimal that, after some place, repeats indefinitely the same sequence of digits (e.g., 5.123144144144144... = 5.123 144 ). An infinite decimal represents

1716-457: The decimal system is a positional numeral system . 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. For example, the decimal expressions 0.8 , 14.89 , 0.00079 , 1.618 , 3.14159 {\displaystyle 0.8,14.89,0.00079,1.618,3.14159} represent

1768-465: The difference of [ x ] n −1 and [ x ] n amounts to 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 x = lim n → ∞ [ x ] n {\textstyle \;x=\lim _{n\rightarrow \infty }[x]_{n}\;} or which

1820-543: 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, The converse is also true: if, at some point in the decimal representation of a number, the same string of digits starts repeating indefinitely, the number is rational. or, dividing both numerator and denominator by 6, ⁠ 692 / 1665 ⁠ . Most modern computer hardware and software systems commonly use

1872-592: 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. 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. The Jewish mathematician Immanuel Bonfils used decimal fractions around 1350 but did not develop any notation to represent them. The Persian mathematician Jamshid al-Kashi used, and claimed to have discovered, decimal fractions in

1924-566: The first time in human history. Greeks adopted the new system, mapping their counting numbers onto two of their alphabets, the Doric and Ionian. In the oldest hieratic texts the individual numerals were clearly written in a ciphered relationship to the Egyptian alphabet. But during the Old Kingdom a series of standardized writings had developed for sign-groups containing more than one numeral, repeated as Roman numerals practiced. However, repetition of

1976-430: The fractions ⁠ 4 / 5 ⁠ , ⁠ 1489 / 100 ⁠ , ⁠ 79 / 100000 ⁠ , ⁠ + 809 / 500 ⁠ and ⁠ + 314159 / 100000 ⁠ , and therefore denote decimal fractions. An example of a fraction that cannot be represented by a decimal expression (with a finite number of digits) is ⁠ 1 / 3 ⁠ , 3 not being a power of 10. More generally,

2028-405: 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. That is,

2080-617: 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.: d N , by d N + 1 , and replacing all subsequent 9s by 0s (see 0.999... ). Any such decimal fraction, i.e.: d n = 0 for n > N , may be converted to its equivalent infinite decimal expansion by replacing d N by d N − 1 and replacing all subsequent 0s by 9s (see 0.999... ). In summary, every real number that

2132-404: The notation of a number in the decimal numeral system. Decimals may sometimes be identified by a decimal separator (usually "." or "," as in 25.9703 or 3,1415 ). Decimal may also refer specifically to the digits after the decimal separator, such as in " 3.14 is the approximation of π to two decimals ". Zero-digits after a decimal separator serve the purpose of signifying the precision of

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2184-412: The number 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

2236-424: 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. Originally and in most uses, a decimal has only a finite number of digits after the decimal separator. However, the decimal system has been extended to infinite decimals for representing any real number , by using an infinite sequence of digits after

2288-462: The reconstructed Middle Egyptian forms of the numerals (which are indicated by a preceding asterisk), the transliteration of the hieroglyphs used to write them, and finally the Coptic numerals which descended from them and which give Egyptologists clues as to the vocalism of the original Egyptian numbers. A breve (˘) in some reconstructed forms indicates a short vowel whose quality remains uncertain;

2340-536: The same numeral for each place-value was not allowed in the hieratic script. As the hieratic writing system developed over time, these sign-groups were further simplified for quick writing; this process continued into Demotic , as well. Two famous mathematical papyri using hieratic script are the Moscow Mathematical Papyrus and the Rhind Mathematical Papyrus . The following table shows

2392-411: The symbol as many times as needed. For instance, a stone carving from Karnak shows the number 4,622 as: Egyptian hieroglyphs could be written in both directions (and even vertically). In this example the symbols decrease in value from top to bottom and from left to right. On the original stone carving, it is right-to-left, and the signs are thus reversed. There was no symbol or concept of zero as

2444-589: The time. As administrative and accounting texts were written on papyrus or ostraca , rather than being carved into hard stone (as were hieroglyphic texts), the vast majority of texts employing the Egyptian numeral system utilize the hieratic script. Instances of numerals written in hieratic can be found as far back as the Early Dynastic Period . The Old Kingdom Abusir Papyri are a particularly important corpus of texts that utilize hieratic numerals. Boyer proved 50 years ago that hieratic script used

2496-659: Was based on 10 . Hittite hieroglyphs (since 15th century BCE) were also strictly decimal. 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, 1000, 2000, 3000, 4000, to 10,000. The world's earliest positional decimal system

2548-473: Was just placed over the beginning of the "denominator": As with most modern day languages, the ancient Egyptian language could also write out numerals as words phonetically, just like one can write thirty instead of "30" in English . The word ( thirty ), for instance, was written as while the numeral ( 30 ) was This was, however, uncommon for most numbers other than one and two and the signs were used most of

2600-463: Was the Chinese rod calculus . 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. 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

2652-593: Was used in Ancient Egypt from around 3000 BC until the early first millennium AD. It was a system of numeration based on multiples of ten, often rounded off to the higher power, written in hieroglyphs . The Egyptians had no concept of a positional notation such as the decimal system . The hieratic form of numerals stressed an exact finite series notation, ciphered one-to-one onto the Egyptian alphabet. The following hieroglyphs were used to denote powers of ten: Multiples of these values were expressed by repeating

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2704-538: Was used to calculate would have taken 148 years to complete the calculations alone. After setting three records, calculating the five trillionth bit, the forty trillionth bit, and the quadrillionth bit, the project ended on September 11, 2000. While the PiHex project calculated the least significant digits of π ever attempted at the time in any base, the second place is held by Peter Trueb who computed some 22+ trillion digits in 2016 and third place by houkouonchi who derived

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