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Addition is one of the four basic operations of arithmetic. The addition of two whole numbers is the total amount of those values combined. For example, in the adjacent picture, there is a combination of three apples and two apples together, making a total of five apples; this observation is equivalent to the mathematical expression "3 + 2 = 5" i.e. "3 add 2 is equal to 5". Besides counting items, addition can be defined on other types of numbers, such as integers, real numbers and complex numbers; this is part of a branch of mathematics. In algebra, another area of mathematics, addition can be performed on abstract objects such as vectors and matrices. Addition has several important properties, it is commutative, meaning that order does not matter, it is associative, meaning that when one adds more than two numbers, the order in which addition is performed does not matter. Repeated addition of 1 is the same as counting. Addition obeys predictable rules concerning related operations such as subtraction and multiplication.

Performing addition is one of the simplest numerical tasks. Addition of small numbers is accessible to toddlers. In primary education, students are taught to add numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient abacus to the modern computer, where research on the most efficient implementations of addition continues to this day. Addition is written using the plus sign "+" between the terms; the result is expressed with an equals sign. For example, 1 + 1 = 2 2 + 2 = 4 1 + 2 = 3 5 + 4 + 2 = 11 3 + 3 + 3 + 3 = 12 There are situations where addition is "understood" though no symbol appears: A whole number followed by a fraction indicates the sum of the two, called a mixed number. For example, 3½ = 3 + ½ = 3.5. This notation can cause confusion since in most other contexts juxtaposition denotes multiplication instead; the sum of a series of related numbers can be expressed through capital sigma notation, which compactly denotes iteration.

For example, ∑ k = 1 5 k 2 = 1 2 + 2 2 + 3 2 + 4 2 + 5 2 = 55. The numbers or the objects to be added in general addition are collectively referred to as the terms, the addends or the summands; this is to be distinguished from factors. Some authors call. In fact, during the Renaissance, many authors did not consider the first addend an "addend" at all. Today, due to the commutative property of addition, "augend" is used, both terms are called addends. All of the above terminology derives from Latin. "Addition" and "add" are English words derived from the Latin verb addere, in turn a compound of ad "to" and dare "to give", from the Proto-Indo-European root *deh₃- "to give". Using the gerundive suffix -nd results in "addend", "thing to be added". From augere "to increase", one gets "augend", "thing to be increased". "Sum" and "summand" derive from the Latin noun summa "the highest, the top" and associated verb summare. This is appropriate not only because the sum of two positive numbers is greater than either, but because it was common for the ancient Greeks and Romans to add upward, contrary to the modern practice of adding downward, so that a sum was higher than the addends.

Addere and summare date back at least to Boethius, if not to earlier Roman writers such as Vitruvius and Frontinus. The Middle English terms "adden" and "adding" were popularized by Chaucer; the plus sign "+" is an abbreviation of the Latin word et, meaning "and". It appears in mathematical works dating back to at least 1489. Addition is used to model many physical processes. For the simple case of adding natural numbers, there are many possible interpretations and more visual representations; the most fundamental interpretation of addition lies in combining sets: When two or more disjoint collections are combined into a single collection, the number of objects in the single collection is the sum of the numbers of objects in the original collections. This interpretation is easy to visualize, with little danger of ambiguity, it is useful in higher mathematics. However, it is not obvious how one should extend this version of addition to include fractional numbers or negative numbers. One possible fix is to consider collections of objects that can be divided, such as pies or, still

1984 San Francisco 49ers season

The 1984 San Francisco 49ers season was their 39th season in the National Football League. The season was highlighted by their second Super Bowl victory; the franchise had their best season with a record of 15 wins and only 1 loss. Quarterback Joe Montana would be awarded the Super Bowl's Most Valuable Player Award for the second time in his career, joining Bart Starr and Terry Bradshaw as the only two-time Super Bowl MVPs; the 1984 49ers became the first team to win fifteen games in the NFL's regular season since the league went to a sixteen-game schedule in 1978. The 49ers, if not for their loss to the Steelers, would have become the 2nd team after the 1972 Miami Dolphins to complete a perfect season, the Niners would have been the first to do so since the NFL expanded to a 16-game schedule; the 1985 Chicago Bears, the 1998 Minnesota Vikings, the 2004 Pittsburgh Steelers, the 2011 Green Bay Packers, the 2015 Carolina Panthers would join the 1984 49ers to finish 15–1, although the 2007 New England Patriots would exceed this feat by finishing the regular season at an unbeaten 16–0.

In the playoffs, the 49ers were the #1 seed. They defeated the Giants 21–10 in the divisional round they shut out the Chicago Bears 23–0, defeated the Miami Dolphins 38–16 in Super Bowl XIX; this 49ers team has gone down as the best in franchise history and many call this season the best in Joe Montana's career. The 1984 San Francisco 49ers held training camp at Sierra College in California. Notes: The 49ers advanced to their second Super Bowl in team history after becoming the first team to win 15 regular season games since the league expanded to a 16-game schedule in 1978. Much of the hype surrounding the team was their offense. Quarterback Joe Montana recorded 279 out of 432 completions for 3,630 yards, 28 touchdowns, only 10 interceptions. Running back Roger Craig was one of the 49ers' major weapons, both rushing and receiving. Craig was the team's second leading rusher with 649 rushing yards and 7 touchdowns, caught 71 passes for 675 yards. Pro Bowl running back Wendell Tyler, who had rushed for a team record 1,262 yards during the regular season, recorded 7 rushing touchdowns, caught 28 passes for 230 yards and 2 touchdown receptions.

Wide receivers Freddie Solomon and Dwight Clark were deep threats, gaining a combined total of 1,617 yards and 16 touchdowns. Up front, 3 of the 49ers' 5 starting offensive linemen, Randy Cross, Fred Quillan, Keith Fahnhorst, had been selected to play in the Pro Bowl. Overall, San Francisco's offense finished the season ranked second in the NFL in scoring and fourth in total yards. Although they did not get as much media attention as the offense, the 49ers defense led the league in fewest points allowed during the regular season. All 4 of the 49ers' starting defensive backs, Ronnie Lott, Eric Wright, Carlton Williamson, Dwight Hicks, were selected to play in the Pro Bowl. Pro Bowl linebacker Keena Turner was a major defensive weapon, recording 2 sacks and 4 interceptions for 51 yards. Defensive end Dwaine Board anchored the line, recording 1 fumble recovery. Notes: Gary Anderson kicked the game-winning field goal in the fourth quarter that would prevent the 49ers from going undefeated. 1Completions/attempts 2Carries 3Long gain 4Receptions 5Interceptions 6Sacks 7Punts 8Kickoff Returns 9Punt Returns Notes: Quarterback Joe Montana threw for 309 yards and 3 touchdown passes as he led the 49ers to a victory, while receiver Dwight Clark caught 9 passes for 112 yards and a touchdown.

The 49ers passed for 228 yards while limiting the Bears to only 37 no points. Joe Montana, Super Bowl Most Valuable Player Joe Montana, All-Pro Selection Joe Montana, NFC Pro Bowl Selection Pre season Local TV Local Radio 49ers on Pro Football Reference 49ers Schedule on

Jim Yarbrough

James Kelley Yarbrough is an American former college and professional football player, an offensive lineman in the National Football League for nine seasons during the 1960s and 1970s. Yarbrough played college football for the University of Florida. A second-round pick in the 1969 NFL Draft, he played his entire professional career for the NFL's Detroit Lions. Yarbrough was born in Charlotte, North Carolina, he grew up in Arcadia, where he attended DeSoto County High School, played high school football for the DeSoto Bulldogs. Yarbrough was an offensive end, doubled on defense as a roving "monster man" linebacker, kicked most of the extra points for the Bulldogs, he was a shot-putter on the DeSoto County High School track team and center on the basketball team. Yarbrough accepted an athletic scholarship to attend the University of Florida in Gainesville, where he played for coach Ray Graves' Florida Gators football team from 1966 to 1968. Memorably, he was a sophomore starter for the Steve Spurrier-led squad that won the first major bowl game in Gators history when they defeated the Georgia Tech Yellow Jackets 27–12 to win the 1967 Orange Bowl.

Yarbrough returned to Florida during the NFL off-season to complete his bachelor's degree in marketing in 1971, he was inducted into the University of Florida Athletic Hall of Fame as a "Gator Great." In a 2006 article series written for The Gainesville Sun, he was ranked among the 100 greatest Gator football players of all time. In the fall of 1999 Yarbrough was named as the Tight End on the Gators All-Century team as chosen by Gator fans and organized by the Gainesville Sun; the Detroit Lions chose Yarbrough in the second round of the 1969 NFL Draft, he played his entire nine-season professional career for the Lions as an offensive tackle from 1969 to 1977. He became a regular starter at left tackle in his third season,1971. Florida Gators football, 1960–69 List of Detroit Lions players List of Florida Gators football players in the NFL List of University of Florida alumni List of University of Florida Athletic Hall of Fame members Carlson, University of Florida Football Vault: The History of the Florida Gators, Whitman Publishing, LLC, Georgia.

ISBN 0-7948-2298-3. Golenbock, Peter, Go Gators! An Oral History of Florida's Pursuit of Gridiron Glory, Legends Publishing, LLC, St. Petersburg, Florida. ISBN 0-9650782-1-3. Hairston, Tales from the Gator Swamp: A Collection of the Greatest Gator Stories Ever Told, Sports Publishing, LLC, Illinois. ISBN 1-58261-514-4. McCarthy, Kevin M. Fightin' Gators: A History of University of Florida Football, Arcadia Publishing, Mount Pleasant, South Carolina. ISBN 978-0-7385-0559-6. McEwen, The Gators: A Story of Florida Football, The Strode Publishers, Alabama. ISBN 0-87397-025-X. Nash, Noel, ed; the Gainesville Sun Presents The Greatest Moments in Florida Gators Football, Sports Publishing, Inc. Champaign, Illinois. ISBN 1-57167-196-X

Kacey Clarke

Kacey Louisa Barnfield credited as Kacey Clarke, is an English actress. As a teenager she played Maddie Gilks in the long running British television series Grange Hill, on which she spent six years; as an adult, her roles have included Crystal in the American action film Resident Evil: Afterlife, Katie Sutherland in British comedy The Inbetweeners. In 2014, Clarke was listed as number 99 in FHM's 100 sexiest women in the world. Clarke was born in the London Borough of Enfield as Kacey Louisa Barnfield to parents Malcolm, a partner at Enfield estate agency, Barnfield's, Karen, she has a sister. Clarke is a second cousin of the actress Victoria Shalet. Clarke splits her time between Los Angeles, maintaining a base in both cities. Clarke's began appearing in advertisements and stage plays. In 2000, Barnfield's won the role of bully Maddie Gilks in Grange Hill, where she found herself the central role in central plots during her four-year stint on the show, It was Clarke's first major acting role. After Grange Hill, Clarke filmed Popcorn with Jodi Albert and Jack Ryder, plays the character of Yukino.

She has appeared in the Sky1 football drama Dream Team. Other roles include Zoe Stringer in Filthy Rich, in which she played Mike Reid's daughter shortly before he died, The Bill, where she played Chloe Fox for 3 episodes in 2004 and appeared again in October 2007 as Kelly Burgess, Casualty in which she appeared as Claudie Waters for 2 episodes on 29 and 30 December 2007. In 2008, Clarke was the face of Clear, appearing in adverts for the skincare brand, she starred in the Road Safety commercial'Mess'. In 2008 she became Galaxy chocolate's'Miss Kiss' to publicize their Christmas Mistletoe Kisses chocolates. In 2009 and 2010 she appeared in the E4 sitcom The Inbetweeners, as Neil's sister Katie in the series 2 episode "A Night Out in London" and series 3's "Will's Dilemma". Clarke made her theatrical film debut as Crystal Waters in the Screen Gem's 3D action film Resident Evil: Afterlife, alongside Milla Jovovich, Wentworth Miller and Ali Larter, it topped the box office in September 2010. Clarke starred in the TV movie Lake Placid 3, where she plays Ellie, this was in 2010.

In 2011 she played the lead role of Kate in Johannes Roberts' Roadkill, a horror film about a group of teens taking an ill-fated RV trip around Ireland. Her performance was well received. Clarke appeared as Barb in Jeremy Leven's 2013 movie Girl on a Bicycle. In June 2011 Variety magazine reported that Clarke would co-star in Glutton, a'3D psychological thriller' directed by David Arquette, playing Virginia the blind neighbour and only friend of a 1,200-pound man. Abraham Benrubi and Patricia Arquette star. Glutton was to begin filming in Canada in summer 2011, though no film under that title has been released. In September 2011 Clarke was female lead of Annabel in Syfy Channel's Jabberwock opposite Battlestar Galactica's Tahmoh Penniket. Kacey Clarke on IMDb

Unit square

In mathematics, a unit square is a square whose sides have length 1. "the" unit square refers to the square in the Cartesian plane with corners at the four points, and. In a Cartesian coordinate system with coordinates the unit square is defined as the square consisting of the points where both x and y lie in a closed unit interval from 0 to 1; that is, the unit square is the Cartesian product I × I, where I denotes the closed unit interval. The unit square can be thought of as a subset of the complex plane, the topological space formed by the complex numbers. In this view, the four corners of the unit square are at the four complex numbers 0, 1, i, 1 + i, it is not known whether any point in the plane is a rational distance from all four vertices of the unit square. However, according to Périat, the only points included in the square of rational distances of the four vertices are on the sides; the distance: x 2 + 2 = x 2 + y 2 − 2 y + 1 = A 2 B 2 − 2 y + 1 = 2 ⟹ y = A B ⟹ x = 0. What is becoming general for the plan.


Donald Baverstock

Donald Leighton Baverstock was a British television producer and executive, born in Cardiff, Wales. He worked for BBC Television in their Talks Department, where he was the Editor of the topical magazine programme Highlight and co-devised and edited its more ambitious and better-remembered successor Tonight, which began in 1957. Baverstock was born in Cardiff, United Kingdom, in January 1924. Baverstock worked on Tonight until 1961, when he was promoted to be the BBC’s Assistant Controller of Programmes across the whole television service and suggested a programme of hymn-singing that led to the creation of Songs of Praise. In early 1963 he succeeded his superior Stuart Hood to become the Controller of Programmes for BBC1, in anticipation of the launch of the station's companion BBC2 the following year. In the same year he requested Sydney Newman to develop a new Saturday evening show for BBC1 which would become Doctor Who. However, soon after the launch of BBC2 in 1964, Controller Michael Peacock began to run into difficulties, BBC Director-General Hugh Greene decided in 1965 that the two men would be better suited to running each other’s channels, took the decision to swap them over.

However, Baverstock felt insulted that he was being asked to take what he saw as a demotion to the lesser channel, refused to take up his new post, instead resigning from the BBC altogether. He subsequently became involved in the establishment of the ITV northern franchise holder Yorkshire Television, becoming the company's first Director of Programmes and overseeing the creation of popular hits such as the soap opera Emmerdale Farm. In 1972, as ITV commissioned the broadcast of the first nationally televised darts tournament with the 1972 News of the World Darts Championship, Baverstock asked Sid Waddell to accompany presenter Peter Jones and commentator Dave Lanning at the 1972 News of the World Championship, to observe the action and to look into creating a programme based on indoor pub sports; the result of this was Waddell creating The Indoor League, which ran from 1972 to 1977, with the first season shown only in the Yorkshire Television region and the following seasons shown across Britain on the ITV network.

The Indoor League was a crucial series in the early years of televised darts. Despite being ensconced in Yorkshire, Baverstock did attempt to return to Wales, at one point applying for the vacant post of Controller of BBC Wales, he died in March 1995. Former colleague Leonard Miall claimed in Baverstock's obituary in the Independent newspaper that the BBC Governors who interviewed him were "put off" by his "casual behaviour". Another factor may have been that, although born in Wales, Baverstock did not speak Welsh - an attribute considered essential for anyone aspiring to become the Controller of BBC Wales. In 2013 the BBC filmed a drama based around the creation and early days of Doctor Who in 1963, as part of the celebrations for the fiftieth anniversary of the series. Baverstock appears as a character in the drama, An Adventure in Space and Time, portrayed by actor Mark Eden; the drama portrays Baverstock at one point wanting Doctor Who to be cancelled after only four episodes. In 1957 Baverstock married Gillian Mary Waters, elder daughter of British children's author Enid Blyton, at St James's Church, Piccadilly