In mathematics, a negative number is a real number, less than zero. Negative numbers represent opposites. If positive represents a movement to the right, negative represents a movement to the left. If positive represents above sea level negative represents below sea level. If positive represents a deposit, negative represents a withdrawal, they are used to represent the magnitude of a loss or deficiency. A debt, owed may be thought of as a negative asset, a decrease in some quantity may be thought of as a negative increase. If a quantity may have either of two opposite senses one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature; the laws of arithmetic for negative numbers ensure that the common sense idea of an opposite is reflected in arithmetic.
For example, − = 3 because the opposite of an opposite is the original value. Negative numbers are written with a minus sign in front. For example, −3 represents a negative quantity with a magnitude of three, is pronounced "minus three" or "negative three". To help tell the difference between a subtraction operation and a negative number the negative sign is placed higher than the minus sign. Conversely, a number, greater than zero is called positive; the positivity of a number may be emphasized by placing a plus sign before it, e.g. +3. In general, the negativity or positivity of a number is referred to as its sign; every real number other than zero is either negative. The positive whole numbers are referred to as natural numbers, while the positive and negative whole numbers are referred to as integers. In bookkeeping, amounts owed are represented by red numbers, or a number in parentheses, as an alternative notation to represent negative numbers. Negative numbers appeared for the first time in history in the Nine Chapters on the Mathematical Art, which in its present form dates from the period of the Chinese Han Dynasty, but may well contain much older material.
Liu Hui established rules for subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers. Islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of negative numbers around the middle of the 19th century. Prior to the concept of negative numbers, mathematicians such as Diophantus considered negative solutions to problems "false" and equations requiring negative solutions were described as absurd; some mathematicians like Leibniz agreed that negative numbers were false, but still used them in calculations. Negative numbers can be thought of as resulting from the subtraction of a larger number from a smaller. For example, negative three is the result of subtracting three from zero: 0 − 3 = −3. In general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers.
For example, 5 − 8 = −3since 8 − 5 = 3. The relationship between negative numbers, positive numbers, zero is expressed in the form of a number line: Numbers appearing farther to the right on this line are greater, while numbers appearing farther to the left are less, thus zero appears in the middle, with the positive numbers to the right and the negative numbers to the left. Note that a negative number with greater magnitude is considered less. For example though 8 is greater than 5, written 8 > 5negative 8 is considered to be less than negative 5: −8 < −5. It follows that any negative number is less than any positive number, so −8 < 5 and −5 < 8. In the context of negative numbers, a number, greater than zero is referred to as positive, thus every real number other than zero is either positive or negative, while zero itself is not considered to have a sign. Positive numbers are sometimes written with a plus sign in front, e.g. +3 denotes a positive three. Because zero is neither positive nor negative, the term nonnegative is sometimes used to refer to a number, either positive or zero, while nonpositive is used to refer to a number, either negative or zero.
Zero is a neutral number. Goal difference in association football and hockey. Plus-minus differential in ice hockey: the difference in total goals scored for the team and against the team when a particular player is on the ice is the player’s +/− rating. Players can have a negative rating. Run differential in baseball: the run differential is negative if the team allows more runs than they scored. British football clubs are deducted points if they enter administration, thus have a negative points total until they have earned at least that many points that season. Lap times in Formula 1 may be given as the difference compared to a previous lap, will be positive if slower and negative if faster. In some athletics events, such as sprint races, the hurdles, the triple jump and the long jump, the wind assistance is measured and recorde
In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most the symbols "0"–"9" to represent values zero to nine, "A"–"F" to represent values ten to fifteen. Hexadecimal numerals are used by computer system designers and programmers, as they provide a more human-friendly representation of binary-coded values; each hexadecimal digit represents four binary digits known as a nibble, half a byte. For example, a single byte can have values ranging from 0000 0000 to 1111 1111 in binary form, which can be more conveniently represented as 00 to FF in hexadecimal. In mathematics, a subscript is used to specify the radix. For example the decimal value 10,995 would be expressed in hexadecimal as 2AF316. In programming, a number of notations are used to support hexadecimal representation involving a prefix or suffix; the prefix 0x is used in C and related languages, which would denote this value by 0x2AF3. Hexadecimal is used in the transfer encoding Base16, in which each byte of the plaintext is broken into two 4-bit values and represented by two hexadecimal digits.
In contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously. A numerical subscript can give the base explicitly: 15910 is decimal 159; some authors prefer a text subscript, such as 159decimal and 159hex, or 159h. In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen: In URIs, character codes are written as hexadecimal pairs prefixed with %: http://www.example.com/name%20with%20spaces where %20 is the space character, ASCII code point 20 in hex, 32 in decimal. In XML and XHTML, characters can be expressed as hexadecimal numeric character references using the notation
ode, thus ’. In the Unicode standard, a character value is represented with U+ followed by the hex value, e.g. U+20AC is the Euro sign. Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits prefixed with #: white, for example, is represented #FFFFFF.
CSS allows 3-hexdigit abbreviations with one hexdigit per component: #FA3 abbreviates #FFAA33. Unix shells, AT&T assembly language and the C programming language use the prefix 0x for numeric constants represented in hex: 0x5A3. Character and string constants may express character codes in hexadecimal with the prefix \x followed by two hex digits:'\x1B' represents the Esc control character. To output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used. In MIME quoted-printable encoding, characters that cannot be represented as literal ASCII characters are represented by their codes as two hexadecimal digits prefixed by an equal to sign =, as in Espa=F1a to send "España". In Intel-derived assembly languages and Modula-2, hexadecimal is denoted with a suffixed H or h: FFh or 05A3H; some implementations require a leading zero when the first hexadecimal digit character is not a decimal digit, so one would write 0FFh instead of FFh Other assembly languages, Delphi, some versions of BASIC, GameMaker Language and Forth use $ as a prefix: $5A3.
Some assembly languages use the notation H'ABCD'. Fortran 95 uses Z'ABCD'. Ada and VHDL enclose hexadecimal numerals in based "numeric quotes": 16#5A3#. For bit vector constants VHDL uses the notation x"5A3". Verilog represents hexadecimal constants in the form 8'hFF, where 8 is the number of bits in the value and FF is the hexadecimal constant; the Smalltalk language uses the prefix 16r: 16r5A3 PostScript and the Bourne shell and its derivatives denote hex with prefix 16#: 16#5A3. For PostScript, binary data can be expressed as unprefixed consecutive hexadecimal pairs: AA213FD51B3801043FBC... Common Lisp uses the prefixes # 16r. Setting the variables *read-base* and *print-base* to 16 can be used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers, thus Hexadecimal numbers can be represented without the #x or #16r prefix code, when the input or output base has been changed to 16. MSX BASIC, QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H: &H5A3 BBC BASIC and Locomotive BASIC use & for hex.
TI-89 and 92 series uses a 0h prefix: 0h5A3 ALGOL 68 uses the prefix 16r to denote hexadecimal numbers: 16r5a3. Binary and octal numbers can be specified similarly; the most common format for hexadecimal on IBM mainframes and midrange computers running the traditional OS's is X'5A3', is used in Assembler, PL/I, COBOL, JCL, scripts and other places. This format was common on
In mathematics, the natural numbers are those used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers"; the natural numbers can, at times, appear as a convenient set of codes. Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers; the natural numbers are a basis from which many other number sets may be built by extension: the integers, by including the neutral element 0 and an additive inverse for each nonzero natural number n. These chains of extensions make the natural numbers canonically embedded in the other number systems.
Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, to contrast the discreteness of counting to the continuity of measurement, established by the real numbers; the most primitive method of representing a natural number is to put down a mark for each object. A set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set; the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers; the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, all the powers of 10 up to over 1 million.
A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, 6 ones. The Babylonians had a place-value system based on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context. A much advance was the development of the idea that 0 can be considered as a number, with its own numeral; the use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica; the use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus, beginning with Dionysius Exiguus in 525, without being denoted by a numeral; the first systematic study of numbers as abstractions is credited to the Greek philosophers Pythagoras and Archimedes.
Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes not as a number at all. Independent studies occurred at around the same time in India and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man". In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not natural but a consequence of definitions. Two classes of such formal definitions were constructed. Set-theoretical definitions of natural numbers were initiated by Frege and he defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox.
Therefore, this formalism was modified so that a natural number is defined as a particular set, any set that can be put into one-to-one correspondence with that set is said to have that number of elements. The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, further explored by Giuseppe Peano, it is based on an axiomatization of the properties of ordinal numbers: each natural number has a
Room 237 is a 2012 American documentary film directed by Rodney Ascher about interpretations and perceived meanings of Stanley Kubrick's film The Shining, adapted from the 1977 novel of the same name by Stephen King. The film includes footage from The Shining and other Kubrick films, along with discussions by a number of Kubrick enthusiasts; the film has nine segments, each segment focusing on different elements within the film which "may reveal hidden clues and hint at a bigger thematic oeuvre." The film was produced by Tim Kirk. The title refers to a room in the haunted hotel featured in The Shining, which a character is warned never to enter; the film was screened in the Directors' Fortnight section at the 2012 Cannes Film Festival and the 2012 Sundance Film Festival. The film's distribution rights were acquired by IFC Midnight and was exhibited theatrically and on VOD on March 29, 2013. Though King's novel and Kubrick's film adaptation depart in some significant details, they share the same premise.
Jack Torrance, a would-be professional writer, takes on a job as the winter caretaker of the sprawling Overlook Hotel in the Colorado mountains. His wife Wendy and son Danny accompany him for the season, the child exhibiting a psychic power called "the shining." Hotel chef Dick Hallorann possesses psychic powers, offers advice to the youngster about managing his unusual abilities. As the months wear on, the isolated family suffers from cabin fever and begin to experience paranormal events. Room 237 is told through voiceovers by people with different theories about The Shining. According to one, the film is about the genocide of Native Americans, according to the story, the hotel was built on a Native American burial mound. For instance, cans of Calumet Baking Powder are noticeable in the background of two important scenes; because a calumet is a ceremonial pipe, the cans featured the image of a Native American, one analyst believed that American imperialism was the subtext of the film. Another theorist believed that Kubrick had directed the footage disseminated by NASA to publicize the Apollo 11 Moon landing.
He believed that there are telltale signs of the use of front projection in NASA's footage and that Kubrick was contracted to produce hoaxed footage of a fake Moon landing. He points to the knitted Apollo 11 sweater that Danny wears and claims that "237" refers to the mean distance of the Earth to the Moon, he refers to the fact that a carpet pattern resembles the Apollo launching pad as evidence that the film is an elaborate apology of sorts for Kubrick's involvement. In particular, the analyst feels that the tirade Jack delivers to Wendy about how she does not understand the duty of work and honoring a contract with an employer portrays Kubrick's own sense of isolation from keeping so big a secret. One analyst connects the Overlook's hedge maze-labyrinth with the mythic story of the Minotaur, believing that a skier in a poster is a minotaur, she bolsters her theory by pointing out that there is no maze in the original book and that an earlier Kubrick film, Killer's Kiss, was made for Minotaur Productions.
Kubrick's unrealized project about the Holocaust, Aryan Papers, suggested to another analyst that The Shining is about that genocide. He connects Jack's sinister recitation of the Big Bad Wolf's refrain to a Disney production where the wolf is an anti-Semitic caricature; the analyst feels that Kubrick embeds a message of hope in Dick's advice to Danny about how to deal with his shining abilities. Dick explains that the images Danny sees are just pictures of the past, they can be forgotten; the analyst feels Kubrick is trying to remind his audience of the Holocaust while at the same time helping them to let go of its horrors. There is an extended sequence. By running the film forwards and backwards at the same time, parallels are created, such as Danny walking in on his father and the previous caretaker as they discuss Danny's murder; the filmmakers do not attempt to promote any of the particular claims made by their interview subjects. Director Rodney Ascher offered his own interpretation in an interview for Complex: My personal take on it is, for one, I don’t think it's nearly as visionary as any one of these folks have found.
I just see it as sort of a story about juggling the responsibilities of your career and family and as cautionary tale of what may happen if you make the wrong choice. And maybe looking at the ghosts as these figures that represent fortune or prestige or things that you might be chasing at the expense of paying proper attention to your family; the film features narration by Bill Blakemore, Geoffrey Cocks, Juli Kearns, John Fell Ryan and Jay Weidner. Buffy Visick appears as the VHS enthusiast; the film contains archive footage featuring Stanley Kubrick, Stephen King, Jack Nicholson, Shelley Duvall, Danny Lloyd, Scatman Crothers, Joe Turkel, Barry Nelson, Philip Stone, Barry Dennen, Keir Dullea, Martin Potter, Tom Cruise, Nicole Kidman. Room 237 opened to general acclaim from critics, it holds a 94% on Rotten Tomatoes, based on 126 critic reviews. At Metacritic, which assigns a weighted average rating out of 100 to reviews from mainstream critics, the film has received an average score of 80 based on 30 critics, considered to be "generally favorable reviews".
Manohla Dargis of The New York Times praised the film as "an ode to movie love at its most deliriously unfettered" and wrote: "The doc positions The Shining as a comparably coiled, thematically overflowing microcosm—standing in for cinema, for history, for obsession, for postmodern theory buckling under the film's heft." Owen Gleiberman of Entertainment Weekly gave the movie an "
On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences cited as Sloane's, is an online database of integer sequences. It was maintained by Neil Sloane while a researcher at AT&T Labs. Foreseeing his retirement from AT&T Labs in 2012 and the need for an independent foundation, Sloane agreed to transfer the intellectual property and hosting of the OEIS to the OEIS Foundation in October 2009. Sloane is president of the OEIS Foundation. OEIS records information on integer sequences of interest to both professional mathematicians and amateurs, is cited; as of September 2018 it contains over 300,000 sequences. Each entry contains the leading terms of the sequence, mathematical motivations, literature links, more, including the option to generate a graph or play a musical representation of the sequence; the database is searchable by subsequence. Neil Sloane started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics; the database was at first stored on punched cards.
He published selections from the database in book form twice: A Handbook of Integer Sequences, containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. The Encyclopedia of Integer Sequences with Simon Plouffe, containing 5,488 sequences and assigned M-numbers from M0000 to M5487; the Encyclopedia includes the references to the corresponding sequences in A Handbook of Integer Sequences as N-numbers from N0001 to N2372 The Encyclopedia includes the A-numbers that are used in the OEIS, whereas the Handbook did not. These books were well received and after the second publication, mathematicians supplied Sloane with a steady flow of new sequences; the collection became unmanageable in book form, when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service, soon after as a web site. As a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998; the database continues to grow at a rate of some 10,000 entries a year.
Sloane has managed'his' sequences for 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database. In 2004, Sloane celebrated the addition of the 100,000th sequence to the database, A100000, which counts the marks on the Ishango bone. In 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS.org was created to simplify the collaboration of the OEIS editors and contributors. The 200,000th sequence, A200000, was added to the database in November 2011. Besides integer sequences, the OEIS catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of rationals are represented by two sequences: the sequence of numerators and the sequence of denominators. For example, the fifth-order Farey sequence, 1 5, 1 4, 1 3, 2 5, 1 2, 3 5, 2 3, 3 4, 4 5, is catalogued as the numerator sequence 1, 1, 1, 2, 1, 3, 2, 3, 4 and the denominator sequence 5, 4, 3, 5, 2, 5, 3, 4, 5.
Important irrational numbers such as π = 3.1415926535897... are catalogued under representative integer sequences such as decimal expansions, binary expansions, or continued fraction expansions. The OEIS was limited to plain ASCII text until 2011, it still uses a linear form of conventional mathematical notation. Greek letters are represented by their full names, e.g. mu for μ, phi for φ. Every sequence is identified by the letter A followed by six digits always referred to with leading zeros, e.g. A000315 rather than A315. Individual terms of sequences are separated by commas. Digit groups are not separated by periods, or spaces. In comments, etc. A represents the nth term of the sequence. Zero is used to represent non-existent sequence elements. For example, A104157 enumerates the "smallest prime of n² consecutive primes to form an n×n magic square of least magic constant, or 0 if no such magic square exists." The value of a is 2. But there is no such 2×2 magic square, so a is 0; this special usage has a solid mathematical basis in certain counting functions.
For example, the totient valence function. There are 4 solutions for 4, but no solutions for 14, hence a of A014197 is 0—there are no solutions. −1 is used for this purpose instead, as in A094076. The OEIS ma
In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: "0" and "1". The base-2 numeral system is a positional notation with a radix of 2; each digit is referred to as a bit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by all modern computers and computer-based devices; the modern binary number system was studied in Europe in the 16th and 17th centuries by Thomas Harriot, Juan Caramuel y Lobkowitz, Gottfried Leibniz. However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt and India. Leibniz was inspired by the Chinese I Ching; the scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions. Horus-Eye fractions are a binary numbering system for fractional quantities of grain, liquids, or other measures, in which a fraction of a hekat is expressed as a sum of the binary fractions 1/2, 1/4, 1/8, 1/16, 1/32, 1/64.
Early forms of this system can be found in documents from the Fifth Dynasty of Egypt 2400 BC, its developed hieroglyphic form dates to the Nineteenth Dynasty of Egypt 1200 BC. The method used for ancient Egyptian multiplication is closely related to binary numbers. In this method, multiplying one number by a second is performed by a sequence of steps in which a value is either doubled or has the first number added back into it; this method can be seen in use, for instance, in the Rhind Mathematical Papyrus, which dates to around 1650 BC. The I Ching dates from the 9th century BC in China; the binary notation in the I Ching is used to interpret its quaternary divination technique. It is based on taoistic duality of yin and yang.eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically.
Viewing the least significant bit on top of single hexagrams in Shao Yong's square and reading along rows either from bottom right to top left with solid lines as 0 and broken lines as 1 or from top left to bottom right with solid lines as 1 and broken lines as 0 hexagrams can be interpreted as sequence from 0 to 63. The Indian scholar Pingala developed a binary system for describing prosody, he used binary numbers in the form of long syllables, making it similar to Morse code. Pingala's Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter; the binary representations in Pingala's system increases towards the right, not to the left like in the binary numbers of the modern, Western positional notation. The residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Asia. Sets of binary combinations similar to the I Ching have been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy.
In the late 13th century Ramon Llull had the ambition to account for all wisdom in every branch of human knowledge of the time. For that purpose he developed a general method or ‘Ars generalis’ based on binary combinations of a number of simple basic principles or categories, for which he has been considered a predecessor of computing science and artificial intelligence. In 1605 Francis Bacon discussed a system whereby letters of the alphabet could be reduced to sequences of binary digits, which could be encoded as scarcely visible variations in the font in any random text. For the general theory of binary encoding, he added that this method could be used with any objects at all: "provided those objects be capable of a twofold difference only. John Napier in 1617 described a system he called location arithmetic for doing binary calculations using a non-positional representation by letters. Thomas Harriot investigated several positional numbering systems, including binary, but did not publish his results.
The first publication of the system in Europe was by Juan Caramuel y Lobkowitz, in 1700. Leibniz studied binary numbering in 1679. Leibniz's system uses 1, like the modern binary numeral system. An example of Leibniz's binary numeral system is as follows: 0 0 0 1 numerical value 20 0 0 1 0 numerical value 21 0 1 0 0 numerical value 22 1 0 0 0 numerical value 23Leibniz interpreted the hexagrams of the I Ching as evidence of binary calculus; as a Sinophile, Leibniz was aware of
The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet. Roman numerals, as used today, employ seven symbols, each with a fixed integer value, as follows: The use of Roman numerals continued long after the decline of the Roman Empire. From the 14th century on, Roman numerals began to be replaced in most contexts by the more convenient Arabic numerals; the original pattern for Roman numerals used the symbols I, V, X as simple tally marks. Each marker for 1 added a unit value up to 5, was added to to make the numbers from 6 to 9: I, II, III, IIII, V, VI, VII, VIII, VIIII, X; the numerals for 4 and 9 proved problematic, are replaced with IV and IX. This feature of Roman numerals is called subtractive notation; the numbers from 1 to 10 are expressed in Roman numerals as follows: I, II, III, IV, V, VI, VII, VIII, IX, X.
The system being decimal and hundreds follow the same underlying pattern. This is the key to understanding Roman numerals: Thus 10 to 100: X, XX, XXX, XL, L, LX, LXX, LXXX, XC, C. Note that 40 and 90 follow the same subtractive pattern as 4 and 9, avoiding the confusing XXXX. 100 to 1000: C, CC, CCC, CD, D, DC, DCC, DCCC, CM, M. Again - 400 and 900 follow the standard subtractive pattern, avoiding CCCC. In the absence of standard symbols for 5,000 and 10,000 the pattern breaks down at this point - in modern usage M is repeated up to three times; the Romans had several ways to indicate larger numbers, but for practical purposes Roman Numerals for numbers larger than 3,999 are if used nowadays, this suffices. M, MM, MMM. Many numbers include hundreds and tens; the Roman numeral system being decimal, each power of ten is added in descending sequence from left to right, as with Arabic numerals. For example: 39 = "Thirty nine" = XXXIX. 246 = "Two hundred and forty six" = CCXLVI. 421 = "Four hundred and twenty one" = CDXXI.
As each power of ten has its own notation there is no need for place keeping zeros, so "missing places" are ignored, as in Latin speech, thus: 160 = "One hundred and sixty" = CLX 207 = "Two hundred and seven" = CCVII 1066 = "A thousand and sixty six" = MLXVI. Roman numerals for large numbers are nowadays seen in the form of year numbers, as in these examples: 1776 = MDCCLXXVI. 1954 = MCMLIV 1990 = MCMXC. 2014 = MMXIV (the year of the games of the XXII Olympic Winter Games The current year is MMXIX. The "standard" forms described above reflect typical modern usage rather than an unchanging and universally accepted convention. Usage in ancient Rome varied and remained inconsistent in medieval times. There is still no official "binding" standard, which makes the elaborate "rules" used in some sources to distinguish between "correct" and "incorrect" forms problematic. "Classical" inscriptions not infrequently use IIII for "4" instead of IV. Other "non-subtractive" forms, such as VIIII for IX, are sometimes seen, although they are less common.
On the numbered gates to the colosseum, for instance, IV is systematically avoided in favour of IIII, but other "subtractives" apply, so that gate 44 is labelled XLIIII. Isaac Asimov speculates that the use of "IV", as the initial letters of "IVPITER" may have been felt to have been impious in this context. Clock faces that use Roman numerals show IIII for four o'clock but IX for nine o'clock, a practice that goes back to early clocks such as the Wells Cathedral clock of the late 14th century. However, this is far from universal: for example, the clock on the Palace of Westminster, Big Ben, uses a "normal" IV. XIIX or IIXX are sometimes used for "18" instead of XVIII; the Latin word for "eighteen" is rendered as the equivalent of "two less than twenty" which may be the source of this usage. The standard forms for 98 and 99 are XCVIII and XCIX, as described in the "decimal pattern" section above, but these numbers are rendered as IIC and IC originally from the Latin duodecentum and undecentum.
Sometimes V and L are not used, with instances such as IIIIII and XXXXXX rather than VI or LX. Most non-standard numerals other than those described above - such as VXL for 45, instead of the standard XLV are modern and may be due to error rather than being genuine variant usage. In the early years of the 20th century, different representations of 900 appeared in several inscribed dates. For instance, 1910 is shown on Admiralty Arch, London, as MDCCCCX rather than MCMX, while on the north entrance to the Saint Louis Art Museum, 1903 is inscribed as MDCDIII rather than MCMIII. Although Roman numerals came to be written with letters