Look up template in Wiktionary, the free dictionary.

Template may mean:

• a stencil, pattern or overlay used in graphic arts (drawing, painting, etc.) and sewing to replicate letters, shapes or designs
• a pre-developed page layout in electronic or paper media used to make new pages with a similar design, pattern, or style
• Form letter, a predefined letter which retains its primary intent when individually customised with variable data or text.

Computer science and information technology[link]

• Template (file format), a standardized non-executable file type used by computer software as a pre-formatted example on which to base other files, especially documents
• Template (C++), a tool for generic programming especially in the C++ language
• Template metaprogramming, a programming technique used by a compiler to generate temporary source code
• Template method, an object-oriented design pattern
• Template processor, a system that combines a template with data to produce an output.
• Template (word processing), a standard document containing layout and styles used to configure word processing software
• Style sheet (web development) or master page on which you can globally edit and format graphic elements and text common to each page of a document
• Web template or web template system, a master page used to produce web pages
• A main document from which mail merge documents are created

Manufacturing[link]

• Mechanical template, used to cut and shape a product, such as:
  • a die (manufacturing)
  • a mold

Molecular genetics[link]

• a strand of DNA which sets the genetic sequence of new strands
• a strand of RNA which translates genes into proteins

Miscellaneous[link]

• Template (racing), a device used in car racing to ensure that the body of the race vehicle adheres to specifications

This disambiguation page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.

The examples and perspective in this article may not represent a worldwide view of the subject. Please improve this article and discuss the issue on the talk page. (September 2008)

"world's tallest billboard" – One Times Square, New York

Billboards on Times Square, New York

A billboard (sometimes also called a hoarding in the UK and many other parts of the world) is a large outdoor advertising structure (a billing board), typically found in high traffic areas such as alongside busy roads. Billboards present large advertisements to passing pedestrians and drivers. Typically showing large, ostensibly witty slogans, and distinctive visuals, billboards are highly visible in the top designated market areas. Bulletins are the largest, most impactful standard-size billboards. Located primarily on major highways, expressways or principal arterials, they command high-density consumer exposure (mostly to vehicular traffic). Bulletins afford greatest visibility due not only to their size, but because they allow creative "customizing" through extensions and embellishments.

Posters are the other common form of billboard advertising, located chiefly in commercial and industrial areas on primary and secondary arterial roads. Posters are a smaller format than bulletins and are viewed principally by residents and commuter traffic, with some pedestrian exposure.

Painted billboards[link]

Almost all these billboards were painted in large studios. The image was projected on the series of panels that made up the billboard, then "pounced" on the board, marking the outlines of the figures or objects. Then, using oil paints, the artists would actually use large brushes to paint the image. Once the panels were installed using large hydraulic booms on trucks, the artists would go up on the installed billboard and touch up the edges between the panels. These large, painted billboards were especially popular in Los Angeles where historic firms such as Foster & Kleiser and Pacific Outdoor Advertising dominated the industry. Eventually, these painted billboards gave way to graphic reproduction, but hand-painted billboards are still in use in some areas where only a single board or two is required. The "Sunset Strip" in Los Angeles is one area where hand-painted billboards can still be found, usually to advertise upcoming films or albums in the heart of the entertainment industry.

Advertising style[link]

Billboard advertisements are designed to catch a person's attention and create a memorable impression very quickly, leaving the reader thinking about the advertisement after they have driven past it. They have to be readable in a very short time because they are usually read while being passed at high speeds. Thus there are usually only a few words, in large print, and a humorous or arresting image in brilliant color.

Some billboard designs spill outside the actual space given to them by the billboard, with parts of figures hanging off the billboard edges or jutting out of the billboard in three dimensions. An example in the United States around the turn of the 21st century were the Chick-fil-A billboards (a chicken sandwich fast food chain), which had three-dimensional cow figures in the act of painting the billboards with misspelled anti-beef slogans such as "frendz don't let frendz eat beef."

The first "scented billboard," an outdoor sign emitting the odors of black pepper and charcoal to suggest a grilled steak, was erected on NC 150 near Mooresville, North Carolina by the Bloom grocery chain. The sign depicted a giant cube of beef being pierced by a large fork that extended to the ground. The scents were emitted between 7–10 a.m. and 4– to 7 pm from May 28, 2010 through June 18, 2010.^[1]

Digital billboards[link]

Digital billboard with changing images, Ypsilanti, MI

Billboard at the closed Forum Hotel in Kraków, Poland. It is the biggest billboard in Europe. It displays a new advertisement every month.

A digital billboard is a billboard that is created from computer programs and software. Digital billboards can be designed to display running text, display several different displays from the same company, and even exist to provide several companies a certain slot of time during the day. Because of the versatility and increased potential revenue for these signs, they are likely to become the standard for the future.

Some companies that create the intelligence behind digital billboards are Four Winds Interactive, Scala, and Helius.

Inflatable billboards[link]

Inflatable billboard in front of a sports stadium

A multi-purpose billboard that works as advertising board, lightning structure and telecommunication antenna

Fast food billboards in Bowling Green, Kentucky

1940s 3AW billboard advertising For the Term of his Natural Life in Melbourne

An inflatable billboard is an inflatable framework with an attached banner ad. Most of them famously appear near sports events or exhibitions. Inflatable billboards can be installed nearly everywhere standing free. They are secured with counter weights and tensioning ropes.^{[citation needed]}

Multi-purpose billboards[link]

Some billboards are not used only for advertising ends; they can be multi-purpose, meaning that they can have more than one function. So, an advertising sign can integrate its main purpose with telecommunications antenna and/or public lighting support. Usually the structure has a steel pole with a coupling flange on the above-fitted advertising billboard structure that can contain telecommunications antennas. The lighting power supply cables and any possible antennas are placed inside of the structure and fastened on appropriate steel wires.

Other types of billboards[link]

Other types of billboards include the Billboard bicycle, which is a billboard attached to the back of a bicycle or the largest mobile billboard, a special advertising trailer to hoist big banners. Mechanical billboards are billboards that display three different billboards at diffent times, because the three advertisements are attached to a conveyor that rolls around inside the billboard. There is also such thing as a three-dimensional billboard, such as the ones at Piccadilly Circus, London, although this type of billboard is arguably mistaken for a simple advertising sign.

Placement of billboards[link]

Some of the most prominent billboards are alongside highways; since passing drivers typically have little to occupy their attention, the impact of the billboard is greater. Billboards are often drivers' primary method of finding lodging, food, and fuel on unfamiliar highways. There were approximately 450,000 billboards on United States highways as of 1991^{[citation needed]}. Somewhere between 5,000 and 15,000 are erected each year. In Europe billboards are a major component and source of income in urban street furniture concepts.

An interesting use of billboards unique to highways was the Burma-Shave advertisements between 1925 and 1963, which had 4- or 5-part messages stretched across multiple signs, keeping the reader hooked by the promise of a punchline at the end. This example is in the National Museum of American History at the Smithsonian Institution:

Shaving brushes

You'll soon see 'em

On a shelf

In some museum

Burma-Shave

These sort of multi-sign advertisements are no longer common, though they are not extinct. One example, advertising for the NCAA, depicts a basketball player aiming a shot on one billboard; on the next one, 90 yards (82 meters) away, is the basket. Another example is the numerous billboards advertising the roadside attraction South of the Border near Dillon, SC, stretching along I-95 for many states.

Many cities have high densities of billboards, especially in places where there is a lot of pedestrian traffic—Times Square in New York City is a good example. Because of the lack of space in cities, these billboards are painted or hung on the sides of buildings and sometimes are free-standing billboards hanging above buildings. Billboards on the sides of buildings create different stylistic opportunities, with artwork that incorporates features of the building into the design, such as using windows as eyes, or for gigantic frescoes that adorn the entire building.

Visual and environmental concerns[link]

The neutrality of this section is disputed. Please see the discussion on the talk page. Please do not remove this message until the dispute is resolved. (July 2009)

The Animal Liberation Front vandalized this Chick-fil-A billboard to support its vegan aims.

Many groups such as Scenic America have complained that billboards on highways cause excessive clearing of trees and intrude on the surrounding landscape, with billboards' bright colors, lights and large fonts making it difficult to focus on anything else, making them a form of visual pollution. Other groups believe that billboards and advertising in general contribute negatively to the mental climate of a culture by promoting products as providing feelings of completeness, wellness and popularity to motivate purchase. One focal point for this sentiment would be the magazine AdBusters, which will often showcase politically motivated billboard and other advertising vandalism, called culture jamming.

In 2000, rooftops in Athens had grown so thick with billboards that it was difficult to see its famous architecture. In preparation for the 2004 Summer Olympics, the city embarked on a successful four-year project demolishing the majority of rooftop billboards to beautify the city for the tourists the games will bring, overcoming resistance from advertisers and building owners. Most of these billboards were illegal, but had been ignored up to then.

In 2007, São Paulo, Brazil instituted a billboard ban because there were no viable regulations of the billboard industry. Today, São Paulo, Brazil, is working with outdoor companies in the region to rebuild the outdoor infrastructure in a way that will reflect the vibrant business climate of the city while adopting good regulations to control growth.

Road safety concerns[link]

The neutrality of this section is disputed. Please see the discussion on the talk page. Please do not remove this message until the dispute is resolved. (July 2009)

In the United States, many cities tried to put laws into effect to ban billboards as early as 1909 (California Supreme Court, Varney & Green vs. Williams) but the First Amendment has made these attempts difficult. A San Diego law championed by Pete Wilson in 1971 cited traffic safety and driver distraction as the reason for the billboard ban, but was narrowly overturned by the Supreme Court in 1981, in part because it banned non-commercial as well as commercial billboards.

Billboards have long been accused of being distracting to drivers and causing accidents. However, this may not necessarily be true, as a study by researchers at the University of North Carolina showed. Released in June 2001, the researchers prepared a thorough report on driver distraction for the AAA Foundation for Traffic Safety. This study said: "The search appears to suggest that some items—such as CB radios, billboards, and temperature controls—are not significant distractions."

Traffic safety experts have studied the relationship between outdoor advertising and traffic accidents since the 1950s, finding no authoritative or scientific evidence that billboards are linked to traffic accidents. However, many of these studies were funded by the Outdoor Advertising Association of America, which has led to accusations of bias. The methodology used in certain studies is also questionable.

The U.S. Department of Transportation, State Department of Transportation and property/casualty insurance companies statistics on fatal accidents indicate no correlation between billboards and traffic accidents. A broad sampling of law enforcement agencies across the country found no evidence to suggest that motor vehicle accidents were caused by billboards. Property and casualty insurance companies have conducted detailed studies of traffic accident records and conclude no correlation between billboards and traffic accidents.

However, studies based on correlations between traffic accidents and billboards face the problem of under-reporting: drivers are unwilling to admit responsibility for a crash, so will not admit to being distracted at a crucial moment. Even given this limitation, some studies have found higher crash rates in the vicinity of advertising using variable message signs^[2] or electronic billboards.^[3]

It is possible that advertising signs in rural areas reduce driver boredom, which many believe is a contribution to highway safety. On the other hand, drivers may fixate on a billboard which unexpectedly appears in a monotonous landscape, and drive straight into it (a phenomenon known as "highway hypnosis").^[4]

Surveys of drivers and road users show that the lighting provided by billboards provide security and visibility to many motorists. The Federal Highway Administration (FHWA) went on record (Federal Register, March 5, 1999) stating that the agency agrees that appropriately regulated billboards do not compromise highway safety. It should be noted that this statement was made before the release of the FHWA report Research review of potential safety effects of electronic billboards on driver attention and distraction^[3] in 2001. What level of regulation is appropriate for billboards in different areas is still under discussion by road safety experts around the world.

Laws limiting billboards[link]

In 1964, the negative impact of the over-proliferation of signage was abundantly evident in Houston, Texas, and it motivated Lady Bird Johnson to ask her husband to create a law. At the same time the outdoor advertising industry was becoming aware that excessive signs, some literally one in front of the other, was bad for business.

In 1965, the Highway Beautification Act was signed into law. The act applied only to "Federal Aid Primary" and "Defense" highways and limited billboards to commercial and industrial zones created by states and municipalities. It required each state to set standards based on "customary use" for the size, lighting and spacing of billboards, and prohibited city and state governments from removing billboards without paying compensation to the owner. The act requires states to maintain "effective control" of billboards or lose 5% of their federal highway dollars.

The act also required the screening of junk yards adjacent to regulated highways.

Around major holidays, volunteer groups erected highway signs offering free coffee at the next rest stop. These were specifically exempted from the limits in the act.

Currently, four states—Vermont, Alaska, Hawaii, and Maine—have prohibited billboards. Vermont's law went into effect in 1968,^[5] Hawaii's law went into effect in 1927,^[6] Maine's law went into effect in 1979,^[7] and Alaska's law went into effect upon its achievement of statehood in 1959.

In the UK, billboards are controlled as adverts as part of the planning system. To display an illegal advert (that is, without planning permission) is a criminal offence with a fine of up to £2500 per offence (per poster). All of the large UK outdoor advertisers such as CBS Outdoor, JCDecaux, Clear Channel, Titan and Primesight have numerous convictions for such crimes.^[8][9]

In Toronto, Canada, a municipal tax on billboards was implemented in April 2010. A portion of the tax will go to help fund arts programs in the city.^[10]

Usages[link]

Highway[link]

Many signs advertise local restaurants and shops in the coming miles, and are crucial to drawing business in small towns. One example is Wall Drug, which in 1931 erected billboards advertising "free ice water" and the town of Wall, South Dakota was essentially built around the 20,000 customers per day those billboards brought in (as of 1981). Some signs were placed at great distances, with slogans such as "only 827 miles to Wall Drug, with FREE ice water." In some areas the signs were so dense that one almost immediately followed the last. This situation changed after the Highway Beautification Act was passed; the proliferation of Wall Drug billboards is sometimes cited as one of the reasons the bill was passed. After the passage of the act, other states (such as Oregon^[11]) embarked on highway beautification efforts.

Railway[link]

Billoard advertising in underground stations, especially, is perhaps a place where they find a greater degree of acceptability and may assist in maintaining a neat, vibrant and safe atmosphere if not too distracting. Museum Station, Sydney has mounted restored 1940's billboard panels along the platforms that are in keeping with its heritage listing.

Big name advertisers[link]

Billboards are also used to advertise national or global brands, particularly in more densely populated urban areas. According to the Outdoor Advertising Association of America, the top three companies advertising on billboards as of 2009 were McDonald's, Verizon Long Distance and Pepsi. A large number of wireless phone companies, movie companies, car manufacturers and banks are high on the list as well.

Tobacco advertising[link]

Mail Pouch Barn advertisement: A bit of Americana in southern Ohio. Mail Pouch painted the barns for free.

Prior to 1999, billboards were a major venue of cigarette advertising; 10% of Michigan billboards advertise alcohol and tobacco, according to the Detroit Free Press.^[12] This is particularly true in countries where tobacco advertisements are not allowed in other media. For example, in the US, tobacco advertising was banned on radio and television in 1971, leaving billboards and magazines as some of the last places tobacco could be advertised. Billboards made the news in America when, in the tobacco settlement of 1999, all cigarette billboards were replaced with anti-smoking messages.^{[citation needed]} In a parody of the Marlboro Man, some billboards depicted cowboys riding on ranches with slogans like "Bob, I miss my lung."

Likely the best-known of the tobacco advertising boards were those for "Mail Pouch" chewing tobacco in the United States during the first half of the 20th century (pictured above). The company agreed to paint two or three sides of a farmer's barn any color he chose in exchange for painting their advertisement on the one or two sides of the structure facing the road. The company has long since abandoned this form of advertising, and none of these advertisements have been painted in many years, but some remain visible on rural highways.

Non-commercial use[link]

Non-commercial advertisement is used around the world by governments and non-profit organisations to obtain donations, volunteer support or change consumer behavior.^[13]

Not all billboards are used for advertising products and services—non-profit groups and government agencies use them to communicate with the public. In 1999 an anonymous person created the God Speaks billboard campaign in Florida "to get people thinking about God", with witty statements signed by God. "Don't make me come down there", "We need to talk" and "Tell the children that I love them" were parts of the campaign, which was picked up by the Outdoor Advertising Association of America and continues today on billboards across the country.

South of Olympia, Washington is the privately owned Uncle Sam billboard. It features conservative, sometimes inflammatory messages, changed on a regular basis. Chehalis farmer Al Hamilton first started the board during the Johnson era, when the government was trying to make him remove his billboards along Interstate 5. He had erected the signs after he lost a legal battle to prevent the building of the freeway across his land. Numerous legal and illegal attempts to remove the Uncle Sam billboard have failed, and it is now in its third location.^[14] One message, attacking a nearby liberal arts college, was photographed, made into a postcard and is sold in the College Bookstore.

Governance[link]

The Traffic Audit Bureau for Media Measurement Inc. (TAB) was established in 1933 as a non-profit organization whose historical mission has been to audit the circulation of out-of-home media in the United States. TAB's role has expanded to lead and/or support other major out of home industry research initiatives. Governed by a tripartite board composed of advertisers, agencies and media companies, the TAB acts as an independent auditor for traffic circulation in accordance to guidelines established by its Board of Directors.

Similarly, in Canada, the Canadian Outdoor Measurement Bureau (COMB) was formed in 1965 as a non-profit organization independently operated by representatives composed of advertisers, advertising agencies and members of the Canadian out-of-home advertising industry. COMB is charged with the verification of traffic circulation for the benefit of the industry and its users.

History[link]

The examples and perspective in this article deal primarily with the United States and do not represent a worldwide view of the subject. Please improve this article and discuss the issue on the talk page. (December 2010)

Early billboards were basically large posters on the sides of buildings, with limited but still appreciable commercial value. As roads and highways multiplied, the billboard business thrived.

• 1794 – Lithography was invented, making real posters possible
• 1835 – Jared Bell was making 9 × 6 posters for the circus in the U.S.
• 1867 – Earliest known billboard rentals (source: OAAA)
• 1872 – International Bill Posters Association of North America was established (now known as the Outdoor Advertising Association of America) as a billboard lobbying group.
• 1889 - The world's first 24 sheet billboard was displayed at the Paris Exposition and later at the 1893 World's Columbian Exposition in Chicago. The format was quickly adopted for various types of advertising, especially for circuses, traveling shows, and movies
• 1908 – The Model T automobile is introduced in the U.S., increasing the number of people using highways and therefore the reach of roadside billboards.
• 1919 - Japanese candy company Glico introduces its building-spanning billboard, the Glico Man
• 1925 – Burma-Shave makes its billboards lining the highways
• 1931 – The Wall Drug billboards start to go up nationwide
• 1960 - The mechanized Kani Doraku billboard is built in Dotonbori, Osaka
• 1965 – The Highway Beautification Act is passed after much campaigning by Lady Bird Johnson
• 1971 – The Public Health Cigarette Smoking Act bans cigarette ads in television and radio, moving that business into billboards
• 1981 – The Supreme Court overturns a San Diego billboard ban, but leaves room open for other cities to ban commercial billboards
• 1986 - Non-television advertising becomes restricted – as now, non-television adverts could not show people smoking. This meant that Benson & Hedges and Silk Cut, amongst other brands – advertised their cigarettes through increasingly indirect and obscure campaigns to a point where they became recognizable.
• 1997 – Tobacco advertising is no longer allowed on outdoor billboards in America
• 2007 – Industry adopts one sheet plastic poster replacement for paper poster billboards and begins phase-out of PVC flexible vinyl, replacing it with eco-plastics such as polyethylene
• 2010 – The first "scented billboard," emitting odors similar to charcoal and black pepper to suggest a steak grilling, was erected in Mooresville, North Carolina by the Bloom grocery chain to promote the sale of beef
• 2010 - Augmented Billboards were introduced in the Transmediale Festival 2010 in Berlin using Artvertiser^[15]

Billboard owners[link]

• Adams Outdoor
• CBS Outdoor
• Airship Ventures
• Clear Channel
• JCDecaux
• Lamar Advertising Company
• Primesight
• Olympus Media

Billboard manufacturers[link]

• The AIRSCREEN Company GmbH & Co. KG
• Daktronics
• YESCO

Notable billboards[link]

• The Citgo Sign, Kenmore Square, Boston
• Cartel Coca-Cola
• Coca Cola Billboard, Kings Cross
• Coca-Cola sign
• Paddy Power Cleeve Hill Sign
• Piccadilly Circus illuminated signs

See also[link]

Wikibooks has a book on the topic of Marketing

• Advertising board
• Billboard bicycle
• Digital billboard
• Ghost sign
• Human billboard
• Marquee (sign)
• Mediascape
• Neon sign
• Poster
• Publicity
• Sales promotion
• Street furniture
• Truckside advertisement

People

• Joel Wachs, Los Angeles City Council member who wrote tough legislation against billboards

References[link]

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1 (one;  /ˈwʌn/ or UK /ˈwɒn/) is a number, a numeral, and the name of the glyph representing that number. It represents a single entity, the unit of counting or measurement. For example, a line segment of "unit length" is a line segment of length 1.

As a number[link]

One, sometimes referred to as unity, is the integer before two and after zero. One is the first non-zero number in the natural numbers as well as the first odd number in the natural numbers.

Any number multiplied by one is the number, as one is the identity for multiplication. As a result, one is its own factorial, its own square, its own cube, and so on. One is also the empty product, as any number multiplied by one is itself, which produces the same result as multiplying by no numbers at all.

As a digit[link]

The glyph used today in the Western world to represent the number 1, a vertical line, often with a serif at the top and sometimes a short horizontal line at the bottom, traces its roots back to the Indians, who wrote 1 as a horizontal line, much like the Chinese character 一. The Gupta wrote it as a curved line, and the Nagari sometimes added a small circle on the left (rotated a quarter turn to the right, this 9-look-alike became the present day numeral 1 in the Gujarati and Punjabi scripts). The Nepali also rotated it to the right but kept the circle small.^[1] This eventually became the top serif in the modern numeral, but the occasional short horizontal line at the bottom probably originates from similarity with the Roman numeral Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{I} . In some European countries (e.g., Germany), the little serif at the top is sometimes extended into a long upstroke, sometimes as long as the vertical line, which can lead to confusion with the glyph for seven in other countries. Where the 1 is written with a long upstroke, the number 7 has a horizontal stroke through the vertical line.

While the shape of the 1 character has an ascender in most modern typefaces, in typefaces with text figures, the character usually is of x-height, as, for example, in .

The 24-hour tower clock in Venice, using J as a symbol for 1

Many older typewriters do not have a separate symbol for 1 and use the lowercase l instead. It is possible to find cases when the uppercase J is used, while it may be for decorative purposes.

Mathematics[link]

Mathematically, 1 is

• in arithmetic (algebra) and calculus, the natural number that follows 0 and precedes 2 and the multiplicative identity of the integers, real numbers and complex numbers;
• more generally, in abstract algebra, the multiplicative identity ("unity"), usually of a ring.

One cannot be used as the base of a positional numeral system; sometimes tallying is referred to as "base 1", since only one mark (the tally) is needed, but this is not a positional notation.

The logarithms base 1 are undefined, since the function 1^x always equals 1 and so has no unique inverse.

In the real-number system, 1 can be represented in two ways as a recurring decimal: as 1.000... and as 0.999... (q.v.).

Formalizations of the natural numbers have their own representations of 1:

• in the Peano axioms, 1 is the successor of 0;
• in Principia Mathematica, 1 is defined as the set of all singletons (sets with one element);
• in the Von Neumann cardinal assignment of natural numbers, 1 is defined as the set {0}.

In a multiplicative group or monoid, the identity element is sometimes denoted 1, especially in abelian groups, but e (from the German Einheit, "unity") is more traditional. However, 1 is especially common for the multiplicative identity of a ring, i.e., when an addition and 0 are also present. When such a ring has characteristic n not equal to 0, the element called 1 has the property that n1 = 1n = 0 (where this 0 is the additive identity of the ring). Important examples are general fields.

One is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few.

In many mathematical and engineering equations, numeric values are typically normalized to fall within the unit interval from 0 to 1, where 1 usually represents the maximum possible value in the range of parameters.

Because of the multiplicative identity, if f(x) is a multiplicative function, then f(1) must equal 1.

It is also the first and second numbers in the Fibonacci sequence and is the first number in many other mathematical sequences. As a matter of convention, Sloane's early Handbook of Integer Sequences added an initial 1 to any sequence that did not already have it and considered these initial 1's in its lexicographic ordering. Sloane's later Encyclopedia of Integer Sequences and its Web counterpart, the On-Line Encyclopedia of Integer Sequences, ignore initial ones in their lexicographic ordering of sequences, because such initial ones often correspond to trivial cases.

One is neither a prime number nor a composite number, but a unit, like -1 and, in the Gaussian integers, i and -i. The fundamental theorem of arithmetic guarantees unique factorization over the integers only up to units (e.g., 4 = 2² = (-1)⁶×1²³×2²).

The definition of a field requires that 1 must not be equal to 0. Thus, there are no fields of characteristic 1. Nevertheless, abstract algebra can consider the field with one element, which is not a singleton and is not a set at all.

One is the only positive integer divisible by exactly one positive integer (whereas prime numbers are divisible by exactly two positive integers, composite numbers are divisible by more than two positive integers, and zero is divisible by all positive integers). One was formerly considered prime by some mathematicians, using the definition that a prime is divisible only by one and itself. However, this complicates the fundamental theorem of arithmetic, so modern definitions exclude units.

One is one of three possible values of the Möbius function: it takes the value one for square-free integers with an even number of distinct prime factors.

One is the only odd number in the range of Euler's totient function φ(x), in the cases x = 1 and x = 2.

One is the only 1-perfect number (see multiply perfect number).

By definition, 1 is the magnitude or absolute value of a unit vector and a unit matrix (more usually called an identity matrix). Note that the term unit matrix is sometimes used to mean something quite different.

By definition, 1 is the probability of an event that is almost certain to occur.

One is the most common leading digit in many sets of data, a consequence of Benford's law.

The ancient Egyptians represented all fractions (with the exception of 2/3 and 3/4) in terms of sums of fractions with numerator 1 and distinct denominators. For example, Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{2}{5} = \frac{1}{3} + \frac{1}{15} . Such representations are popularly known as Egyptian Fractions or Unit Fractions.

The Generating Function that has all coefficients 1 is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots .

This power series converges and has finite value if and only if, Failed to parse (Missing texvc executable; please see math/README to configure.): | x | < 1 .

Table of basic calculations[link]

In technology[link]

• The resin identification code used in recycling to identify polyethylene terephthalate
• Used in binary code along with 0

In science[link]

• The atomic number of hydrogen

In philosophy[link]

In the philosophy of Plotinus and a number of other neoplatonists, The One is the ultimate reality and source of all existence.

In fiction[link]

• The characters Thomas the Tank Engine, Skarloey and Godred from The Railway Series and Thomas and Friends have this number.
• Numbuh 1, a character from Codename: Kids Next Door.
• The character Francesco Bernoulli from Cars 2 is numbered 1.
• Mr. One from One Piece.
• The General Lee from The Dukes of Hazzard is numbered 01.
• Jean-Luc Picard, Captain of the Enterprise-D in Star Trek: The Next Generation, refers to his second-in-command, Will Riker as Number One.

In sports[link]

• 1 is the lowest number permitted for use by players of the National Hockey League (NHL), as the league has banned 00 and 0. (The highest number permitted is 98.)
• Many sports use 1 as their standard scoring increment—examples include goals in a large number of sports, runs in baseball and cricket, and points in volleyball. Examples where 1 is a non-standard increment, or used for one of several possible classes of scores, are listed below by sport.
• In Australian rules football, 1 point is awarded to the attacking team for a behind, scored when:
  • The ball is kicked by the attacking team and passes between a goal post (taller post) and the nearest behind post (shorter post) on the defensive side of the field without touching the behind post.
  • The ball passes between the defending team's goal posts, but either (1) was not kicked by the attacking team or (2) hit a goal post.
  • The defending team deliberately forces the ball between any two of its own posts. This particular score is officially called a "rushed behind".
• In baseball scoring, the number 1 is assigned to the pitcher.
• In basketball:
  • 1 point is awarded for a successful free throw.
  • The number 1 is used to designate the point guard position.
• In football (soccer):
  • 1 point is awarded for a successful goal
  • The number 1 is occasionally given to the goalkeeper
• In Gaelic football, hurling and camogie, a "point", with a scoring value of 1, is awarded when the attacking team legally sends the ball over the opponent's crossbar (above the goal).
• In gridiron football codes, one point is awarded under the following circumstances:
  • In almost all leagues, for a successful place kick after a touchdown. In American football, the score is formally known as a "try", although the terms "extra point", "conversion", "PAT" (for "point after touchdown"), and "point after" are far more commonly used. In Canadian football, the score is formally and popularly called a "convert". Conversions can also be scored by the now-rare drop kick; in standard American and Canadian football, such a conversion is worth 1 point, while most forms of indoor football, including the Arena Football League, award 2 points for a drop-kicked conversion.
  • In six-man football, one point is awarded for a successful conversion from scrimmage after a touchdown. Note that in standard 11-man (American) or 12-man (Canadian) football, place kicks are worth 1 point and conversions from scrimmage worth 2; this is reversed in six-man because the reduced number of players makes kicked conversions much more difficult.
  • In Canadian football only, a single or "rouge" is awarded when the ball is legally kicked into the opponent's end zone (except for a successful field goal), and the receiving team does not return, or kick, the ball out of its end zone. (In American football, the same play would result in a touchback and no points.)
  • Some forms of indoor football in the U.S. award a "single", similar to the Canadian score, on kickoffs only.
• In rugby league:
  • A drop goal is worth 1 point.
  • In most competitions (though not the European Super League, which uses static squad numbering), the starting fullback wears jersey number 1.
• In rugby union:
  • The starting loosehead prop wears the jersey number 1.
  • In the early years of the sport (prior to 1890), conversions, penalties, drop goals, and goals from mark were all worth 1 point. At that time, a try was worth no points, only giving the attacking team the right to attempt a conversion. In 1890–91, tries were rewarded with 1 point, while all other scores were increased in value. After that time, all scores have been worth at least 2 points (the goal from mark was abolished in 1977).
• The jersey number 1 has been retired by several North American sports teams in honor of past playing greats or other key figures (or, in one case, a team's fans):
  • In Major League Baseball:
    • The Boston Red Sox, for Hall of Famer Bobby Doerr.
    • The Cincinnati Reds, for manager Fred Hutchinson.
    • The Los Angeles Dodgers, for Hall of Famer Pee Wee Reese.
    • The New York Yankees, for Billy Martin, who both played for and managed the team.
    • The Philadelphia Phillies, for Hall of Famer Richie Ashburn.
    • The Pittsburgh Pirates, for manager Billy Meyer.
    • The St. Louis Cardinals, for Hall of Famer Ozzie Smith.
  • In the NBA:
    • The Boston Celtics, for founding owner Walter Brown, a member of the Hall of Fame as a contributor.
    • The Milwaukee Bucks, for Hall of Fame player Oscar Robertson.
    • The Portland Trail Blazers, for founding owner Larry Weinberg. Unlike most numbers so honored, this number remains in circulation for players.
    • The Sacramento Kings, for Hall of Fame player Nate Archibald, honoring the number's retirement when the team was known as the Kansas City Kings.
    • The Seattle SuperSonics, for Gus Williams. The team has since relocated to become the Oklahoma City Thunder, but the Thunder have yet to issue any number retired by the franchise in Seattle.
    • The Utah Jazz, for Frank Layden, who served the team first as head coach and then as president.
  • In the NFL:
    • The New York Giants, for Hall of Famer Ray Flaherty.
    • The Tennessee Titans, for Hall of Famer Warren Moon, who played for the team in its past incarnation as the Houston Oilers.
  • In the NHL:
    • The Chicago Blackhawks, for Hall of Famer Glenn Hall.
    • The Detroit Red Wings, for Hall of Famer Terry Sawchuk.
    • The Montreal Canadiens, for Hall of Famer Jacques Plante.
    • The New York Rangers, for Hall of Famer Eddie Giacomin.
    • The Minnesota Wild, for their fans.
    • The Philadelphia Flyers, for Hall of Famer Bernie Parent.
    • The Toronto Maple Leafs have a policy of not retiring numbers unless the player honoured either died or suffered a career-ending incident while a member of the team. Other players whose numbers would otherwise be retired instead have their numbers enshrined by the team as "Honoured Numbers", which remain in circulation for future players. The number 1 is currently honoured for Hall of Famers Johnny Bower and Turk Broda.

See also[link]

Mathematics portal

• −1
• One (word)
• Root of unity

Notes[link]

Wikimedia Commons has media related to: 1 (number) (category)

Look up one in Wiktionary, the free dictionary.

vep:1 (lugu)

A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers.

Mathematical operations are certain procedures that take one or more numbers as input and produce a number as output. Unary operations take a single input number and produce a single output number. For example, the successor operation adds one to an integer, thus the successor of 4 is 5. Binary operations take two input numbers and produce a single output number. Examples of binary operations include addition, subtraction, multiplication, division, and exponentiation. The study of numerical operations is called arithmetic.

A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels (telephone numbers), for ordering (serial numbers), and for codes (e.g., ISBNs).

In common use, the word number can mean the abstract object, the symbol, or the word for the number.

Classification of numbers[link]

Different types of numbers are used in many cases. Numbers can be classified into sets, called number systems. (For different methods of expressing numbers with symbols, such as the Roman numerals, see numeral systems.)

Natural numbers[link]

The most familiar numbers are the natural numbers or counting numbers: one, two, three, and so on. Traditionally, the sequence of natural numbers started with 1 (0 was not even considered a number for the Ancient Greeks.) However, in the 19th century, set theorists and other mathematicians started including 0 (cardinality of the empty set, i.e. 0 elements, where 0 is thus the smallest cardinal number) in the set of natural numbers.^{[citation needed]} Today, different mathematicians use the term to describe both sets, including zero or not. The mathematical symbol for the set of all natural numbers is N, also written Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{N} .

In the base ten numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In this base ten system, the rightmost digit of a natural number has a place value of one, and every other digit has a place value ten times that of the place value of the digit to its right.

In set theory, which is capable of acting as an axiomatic foundation for modern mathematics,^[1] natural numbers can be represented by classes of equivalent sets. For instance, the number 3 can be represented as the class of all sets that have exactly three elements. Alternatively, in Peano Arithmetic, the number 3 is represented as sss0, where s is the "successor" function (i.e., 3 is the third successor of 0). Many different representations are possible; all that is needed to formally represent 3 is to inscribe a certain symbol or pattern of symbols three times.

Integers[link]

The negative of a positive integer is defined as a number that produces zero when it is added to the corresponding positive integer. Negative numbers are usually written with a negative sign (a minus sign). As an example, the negative of 7 is written −7, and 7 + (−7) = 0. When the set of negative numbers is combined with the set of natural numbers (which includes zero), the result is defined as the set of integer numbers, also called integers, Z also written Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z} . Here the letter Z comes from German Zahl, meaning "number".

The set of integers forms a ring with operations addition and multiplication.^[2]

Rational numbers[link]

A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero natural number denominator. Fractions are written as two numbers, the numerator and the denominator, with a dividing bar between them. In the fraction written ^m⁄_n or

Failed to parse (Missing texvc executable; please see math/README to configure.): m \over n \,

m represents equal parts, where n equal parts of that size make up one whole. Two different fractions may correspond to the same rational number; for example ¹⁄₂ and ²⁄₄ are equal, that is:

Failed to parse (Missing texvc executable; please see math/README to configure.): {1 \over 2} = {2 \over 4}.\,

If the absolute value of m is greater than n, then the absolute value of the fraction is greater than 1. Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or zero. The set of all rational numbers includes the integers, since every integer can be written as a fraction with denominator 1. For example −7 can be written ⁻⁷⁄₁. The symbol for the rational numbers is Q (for quotient), also written Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Q} .

Real numbers[link]

The real numbers include all of the measuring numbers. Real numbers are usually written using decimal numerals, in which a decimal point is placed to the right of the digit with place value one. Each digit to the right of the decimal point has a place value one-tenth of the place value of the digit to its left. Thus

Failed to parse (Missing texvc executable; please see math/README to configure.): 123.456\,

represents 1 hundred, 2 tens, 3 ones, 4 tenths, 5 hundredths, and 6 thousandths. In saying the number, the decimal is read "point", thus: "one two three point four five six ". In the US and UK and a number of other countries, the decimal point is represented by a period, whereas in continental Europe and certain other countries the decimal point is represented by a comma. Zero is often written as 0.0 when it must be treated as a real number rather than an integer. In the US and UK a number between −1 and 1 is always written with a leading zero to emphasize the decimal. Negative real numbers are written with a preceding minus sign:

Failed to parse (Missing texvc executable; please see math/README to configure.): -123.456.\,

Every rational number is also a real number. It is not the case, however, that every real number is rational. If a real number cannot be written as a fraction of two integers, it is called irrational. A decimal that can be written as a fraction either ends (terminates) or forever repeats, because it is the answer to a problem in division. Thus the real number 0.5 can be written as ¹⁄₂ and the real number 0.333... (forever repeating threes, otherwise written 0.3) can be written as ¹⁄₃. On the other hand, the real number π (pi), the ratio of the circumference of any circle to its diameter, is

Failed to parse (Missing texvc executable; please see math/README to configure.): \pi = 3.14159265358979\dots.\,

Since the decimal neither ends nor forever repeats, it cannot be written as a fraction, and is an example of an irrational number. Other irrational numbers include

Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{2} = 1.41421356237 \dots\,

(the square root of 2, that is, the positive number whose square is 2).

Thus 1.0 and 0.999... are two different decimal numerals representing the natural number 1. There are infinitely many other ways of representing the number 1, for example ²⁄₂, ³⁄₃, 1.00, 1.000, and so on.

Every real number is either rational or irrational. Every real number corresponds to a point on the number line. The real numbers also have an important but highly technical property called the least upper bound property. The symbol for the real numbers is R, also written as Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R} .

When a real number represents a measurement, there is always a margin of error. This is often indicated by rounding or truncating a decimal, so that digits that suggest a greater accuracy than the measurement itself are removed. The remaining digits are called significant digits. For example, measurements with a ruler can seldom be made without a margin of error of at least 0.001 meters. If the sides of a rectangle are measured as 1.23 meters and 4.56 meters, then multiplication gives an area for the rectangle of 5.6088 square meters. Since only the first two digits after the decimal place are significant, this is usually rounded to 5.61.

In abstract algebra, it can be shown that any complete ordered field is isomorphic to the real numbers. The real numbers are not, however, an algebraically closed field.

Complex numbers[link]

Moving to a greater level of abstraction, the real numbers can be extended to the complex numbers. This set of numbers arose, historically, from trying to find closed formulas for the roots of cubic and quartic polynomials. This led to expressions involving the square roots of negative numbers, and eventually to the definition of a new number: the square root of negative one, denoted by i, a symbol assigned by Leonhard Euler, and called the imaginary unit. The complex numbers consist of all numbers of the form

Failed to parse (Missing texvc executable; please see math/README to configure.): \,a + b i

where a and b are real numbers. In the expression a + bi, the real number a is called the real part and b is called the imaginary part. If the real part of a complex number is zero, then the number is called an imaginary number or is referred to as purely imaginary; if the imaginary part is zero, then the number is a real number. Thus the real numbers are a subset of the complex numbers. If the real and imaginary parts of a complex number are both integers, then the number is called a Gaussian integer. The symbol for the complex numbers is C or Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{C} .

In abstract algebra, the complex numbers are an example of an algebraically closed field, meaning that every polynomial with complex coefficients can be factored into linear factors. Like the real number system, the complex number system is a field and is complete, but unlike the real numbers it is not ordered. That is, there is no meaning in saying that i is greater than 1, nor is there any meaning in saying that i is less than 1. In technical terms, the complex numbers lack the trichotomy property.

Complex numbers correspond to points on the complex plane, sometimes called the Argand plane.

Each of the number systems mentioned above is a proper subset of the next number system. Symbolically, Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} .

Computable numbers[link]

Moving to problems of computation, the computable numbers are determined in the set of the real numbers. The computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. Equivalent definitions can be given using μ-recursive functions, Turing machines or λ-calculus as the formal representation of algorithms. The computable numbers form a real closed field and can be used in the place of real numbers for many, but not all, mathematical purposes.

Other types[link]

Algebraic numbers are those that can be expressed as the solution to a polynomial equation with integer coefficients. The complement of the algebraic numbers are the transcendental numbers.

Hyperreal numbers are used in non-standard analysis. The hyperreals, or nonstandard reals (usually denoted as *R), denote an ordered field that is a proper extension of the ordered field of real numbers R and satisfies the transfer principle. This principle allows true first order statements about R to be reinterpreted as true first order statements about *R.

Superreal and surreal numbers extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form fields.

The p-adic numbers may have infinitely long expansions to the left of the decimal point, in the same way that real numbers may have infinitely long expansions to the right. The number system that results depends on what base is used for the digits: any base is possible, but a prime number base provides the best mathematical properties.

For dealing with infinite collections, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. The former gives the ordering of the collection, while the latter gives its size. For the finite set, the ordinal and cardinal numbers are equivalent, but they differ in the infinite case.

A relation number is defined as the class of relations consisting of all those relations that are similar to one member of the class.^[3]

Sets of numbers that are not subsets of the complex numbers are sometimes called hypercomplex numbers. They include the quaternions H, invented by Sir William Rowan Hamilton, in which multiplication is not commutative, and the octonions, in which multiplication is not associative. Elements of function fields of non-zero characteristic behave in some ways like numbers and are often regarded as numbers by number theorists.

Specific uses[link]

There are also other sets of numbers with specialized uses. Some are subsets of the complex numbers. For example, algebraic numbers are the roots of polynomials with rational coefficients. Complex numbers that are not algebraic are called transcendental numbers.

An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2. (The old-fashioned term "evenly divisible" is now almost always shortened to "divisible".) A formal definition of an odd number is that it is an integer of the form n = 2k + 1, where k is an integer. An even number has the form n = 2k where k is an integer.

A perfect number is a positive integer that is the sum of its proper positive divisors—the sum of the positive divisors not including the number itself. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors, or σ(n) = 2 n. The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors and 1 + 2 + 3 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next perfect numbers are 496 and 8128 (sequence A000396 in OEIS). These first four perfect numbers were the only ones known to early Greek mathematics.

A figurate number is a number that can be represented as a regular and discrete geometric pattern (e.g. dots). If the pattern is polytopic, the figurate is labeled a polytopic number, and may be a polygonal number or a polyhedral number. Polytopic numbers for r = 2, 3, and 4 are:

• P₂(n) = ¹⁄₂ n(n + 1) (triangular numbers)
• P₃(n) = ¹⁄₆ n(n + 1)(n + 2) (tetrahedral numbers)
• P₄(n) = ¹⁄₂₄ n(n + 1)(n + 2)(n + 3) (pentatopic numbers)

Numerals[link]

Numbers should be distinguished from numerals, the symbols used to represent numbers. Boyer showed that Egyptians created the first ciphered numeral system.^{[citation needed]} Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. The number five can be represented by both the base ten numeral '5', by the Roman numeral 'Ⅴ' and ciphered letters. Notations used to represent numbers are discussed in the article numeral systems. An important development in the history of numerals was the development of a positional system, like modern decimals, which can represent very large numbers. The Roman numerals require extra symbols for larger numbers.

History[link]

First use of numbers[link]

Bones and other artifacts have been discovered with marks cut into them that many believe are tally marks.^[4] These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals.

A tallying system has no concept of place value (as in modern decimal notation), which limits its representation of large numbers. Nonetheless tallying systems are considered the first kind of abstract numeral system.

The first known system with place value was the Mesopotamian base 60 system (ca. 3400 BC) and the earliest known base 10 system dates to 3100 BC in Egypt.^[5]

[edit] Zero

The use of zero as a number should be distinguished from its use as a placeholder numeral in place-value systems. Many ancient texts used zero. Babylonian and Egyptian texts used it. Egyptians used the word nfr to denote zero balance in double entry accounting entries. Indian texts used a Sanskrit word Shunye to refer to the concept of void. In mathematics texts this word often refers to the number zero.^[6]

Records show that the Ancient Greeks seemed unsure about the status of zero as a number: they asked themselves "how can 'nothing' be something?" leading to interesting philosophical and, by the Medieval period, religious arguments about the nature and existence of zero and the vacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero. (The ancient Greeks even questioned whether 1 was a number.)

The late Olmec people of south-central Mexico began to use a true zero (a shell glyph) in the New World possibly by the 4th century BC but certainly by 40 BC, which became an integral part of Maya numerals and the Maya calendar. Mayan arithmetic used base 4 and base 5 written as base 20. Sanchez in 1961 reported a base 4, base 5 'finger' abacus.

By 130 AD, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals. Because it was used alone, not as just a placeholder, this Hellenistic zero was the first documented use of a true zero in the Old World. In later Byzantine manuscripts of his Syntaxis Mathematica (Almagest), the Hellenistic zero had morphed into the Greek letter omicron (otherwise meaning 70).

Another true zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus), but as a word, nulla meaning nothing, not as a symbol. When division produced zero as a remainder, nihil, also meaning nothing, was used. These medieval zeros were used by all future medieval computists (calculators of Easter). An isolated use of their initial, N, was used in a table of Roman numerals by Bede or a colleague about 725, a true zero symbol.

An early documented use of the zero by Brahmagupta (in the Brahmasphutasiddhanta) dates to 628. He treated zero as a number and discussed operations involving it, including division. By this time (the 7th century) the concept had clearly reached Cambodia as Khmer numerals, and documentation shows the idea later spreading to China and the Islamic world.

[edit] Negative numbers

The abstract concept of negative numbers was recognised as early as 100 BC – 50 BC. The Chinese Nine Chapters on the Mathematical Art (Chinese: Jiu-zhang Suanshu) contains methods for finding the areas of figures; red rods were used to denote positive coefficients, black for negative.^[7] This is the earliest known mention of negative numbers in the East; the first reference in a Western work was in the 3rd century in Greece. Diophantus referred to the equation equivalent to Failed to parse (Missing texvc executable; please see math/README to configure.): 4x + 20 = 0

(the solution is negative) in Arithmetica, saying that the equation gave an absurd result.

During the 600s, negative numbers were in use in India to represent debts. Diophantus’ previous reference was discussed more explicitly by Indian mathematician Brahmagupta, in Brahma-Sphuta-Siddhanta 628, who used negative numbers to produce the general form quadratic formula that remains in use today. However, in the 12th century in India, Bhaskara gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots."

European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debts (chapter 13 of Liber Abaci, 1202) and later as losses (in Flos). At the same time, the Chinese were indicating negative numbers either by drawing a diagonal stroke through the right-most nonzero digit of the corresponding positive number's numeral.^[8] The first use of negative numbers in a European work was by Chuquet during the 15th century. He used them as exponents, but referred to them as “absurd numbers”.

As recently as the 18th century, it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless, just as René Descartes did with negative solutions in a Cartesian coordinate system.

[edit] Rational numbers

It is likely that the concept of fractional numbers dates to prehistoric times. The Ancient Egyptians used their Egyptian fraction notation for rational numbers in mathematical texts such as the Rhind Mathematical Papyrus and the Kahun Papyrus. Classical Greek and Indian mathematicians made studies of the theory of rational numbers, as part of the general study of number theory. The best known of these is Euclid's Elements, dating to roughly 300 BC. Of the Indian texts, the most relevant is the Sthananga Sutra, which also covers number theory as part of a general study of mathematics.

The concept of decimal fractions is closely linked with decimal place-value notation; the two seem to have developed in tandem. For example, it is common for the Jain math sutras to include calculations of decimal-fraction approximations to pi or the square root of two. Similarly, Babylonian math texts had always used sexagesimal (base 60) fractions with great frequency.

[edit] Irrational numbers

The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between 800–500 BC.^[9] The first existence proofs of irrational numbers is usually attributed to Pythagoras, more specifically to the Pythagorean Hippasus of Metapontum, who produced a (most likely geometrical) proof of the irrationality of the square root of 2. The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction. However Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. He could not disprove their existence through logic, but he could not accept irrational numbers, so he sentenced Hippasus to death by drowning.

The sixteenth century brought final European acceptance of negative integral and fractional numbers. By the seventeenth century, mathematicians generally used decimal fractions with modern notation. It was not, however, until the nineteenth century that mathematicians separated irrationals into algebraic and transcendental parts, and once more undertook scientific study of irrationals. It had remained almost dormant since Euclid. 1872 brought publication of the theories of Karl Weierstrass (by his pupil Kossak), Heine (Crelle, 74), Georg Cantor (Annalen, 5), and Richard Dedekind. In 1869, Méray had taken the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method was completely set forth by Salvatore Pincherle (1880), and Dedekind's has received additional prominence through the author's later work (1888) and endorsement by Paul Tannery (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of real numbers, separating all rational numbers into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, Kronecker (Crelle, 101), and Méray.

Continued fractions, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler, and at the opening of the nineteenth century were brought into prominence through the writings of Joseph Louis Lagrange. Other noteworthy contributions have been made by Druckenmüller (1837), Kunze (1857), Lemke (1870), and Günther (1872). Ramus (1855) first connected the subject with determinants, resulting, with the subsequent contributions of Heine, Möbius, and Günther, in the theory of Kettenbruchdeterminanten. Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject.

[edit] Transcendental numbers and reals

The first results concerning transcendental numbers were Lambert's 1761 proof that π cannot be rational, and also that eⁿ is irrational if n is rational (unless n = 0). (The constant e was first referred to in Napier's 1618 work on logarithms.) Legendre extended this proof to show that π is not the square root of a rational number. The search for roots of quintic and higher degree equations was an important development, the Abel–Ruffini theorem (Ruffini 1799, Abel 1824) showed that they could not be solved by radicals (formula involving only arithmetical operations and roots). Hence it was necessary to consider the wider set of algebraic numbers (all solutions to polynomial equations). Galois (1832) linked polynomial equations to group theory giving rise to the field of Galois theory.

The existence of transcendental numbers^[10] was first established by Liouville (1844, 1851). Hermite proved in 1873 that e is transcendental and Lindemann proved in 1882 that π is transcendental. Finally Cantor shows that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite, so there is an uncountably infinite number of transcendental numbers.

[edit] Infinity and infinitesimals

The earliest known conception of mathematical infinity appears in the Yajur Veda, an ancient Indian script, which at one point states, "If you remove a part from infinity or add a part to infinity, still what remains is infinity." Infinity was a popular topic of philosophical study among the Jain mathematicians c. 400 BC. They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.

Aristotle defined the traditional Western notion of mathematical infinity. He distinguished between actual infinity and potential infinity—the general consensus being that only the latter had true value. Galileo's Two New Sciences discussed the idea of one-to-one correspondences between infinite sets. But the next major advance in the theory was made by Georg Cantor; in 1895 he published a book about his new set theory, introducing, among other things, transfinite numbers and formulating the continuum hypothesis. This was the first mathematical model that represented infinity by numbers and gave rules for operating with these infinite numbers.

In the 1960s, Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of infinitesimal calculus by Newton and Leibniz.

A modern geometrical version of infinity is given by projective geometry, which introduces "ideal points at infinity," one for each spatial direction. Each family of parallel lines in a given direction is postulated to converge to the corresponding ideal point. This is closely related to the idea of vanishing points in perspective drawing.

[edit] Complex numbers

The earliest fleeting reference to square roots of negative numbers occurred in the work of the mathematician and inventor Heron of Alexandria in the 1st century AD, when he considered the volume of an impossible frustum of a pyramid. They became more prominent when in the 16th century closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians such as Niccolo Fontana Tartaglia and Gerolamo Cardano. It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers.

This was doubly unsettling since they did not even consider negative numbers to be on firm ground at the time. When René Descartes coined the term "imaginary" for these quantities in 1637, he intended it as derogatory. (See imaginary number for a discussion of the "reality" of complex numbers.) A further source of confusion was that the equation

Failed to parse (Missing texvc executable; please see math/README to configure.): \left ( \sqrt{-1}\right )^2 =\sqrt{-1}\sqrt{-1}=-1

seemed capriciously inconsistent with the algebraic identity

Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{a}\sqrt{b}=\sqrt{ab},

which is valid for positive real numbers a and b, and was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity, and the related identity

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{\sqrt{a}}=\sqrt{\frac{1}{a}}

in the case when both a and b are negative even bedeviled Euler. This difficulty eventually led him to the convention of using the special symbol i in place of Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{-1}

to guard against this mistake.

The 18th century saw the work of Abraham de Moivre and Leonhard Euler. de Moivre's formula (1730) states:

Failed to parse (Missing texvc executable; please see math/README to configure.): (\cos \theta + i\sin \theta)^{n} = \cos n \theta + i\sin n \theta \,

and to Euler (1748) Euler's formula of complex analysis:

Failed to parse (Missing texvc executable; please see math/README to configure.): \cos \theta + i\sin \theta = e ^{i\theta }. \,

The existence of complex numbers was not completely accepted until Caspar Wessel described the geometrical interpretation in 1799. Carl Friedrich Gauss rediscovered and popularized it several years later, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis's De Algebra tractatus.

Also in 1799, Gauss provided the first generally accepted proof of the fundamental theorem of algebra, showing that every polynomial over the complex numbers has a full set of solutions in that realm. The general acceptance of the theory of complex numbers is due to the labors of Augustin Louis Cauchy and Niels Henrik Abel, and especially the latter, who was the first to boldly use complex numbers with a success that is well known.

Gauss studied complex numbers of the form a + bi, where a and b are integral, or rational (and i is one of the two roots of x² + 1 = 0). His student, Gotthold Eisenstein, studied the type a + bω, where ω is a complex root of x³ − 1 = 0. Other such classes (called cyclotomic fields) of complex numbers derive from the roots of unity x^k − 1 = 0 for higher values of k. This generalization is largely due to Ernst Kummer, who also invented ideal numbers, which were expressed as geometrical entities by Felix Klein in 1893. The general theory of fields was created by Évariste Galois, who studied the fields generated by the roots of any polynomial equation F(x) = 0.

In 1850 Victor Alexandre Puiseux took the key step of distinguishing between poles and branch points, and introduced the concept of essential singular points. This eventually led to the concept of the extended complex plane.

[edit] Prime numbers

Prime numbers have been studied throughout recorded history. Euclid devoted one book of the Elements to the theory of primes; in it he proved the infinitude of the primes and the fundamental theorem of arithmetic, and presented the Euclidean algorithm for finding the greatest common divisor of two numbers.

In 240 BC, Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime numbers. But most further development of the theory of primes in Europe dates to the Renaissance and later eras.

In 1796, Adrien-Marie Legendre conjectured the prime number theorem, describing the asymptotic distribution of primes. Other results concerning the distribution of the primes include Euler's proof that the sum of the reciprocals of the primes diverges, and the Goldbach conjecture, which claims that any sufficiently large even number is the sum of two primes. Yet another conjecture related to the distribution of prime numbers is the Riemann hypothesis, formulated by Bernhard Riemann in 1859. The prime number theorem was finally proved by Jacques Hadamard and Charles de la Vallée-Poussin in 1896. Goldbach and Riemann's conjectures remain unproven and unrefuted.

Word alternatives[link]

Some numbers traditionally have alternative words to express them, including the following:

• Nothing: 0
• Single: 1
• Pair, couple, brace: 2
• Trio: 3
• Half-Dozen: 6
• Decade: 10
• Dozen: 12
• Baker's dozen: 13
• Score: 20
• Half-century: 50
• Century: 100
• Gross: 144
• Ream: 480 (old measure) 500 (new measure)
• Millenium: 1000
• Great gross: 1728
• "n-figure", as in digit, generally for larger-number ranges, also written without a hyphen; often used in financial discussion. For example

• "five-figure": 10,000 to 99,999 (five digits); ten-thousands
• "six-figure": 100,000 to 999,999 (six digits); hundred-thousands
• "seven-figure": 1,000,000 to 9,999,999 (seven digits); millions

See also[link]

Wikimedia Commons has media related to: Numbers

Notes[link]

References[link]

• Tobias Dantzig, Number, the language of science; a critical survey written for the cultured non-mathematician, New York, The Macmillan company, 1930.
• Erich Friedman, What's special about this number?
• Steven Galovich, Introduction to Mathematical Structures, Harcourt Brace Javanovich, 23 January 1989, ISBN 0-15-543468-3.
• Paul Halmos, Naive Set Theory, Springer, 1974, ISBN 0-387-90092-6.
• Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972.
• Alfred North Whitehead and Bertrand Russell, Principia Mathematica to *56, Cambridge University Press, 1910.
• George I. Sanchez, Arithmetic in Maya,Austin-Texas, 1961.

External links[link]

Look up number in Wiktionary, the free dictionary.

Wikiversity has learning materials about Primary mathematics:Numbers

• Nechaev, V.I. (2001), "Number", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4, http://www.encyclopediaofmath.org/index.php?title=Number 
• Mesopotamian and Germanic numbers
• BBC Radio 4, In Our Time: Negative Numbers
• '4000 Years of Numbers', lecture by Robin Wilson, 07/11/07, Gresham College (available for download as MP3 or MP4, and as a text file).
• http://planetmath.org/encyclopedia/MayanMath2.html
• "What's the World's Favorite Number?". 2011-06-22. http://www.npr.org/blogs/krulwich/2011/07/22/138493147/what-s-your-favorite-number-world-wide-survey-v1. Retrieved 2011-09-17. ; "Cuddling With 9, Smooching With 8, Winking At 7". 2011-08-11. http://www.npr.org/templates/transcript/transcript.php?storyId=139797360. Retrieved 2011-09-17.