Computers have a funny, uneasy relationship to decimal numbers, however.
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This binary code is then rewritten as decimal numbers, which are then converted into portable game notation, a computer-readable equivalent of algebraic notation used by many computer chess games.
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The exponential values required to scale most decimal numbers (that is, determine their decimal point locations) are usually a lot less what can be represented by five bits, as are allocated in a 26-bit float, but the current standard tries to prepare for the worst case.
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Decimal numbers are an essential part of the metric system, with only one base unit and multiples created on the decimal base, the figures remain the same. This simplifies calculations. Although the Indians used decimal numbers for mathematical computations, it was Simon Stevin who in 1585 first advocated the use of decimal numbers for everyday purposes in his booklet De Thiende (old Dutch for 'the tenth'). He also declared that it would only be a matter of time before decimal numbers were used for currencies and measurements.
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Decimal numbers are an essential part of the metric system, with only one base unit and multiples created on the decimal base, the figures remain the same. This simplifies calculations. Although the Indians used decimal numbers for mathematical computations, it was Simon Stevin who in 1585 first advocated the use of decimal numbers for everyday purposes in his booklet De Thiende (Middle Dutch for 'The Tenth'). He also declared that it would only be a matter of time before decimal numbers were used for currencies and measurements.
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In 1840, Augustin Cauchy also expressed preference for such modified decimal numbers to reduce errors in computation.
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Reverse DNS lookups for IPv4 addresses use the special domain `in-addr.arpa`. In this domain, an IPv4 address is represented as a concatenated sequence of four decimal numbers, separated by dots, to which is appended the second level domain suffix `.in-addr.arpa`. The four decimal numbers are obtained by splitting the 32-bit IPv4 address into four octets and converting each octet into a decimal number. These decimal numbers are then concatenated in the order: least significant octet first (leftmost), to most significant octet last (rightmost).
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This system has been extended to represent some non-integer numbers, called decimal fractions or decimal numbers, for forming the decimal numeral system.
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Binary- coded decimal (BCD) numbers are used for storing decimal numbers, especially in financial software. The opcodes mentioned above give the x86 rudimentary BCD support.
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The base b may also be indicated by the phrase "base-b". So binary numbers are "base-2"; octal numbers are "base-8"; decimal numbers are "base-10"; and so on. To a given radix b the set of digits {0, 1, ..., b−2, b−1} is called the standard set of digits. Thus, binary numbers have digits {0, 1}; decimal numbers have digits and so on.
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The Devanagari numerals are the symbols used to write numbers in the Devanagari script, the predominant script in India. They are used to write decimal numbers, instead of the Western Arabic numerals.
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While the LMC does reflect the actual workings of binary processors, the simplicity of decimal numbers was chosen to minimize the complexity for students who may not be comfortable working in binary/hexadecimal.
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Dot-decimal notation is a presentation format for numerical data. It consists of a string of decimal numbers, using the full stop (dot) as a separation character. A common use of dot-decimal notation is in information technology where it is a method of writing numbers in octet-grouped base-10 (decimal) numbers. In computer networking, Internet Protocol Version 4 (IPv4) addresses are commonly written using the quad-dotted notation of four decimal integers, ranging from 0 to 255 each.
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Dot-decimal notation is a presentation format for numerical data expressed as a string of decimal numbers each separated by a full stop. For example, the hexadecimal number 0xFF000000 may be expressed in dot-decimal notation as 255.0.0.0. In computer networking, the notation is associated with the specific use of quad-dotted notation to represent IPv4 addresses. and used as a synonym for dotted quad notation, Object identifiers use a style of dot-decimal notation to represent an arbitrarily deep hierarchy of objects identified by decimal numbers.
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The suanpan is a 2:5 abacus: two heaven beads and five earth beads. If one compares the suanpan to the soroban which is a 1:4 abacus, one might think there are two "extra" beads in each column. In fact, to represent decimal numbers and add or subtract such numbers, one strictly needs only one upper bead and four lower beads on each column. Some "old" methods to multiply or divide decimal numbers use those extra beads like the "Extra Bead technique" or "Suspended Bead technique".
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Decimal numbers written with leading zeros will be interpreted as octal by languages that follow this convention and will generate errors (not just unexpected results) if they contain "8" or "9", since these digits do not exist in octal. This behavior can be quite a nuisance when working with sequences of strings with embedded, zero-padded decimal numbers (typically file names) to facilitate alphabetical sorting (see above) or when validating inputs from users who would not know that adding a leading zero triggers this base conversion.
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An even parity bit follows, then a stop bit at 1,200 Hz. For example, the bits corresponding the decimal numbers 44 or 32 imply that the mode is Martin M1, whereas the number 60 represents Scottie S1.
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He defines the lexicographical order and an addition operation, noting that 0.999... < 1 simply because 0 < 1 in the ones place, but for any nonterminating x, one has 0.999... + x = 1 + x. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to . After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.Richman pp. 397–399 In the process of defining multiplication, Richman also defines another system he calls "cut D", which is the set of Dedekind cuts of decimal fractions.
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Colloquially, decimal numbers are formed by saying (, Pali for 'tenth') where the decimal separator is located. For example, 10.1 is (). Half (1/2) is expressed primarily by (), although , and are also used. Quarter (1/4) is expressed with () or .
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At the end, he tied (having exactly the same decimal numbers) the third prize with his brother Dong-Hyek Lim (while there was no second prize) as the first Koreans in the history to be on the first five list.
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Memory cells hold signed decimal numbers from 0 to ±999 and are written with a pencil. Cells are erased with an eraser. A “bug” is provided to act as a program counter, and is placed in a hole beside the current memory cell.
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Intercom 1000 even has an optional double-precision version. As mentioned above the machine uses hexadecimal numbers, but the user never has to deal with this in normal programming. The user programs use the decimal numbers while the OS resides in the higher addresses.
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The script also includes a set of decimal numbers. Layout of the Rohingya virtual keyboard. A virtual keyboard was developed by Google for the Rohingya language in 2019 and allows users to type directly in Rohingya script. The Rohingya Unicode keyboard layout can be found here.
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Similar examples also work in other bases. For instance, the two binary numbers whose nonzero bits are in the same positions as the nonzero digits of the two decimal numbers above are also irrational reciprocals.. See in particular the discussion following Theorem 4.2. These binary and decimal numbers, and the numbers defined in the same way for any other base by repeating a single nonzero digit in the positions given by the Moser–de Bruijn sequence, are transcendental numbers. Their transcendence can be proven from the fact that the long strings of zeros in their digits allow them to be approximated more accurately by rational numbers than would be allowed by Roth's theorem if they were algebraic numbers.
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The Intel BCD opcodes are a set of x86 instructions that operate with binary coded decimal numbers. The radix used for the representation of numbers in the x86 processors is 2. This is called a binary numeral system. However, the x86 processors do have limited support for the decimal numeral system.
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The webserver software developed by Microsoft, Microsoft's Internet Information Services (IIS), returns a set of substatus codes with its 404 responses. The substatus codes take the form of decimal numbers appended to the 404 status code. The substatus codes are not officially recognized by IANA and are not returned by non-Microsoft servers.
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The input and output were in decimal numbers, with a decimal exponent and the units had special machinery for converting these to and from binary numbers. The input and output instructions would be read or written as floating-point numbers. The program tape was 35 mm film with the instructions encoded in punched holes.
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Since overbars designate repeating decimal numbers, the word foobar also signifies a number in hexadecimal, a base typically used in low level programming. That number is 0.F00F00F00 ... , which equals the decimal fraction 256/273. The term foobar was propagated through computer science circles in the 1960s and early 1970s by system manuals from Digital Equipment Corporation.
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By the end of the 7th century, decimal numbers begin to appear in inscriptions in Southeast Asia as well as in India. Some scholars hold that they appeared even earlier. A 6th century copper-plate grant at Mankani bearing the numeral 346 (corresponding to 594 CE) is often cited. But its reliability is subject to dispute.
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Assuming the era as Kalachuri era, Taralasvamin would have been a contemporary of Shankaragana. However, Taralasvamin and Nanna are not mentioned in other Kalachuri records. Also, unlike other Kalachuri inscriptions, the date in this inscription is mentioned in decimal numbers. Moreover, some expressions in the inscription appear to have been borrowed from the 7th century Sendraka inscriptions.
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The more power items collected, the higher the power gauge rises. The power gauge takes on values from 0 to 5, and decimal numbers in between. The number of options the player has corresponds to the integer part of the power gauge – e.g., if the player has a power level of 2.75, he would have two options.
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60 is the smallest number that is divisible by every number from 1 to 6; that is, it is the lowest common multiple of 1, 2, 3, 4, 5, and 6. :In this article, all sexagesimal digits are represented as decimal numbers, except where otherwise noted. For example, 10 means the number ten and 60 means the number sixty.
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Since the common logarithm of a power of is exactly the exponent, the characteristic is an integer number, which makes the common logarithm exceptionally useful in dealing with decimal numbers. For numbers less than the characteristic makes the resulting logarithm negative, as required.E. R. Hedrick, Logarithmic and Trigonometric Tables (Macmillan, New York, 1913). See common logarithm for details on the use of characteristics and mantissas.
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Traditional abacus showing 52 in decimal Inupiaq abacus to use with the Kaktovik numerals The students that invented the numerals also developed an Iñupiaq abacus in their shop. The abacus helped to convert decimal numbers into the new base-20 numerals. The upper section of the Abacus with three beads representing the sub bases also showed the non-standard positional numeral systems in their upper sectors.
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This definition is equivalent to the definition of decimal numbers as the limits of their summed components, which, in the case of 0.999..., is the limit of the sequence (0.9, 0.99, 0.999, ...). The equivalence is due to bounded increasing sequences having their limit always equal to their least upper bound. This number is equal to 1. In other words, "0.999..." and "1" represent the same number.
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The previous explanation is not a proof, as one cannot define properly the relationship between a number and its representation as a point on the number line. For the accuracy of the proof, the number , with nines after the decimal point, is denoted . Thus , , , and so on. As , with digits after the decimal point, the addition rule for decimal numbers implies and for every positive integer .
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In this case a number system with 36 symbols is used, > with the case-insensitive 'a' through 'z' equal to the decimal numbers 0 > through 25, and '0' through '9' equal to the decimal numbers 26 through 35. > Thus "kva", corresponds to the decimal number string "10 21 0". To decode this string of symbols, a sequence of thresholds will be needed, in this case (1, 1, 26). The threshold starts out as 1 and the weight is 1. The first symbol is the units place value; 'k' (=10) with a weight of 1 equals 10. After this, the threshold value is adjusted; in this case the threshold is again 1. The second symbol has a place value of 36 minus the previous threshold value, in this case, 35. Therefore, the sum of the first two symbols 'k' (=10) and 'v' (=21) is 10 × 1 + 21 × 35.
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Sony CXQ70108D 8 MHz The V20 Instruction Set Architecture (ISA) included several instructions not executed by the 8088. These included instructions for bit manipulation, packed BCD operations, multiplication, and division. They also included new real-mode instructions from the Intel 80286. The `ADD4S`, `SUB4S`, and `CMP4S` instructions were able to add, subtract, and compare huge packed binary-coded decimal numbers stored in memory. Instructions `ROL4` and `ROR4` rotate four- bit nibbles.
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Decimal numbers, while arguably more useful to work with when performing calculations, sometimes lack the precision that common fractions have. Sometimes an infinite repeating decimal is required to reach the same precision. Thus, it is often useful to convert repeating decimals into fractions. The preferred way to indicate a repeating decimal is to place a bar (known as a vinculum) over the digits that repeat, for example 0.
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Tom Tooman is an alternate reality game that CBS ran in conjunction with Jericho, beginning in August 2007. The game began with a cryptic letter posted on a website, supposedly from a Tom Tooman of Lame Deer, Montana. This letter was accompanied by a series of bar codes, some with decimal numbers and others with Mayan numbers. These numbers were used to create an IP address for a second website.
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Colloquially, decimal numbers are formed by saying chut (จุด, dot) where the decimal separator is located. For example, 1.01 is nueng chut sun nueng (หนึ่งจุดศูนย์หนึ่ง). Fractional numbers are formed by placing nai (ใน, in, of) between the numerator and denominator or using [set] x suan y ([เศษ] x ส่วน y, x parts of the whole y) to clearly indicate. For example, is nueng nai sam (หนึ่งในสาม) or [set] nueng suan sam ([เศษ]หนึ่งส่วนสาม).
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Just as decimal numbers can be represented in exponential notation, so too can hexadecimal numbers. By convention, the letter P (or p, for "power") represents times two raised to the power of, whereas E (or e) serves a similar purpose in decimal as part of the E notation. The number after the P is decimal and represents the binary exponent. Increasing the exponent by 1 multiplies by 2, not 16. 10.0p1 = 8.0p2 = 4.0p3 = 2.0p4 = 1.0p5.
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He participated in Art by Telephone at the Museum of Contemporary Art, Chicago in 1969. In 1970, Armajani contributed two works to the Museum of Modern Art exhibition Information: first, A Number Between Zero and One, a high column filled with computer printouts of individual decimal numbers; and second, North Dakota Tower, a proposed spire high and wide calculated to cast a narrow shadow over the entire length of North Dakota from east to west.
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Together they designed the programming language for the Ferranti Mark 1. She wrote the Programmers Handbook for the Ferranti Mark 1 in 1951. Whilst Turing worked on Scheme A, an early operating system, Popplewell proposed Scheme B, which allowed for decimal numbers, in 1952. Popplewell went on to become an advisor and administrator in the newly formed University of Manchester Computing Service where she was remembered as a 'universally liked' mother- figure.
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Formal instruction in numeracy begins at age 6/7 with the four primary operations of arithmetic. Fractions are introduced at age 9/10, decimal numbers and proportions at age 10/11, percentages and rates of interest at age 11/12, algebra at age 12/13. At the secondary level, topics include algebra, geometry, conics, trigonometry, probability, combinatorics and calculus. Descriptive geometry and projective geometry are introduced at age 15/16 and 16/17, respectively.
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A binary clock might use LEDs to express binary values. In this clock, each column of LEDs shows a binary-coded decimal numeral of the traditional sexagesimal time. In computing and electronic systems, binary-coded decimal (BCD) is a class of binary encodings of decimal numbers where each digit is represented by a fixed number of bits, usually four or eight. Sometimes, special bit patterns are used for a sign or other indications (e.g.
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In the first of the Supreme Court's computer software decisions (the "patent-eligibility trilogy"), Gottschalk v. Benson,. the Court reversed the CCPA's reversal of a Patent Office decision, thus denying a patent on an algorithm for converting binary-coded decimal numbers into pure binary numbers. In so ruling, the Court looked back to 19th century decisions such as O'Reilly v. Morse,. which held that abstract ideas could not be made the subject of patents.
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If a new station is added between stations of, for example, number 08 and 09, branch numbers such as "08-1" are assigned in order not to change the existing station numbers, especially in western Japan. The same method is used by city buses and highway interchanges in Japan. As an exception, JR West takes the approach to assign decimal numbers like "08.5". Meanwhile, there are cases where the station numbers are shifted and renumbered.
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The term bounded poset is sometimes used to refer to a partially ordered set that has both a least and a greatest element. Hence it is important to distinguish between a bounded-complete poset and a bounded complete partial order (cpo). For a typical example of a bounded-complete poset, consider the set of all finite decimal numbers starting with "0." (like 0.1, 0.234, 0.122) together with all infinite such numbers (like the decimal representation 0.1111... of 1/9).
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He wrote two texts on mathematics and astronomy: The Brahma Sphuta Siddhanta in 628, and the Khandakhadyaka in 665. He made seminal contributions to mathematics, including the first mathematical treatment of zero, rules for manipulating positive and negative numbers, as well as algorithms for algebraic operations on decimal numbers. His work on astronomy and mathematics was transmitted to the court of the Abbasid Caliph Al-Mansur (r. 754-775 CE), who had the Indian astronomical texts translated into Arabic.
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Decomposition of an IPv4 address from dot-decimal notation to its binary value. An IPv4 address has a size of 32 bits, which limits the address space to (232) addresses. Of this number, some addresses are reserved for special purposes such as private networks (~18 million addresses) and multicast addressing (~270 million addresses). IPv4 addresses are usually represented in dot-decimal notation, consisting of four decimal numbers, each ranging from 0 to 255, separated by dots, e.g.
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IPv4 addresses may be represented in any notation expressing a 32-bit integer value. They are most often written in dot-decimal notation, which consists of four octets of the address expressed individually in decimal numbers and separated by periods. For example, the quad-dotted IP address 192.0.2.235 represents the 32-bit decimal number 3221226219, which in hexadecimal format is 0xC00002EB. This may also be expressed in dotted hex format as 0xC0.0x00.0x02.0xEB, or with octal byte values as 0300.0000.0002.0353.
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Further words can be made by treating some of the decimal numbers as letters - the digit "`0`" can represent the letter "O", and "`1`" can represent the letters "I" or "L". Less commonly, "`5`" can represent "S", "`7`" represent "T", "`12`" represent "R" and "`6`" or "`9`" can represent "G" or "g", respectively. Numbers such as `2`, `4` or `8` can be used in a manner similar to leet or rebuses; e.g. the word "defecate" can be expressed either as `DEFECA7E` or `DEFEC8`.
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Now these elements can be ordered based on the prefix order of words: a decimal number n is below some other number m if there is some string of digits w such that nw = m. For example, 0.2 is below 0.234, since one can obtain the latter by appending the string "34" to 0.2. The infinite decimal numbers are the maximal elements within this order. In general, subsets of this order do not have least upper bounds: just consider the set {0.1, 0.3}.
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Her work, printed by Simon Stevin, goes well beyond what was provided at the time in calculation manuals. She wrote, Her work introduced two fundamental innovations: the decimal point (today called the virgule in French) to separate the mantissa of the decimal parts, as well as the use of a zero in the decimal part to indicate that a place is absent. In doing this, she gave form to current decimal numbers. She named the zeroes nuls as the Germans were doing.
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One of the greatest achievements of Georg Cantor was the construction of a one-to-one correspondence between the points of a square and the points of one of its edges by means of continued fractions. Kőnig found a simple method involving decimal numbers which had escaped Cantor. In 1904, at the third International Congress of Mathematicians at Heidelberg, Kőnig gave a talk to disprove Cantor's continuum hypothesis. The announcement was a sensation and was widely reported by the press.
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Very long numbers can be further grouped by doubling up separators. Typically decimal numbers (base-10) are grouped in three digit groups (representing one of 1000 possible values), binary numbers (base-2) in four digit groups (one nibble, representing one of 16 possible values), and hexadecimal numbers (base-16) in two digit groups (each digit is one nibble, so two digits are one byte, representing one of 256 possible values). Numbers from other systems (such as id numbers) are grouped following whatever convention is in use.
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Z11 in the Technical Museum in Vienna The Z11 was a computer, the first serially produced machine of the Zuse KG. Weighing , in 1955 it was built with relays and stepwise relays. Beginning in 1957 the Z11 could be programmed by punched tapes. It consumed 2 kW of electricity, and operated mechanically at a frequency of 10 to 20 Hz. Both input and output were in decimal numbers, and it used floating point arithmetic. The Z11 was first presented on the Hanover Messe in 1957.
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The ten litų () note (LTL 10) was the lowest value of Lithuanian banknotes and has been used since 1922 when Lithuania became independent from German forces after World War I. The note measures 135x65mm, just like all banknotes in Lithuania. The ten litų banknotes show the flight of airplane Lituanica by Steponas Darius and Stasys Girėnas. Word litų is a genitive case of word litai, which is plural of litas. Plural genitive case is used with decimal numbers (10, 20, 50 and so on).
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An IP address (version 4) in both dot-decimal notation and binary code An IPv4 address has 32 bits. For purposes of representation, the bits may be divided into four octets written in decimal numbers, ranging from 0 to 255, concatenated as a character string with full stop delimiters between each number. For example, the address of the loopback interface, usually assigned the host name localhost, is 127.0.0.1. It consists of the four octets, written in binary notation: 01111111, 00000000, 00000000, and 00000001. The 32-bit number is represented in hexadecimal notation as 0x7F000001.
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Technically, binary-coded decimal describes the encoding of decimal numbers where each decimal digit is represented by a fixed number of bits, usually four. With the introduction of the IBM card in 1928, IBM created a code capable of representing alphanumeric information, later adopted by other manufacturers. This code represents the numbers 0-9 by a single punch, and uses multiple punches for upper-case letters and special characters. A letter has two punches (zone [12,11,0] + digit [1–9]); most special characters have two or three punches (zone [12,11,0,or none] + digit [2–7] + 8).
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The idea of measurement and currency systems where units are related by factors of ten was suggested by Simon Stevin who in 1585 first advocated the use of decimal numbers for everyday purposes. The Metric system was developed in France in the 1790s as part of the reforms introduced during the French Revolution. Its adoption was gradual, both within France and in other countries, but its use is nearly universal today. One aspect of measurement decimalisation was the introduction of metric prefixes to derive bigger and smaller sizes from base unit names.
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The effect can be demonstrated with decimal numbers. The following example demonstrates loss of significance for a decimal floating-point data type with 10 significant digits: Consider the decimal number x = 0.1234567891234567890 A floating-point representation of this number on a machine that keeps 10 floating-point digits would be y = 0.1234567891 which is fairly close when measuring the error as a percentage of the value. It is very different when measured in order of precision. The value 'x' is accurate to , while the value 'y' is only accurate to .
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The calculator can be set to display values in binary, octal, or hexadecimal form, as well as the default decimal. When a non-decimal base is selected, calculation results are truncated to integers. Regardless of which display base is set, non- decimal numbers must be entered with a suffix indicating their base, which involves three or more extra keystrokes. When hexadecimal is selected, the row of six keys normally used for floating-point functions (trigonometry, logarithms, exponentiation, etc.) are instead allocated to the hex digits A to F (although they are physically labelled to ).
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Almost all JIS X 0208 graphic character codes are represented with two bytes of at least seven bits each. However, every control character, as well as the plain space - although not the ideographic space - is represented with a one-byte code. In order to represent the of a one-byte code, two decimal numbers – a column number and a line number – are used. Three high-order bits out of seven or four high-order bits out of eight, counting from zero to seven or from zero to fifteen respectively, form the column number.
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The invention in this case was a method of programming a general-purpose digital computer using an algorithm to convert binary-coded decimal numbers into pure binary numbers. The Supreme Court noted that phenomena of nature, mental processes and abstract intellectual concepts were not patentable, since they were the basic tools of scientific and technological work. However, new and useful inventions derived from such discoveries are patentable. The Court found that the discovery in Benson was unpatentable since the invention, an algorithm, was no more than abstract mathematics.
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In the early 1950s, most computers were built for specific numerical processing tasks, and many machines used decimal numbers as their basic number system; that is, the mathematical functions of the machines worked in base-10 instead of base-2 as is common today. These were not merely binary coded decimal (BCD). Most machines had ten vacuum tubes per digit in each processor register. Some early Soviet computer designers implemented systems based on ternary logic; that is, a bit could have three states: +1, 0, or -1, corresponding to positive, zero, or negative voltage.
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Minicomputers to the right of a vertical bar (placed to the right of the first board, representing a decimal point) may be used to represent decimal numbers. Students are instructed to represent values on the Minicomputers by adding checkers to the proper squares. To do this only requires a memorization of representations for the digits zero through nine, although non-standard representations are possible since squares can hold more than one checker. Each checker is worth the value of the square it is in, and the sum of the checkers on the board(s) determine the overall value represented.
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To convert a decimal fraction to octal, multiply by 8; the integer part of the result is the first digit of the octal fraction. Repeat the process with the fractional part of the result, until it is null or within acceptable error bounds. Example: Convert 0.1640625 to octal: :0.1640625 × 8 = 1.3125 = 1 + 0.3125 :0.3125 × 8 = 2.5 = 2 + 0.5 :0.5 × 8 = 4.0 = 4 + 0 Therefore, 0.164062510 = 0.1248. These two methods can be combined to handle decimal numbers with both integer and fractional parts, using the first on the integer part and the second on the fractional part.
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The Archimedean property: any point x before the finish line lies between two of the points P_n (inclusive). There is an elementary proof of the equation , which uses just the mathematical tools of comparison and addition of (finite) decimal numbers, without any reference to more advanced topics such as series, limits, formal construction of real numbers, etc. The proof, an exercise given by , is a direct formalization of the intuitive fact that, if one draws 0.9, 0.99, 0.999, etc. on the number line there is no room left for placing a number between them and 1.
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The ccTalk protocol is a byte-oriented protocol. The series of bytes in a message -- represented above as a series of decimal numbers -- is transmitted as 8-N-1. Many devices have single electrical connector that carries both power (typically +12 V or +24 V) and the ccTalk data over a total of 4 wires. To reduce cost, for short interconnection distances CPI recommends sending ccTalk data over an unbalanced multi-drop open-collector interface: both transmit and receive messages occur on the same bi-directional serial DATA line at TTL level, driven through an open-collector NPN transistor.
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Lonclass and UDC (like DDC) are expressed using codes based on decimal numbers. Unlike DDC, the Lonclass and UDC codes use additional punctuation to express patterns of relationships and re-usable qualifiers. While Lonclass makes a few structural adjustments to the UDC system to support its emphasis on TV and radio content, its main distinction is in the actual set of topics that are recorded within its authority file and within specific BBC catalogue records. Unlike UDC and DDC, which are widely used across the library community, Lonclass has remained a BBC-internal system since its creation in the 1960s.
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The unusual use of decimal numbers as memory addresses was initially no problem; it merely involved using 1-in-5 rather than 1-in-8 decoder logic in the core memory's row selects and bank selects. But later machines used standard memory chips that expected binary addresses. Each 1000-byte block of logical memory could be trivially mapped onto a subset of 1024 bytes in a chip with only 2.3% waste. But for denser chips and larger total memories, the entire decimal address had to be crunched into a shorter quasi binary form before sending the address to the chips, and done again for each cache or memory cycle.
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Example: The addition of two decimal numbers A typical example of carry is in the following pencil-and-paper addition: ¹ 27 \+ 59 \---- 86 7 + 9 = 16, and the digit 1 is the carry. The opposite is a borrow, as in −1 47 − 19 \---- 28 Here, , so try , and the 10 is got by taking ("borrowing") 1 from the next digit to the left. There are two ways in which this is commonly taught: # The ten is moved from the next digit left, leaving in this example in the tens column. According to this method, the term "borrow" is a misnomer, since the ten is never paid back.
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If shares in a company are traded in a financial market, the market price of the shares is used in certain financial ratios. Ratios can be expressed as a decimal value, such as 0.10, or given as an equivalent percent value, such as 10%. Some ratios are usually quoted as percentages, especially ratios that are usually or always less than 1, such as earnings yield, while others are usually quoted as decimal numbers, especially ratios that are usually more than 1, such as P/E ratio; these latter are also called multiples. Given any ratio, one can take its reciprocal; if the ratio was above 1, the reciprocal will be below 1, and conversely.
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Fixed-point decimal numbers are supported by some programming languages (such as COBOL, PL/I and Ada). These languages allow the programmer to specify an implicit decimal point in front of one of the digits. For example, a packed decimal value encoded with the bytes 12 34 56 7C represents the fixed-point value +1,234.567 when the implied decimal point is located between the 4th and 5th digits: 12 34 56 7C 12 34.56 7+ The decimal point is not actually stored in memory, as the packed BCD storage format does not provide for it. Its location is simply known to the compiler, and the generated code acts accordingly for the various arithmetic operations.
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In arithmetic and algebra the sixth power of a number n is the result of multiplying six instances of n together. So: :. Sixth powers are also formed by multiplying a number by its fifth power, the square of a number by its fourth power, or the cube of a number by itself, by taking a square to the third power, or by squaring a cube. The sequence of sixth powers of integers is: :0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1000000, 1771561, 2985984, 4826809, 7529536, 11390625, 16777216, 24137569, 34012224, 47045881, 64000000, 85766121, 113379904, 148035889, 191102976, 244140625, 308915776, 387420489, 481890304, ... They include the significant decimal numbers 106 (a million), 1006 (a short-scale trillion and long-scale billion), and 10006 (a long-scale trillion).
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An example of this construction is given in the illustration, in which the left part is represented graphically as a shape and the right part is represented as a color; in this example, each cell is updated with the shape of its left neighbor and the color of its right neighbor. Then this automaton is reversible: the values on the left side of each pair migrate rightwards and the values on the right side migrate leftwards, so the prior state of each cell can be recovered by looking for these values in neighboring cells. The operation used to combine pairs of states in this automaton forms an algebraic structure known as a rectangular band. Multiplication of decimal numbers by two or by five can be performed by a one-dimensional reversible cellular automaton with ten states per cell (the ten decimal digits).
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Each digit of the product depends only on a neighborhood of two digits in the given number: the digit in the same position and the digit one position to the right. More generally, multiplication or division of doubly infinite digit sequences in any radix , by a multiplier or divisor all of whose prime factors are also prime factors of , is an operation that forms a cellular automaton because it depends only on a bounded number of nearby digits, and is reversible because of the existence of multiplicative inverses., p. 1093. Multiplication by other values (for instance, multiplication of decimal numbers by three) remains reversible, but does not define a cellular automaton, because there is no fixed bound on the number of digits in the initial value that are needed to determine a single digit in the result.
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However, more complicated coding schemes allow a greater amount of information to be sent across the same channel, by using codewords of length greater than one. For instance, suppose that in two consecutive steps the sender transmits one of the five code words "11", "23", "35", "54", or "42". (Here, the quotation marks indicate that these words should be interpreted as strings of symbols, not as decimal numbers.) Each pair of these code words includes at least one position where its values differ by two or more modulo 5; for instance, "11" and "23" differ by two in their second position, while "23" and "42" differ by two in their first position. Therefore, a recipient of one of these code words will always be able to determine unambiguously which one was sent: no two of these code words can be confused with each other.
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