Ashley John Callus  (born 10 March 1979) is an Australian former sprint freestyle swimmer, who won a gold medal in the 4 × 100 m freestyle relay at the 2000 Sydney Olympics.

Coming from Brisbane, Queensland, and of Maltese origin, Callus, graduated in Iona College, he was coached by Chris Urquhart, spent the beginning of his career as the understudy of fellow Australians Michael Klim and Chris Fydler. After finishing fourth in the 100 m freestyle at the 2000 Australian Championships, Callus was selected to make his debut at the age of 21 at the 2000 Sydney Olympics in the 4 × 100 m freestyle relay.

2000 Olympics
Swimming with Klim, Fydler and Ian Thorpe, the Australians were not expected to win, as the United States had never lost the event at Olympic level. However, cheered on by a raucous home crowd, Klim set a world record 48.18 s in the leadoff leg, before Fydler maintained the lead, handing Callus a half-body length lead over Jason Lezak. Callus did well to stave off the faster and more experienced Lezak, giving Thorpe a slender lead. The fast-finishing Thorpe managed to overhaul the fast-starting Gary Hall Jr. by a handspan, sealing a gold medal in a world record time of 3 min 13.67 s, almost two seconds below the previous world mark.

At the 2001 World Championships in Fukuoka, Japan, he was again part of the 4 × 100 m freestyle relay team which won gold ahead of the United States, and was also a finalist in the 100 m freestyle. The following year, 2002 saw his finest moment as he won individual gold at the 2002 World Short Course Championships in Moscow, Russia in the 100 m freestyle.

2004 Olympics
In 2004, Callus qualified to represent Australia for both the 50 m and 100 m freestyle, as well as the relays at the National Trials, swimming to his usual standards. However, at the Athens Olympics, Callus swum much slower in the heats of his individual events and was a long margin from even qualifying for the semi-finals. He was ignominiously dropped from the final of the 4 × 100 m freestyle relay, where Australia came sixth. It was revealed that he had contracted the Epstein-Barr virus and he took a year off to recover, returning in late 2005, declaring that he had completely overcome his illness. He has gained selection for the 2006 Commonwealth Games in Melbourne.

At the 2006 Commonwealth Games, Callus anchored the 4 × 100 m freestyle relay team to silver, and also reached the final of both the 50 m and 100 m freestyle, although he was considerably outside his best time in both events. He took a new gold for Australia in the 2006 World Short Course Championships at 4 × 100 m medley relay, in Shanghai, China, April 2006.

2008 Olympics
At the 2008 Summer Olympics, Callus was part of the Australian 4 × 100 m freestyle relay team, which won the bronze medal in the event. He also took part in the 50m freestyle finishing a close 4th. In a post swim interview, he said he would not return for a 4th Olympics.

Successful Return
Callus made a return in 2009 at the Aus trials, despite clocking thunderous times in the prelims and semi-finals, he failed to beat the younger generation of sprinters in Eamon Sullivan & Matthew Abood. In a post swim interview he claimed due to the slow real estate market he had more free time to return to the pool.

After what most swimming fans assumed was a quiet retirement, Callus made an unexpected return to the Australian short course championships, clocking a new Australian record in the 50m freestyle.

On 26 November at the 2009 International AIS meet, Callus broke Sullivan's Australian record (former world record) in the men's 50m freestyle to clock 21.24 in the mourning prelims. For the second time in a day, taking 0.05 off the mark in the finals and bringing it down to 21.19, making him the 3rd fastest swimmer in history. "At the end of the day it's great to do fast times and see what I can possibly do, and that would have won a medal at Worlds," said Callus.

At the 2010 Telstra Trials he won the 50m freestyle in a time of 22.09, at the 2010 Pan Pacs he finished 7th in 22.15. At the 2010 Commonwealth Games he was disqualified in the semi-finals. "It's one of those things," he said. "That's bitterly disappointing ... I could have been up on the dais because I knew I was in good form." Asked if that was his last swim: "Definitely in the Commonwealth Games format. As they say, shit happens."

Career best times

See also
 List of Commonwealth Games medallists in swimming (men)
 List of Olympic medalists in swimming (men)
 World record progression 4 × 100 metres freestyle relay

References

External links

1979 births
Living people
Sportsmen from Queensland
Swimmers at the 2000 Summer Olympics
Swimmers at the 2004 Summer Olympics
Swimmers at the 2008 Summer Olympics
Olympic swimmers for Australia
Australian male freestyle swimmers
Olympic gold medalists for Australia
Olympic bronze medalists for Australia
Commonwealth Games gold medallists for Australia
Commonwealth Games silver medallists for Australia
Australian people of Maltese descent
Swimmers at the 2002 Commonwealth Games
Swimmers at the 2006 Commonwealth Games
World record setters in swimming
Olympic bronze medalists in swimming
World Aquatics Championships medalists in swimming
Medalists at the FINA World Swimming Championships (25 m)
Medalists at the 2008 Summer Olympics
Swimmers from Brisbane
Medalists at the 2000 Summer Olympics
Olympic gold medalists in swimming
Commonwealth Games medallists in swimming
Goodwill Games medalists in swimming
Recipients of the Medal of the Order of Australia
Competitors at the 2001 Goodwill Games
People from Redland City
21st-century Australian people
Medallists at the 2002 Commonwealth Games
Medallists at the 2006 Commonwealth Games

Ashley Callus


Oliver Heaviside (Camden, London 18 May 1850 – Torquay, Devon 3 February 1925) was a self-taught English electrical engineer, mathematician, and physicist.

Although he was well known in the scientific world during his life, it was not until after his death that people realized how much he had changed the study of mathematics and physics.  He used complex numbers to study electrical circuits and did a lot of work on electromagnetic wave theory which made it necessary to find new ways of working with differential equations.  He did work on  Maxwell's field equations and vector analysis. In fact, he was the inventor of Vector Calculus. Part of the ionosphere is co-named after him: the Kennelly-Heaviside layer.  His personality was very eccentric and he is a good example of what many people think of as a mad scientist.

Life
Heaviside was born in Camden Town, London. His father was a skilled wood engraver.  His uncle, Sir Charles Wheatstone (1802-1875), co-invented the telegraph and made famous experiments in 1837 from Euston Station.

Heaviside was short and he had red-hair.  When he was young he suffered from scarlet fever which made him rather deaf, especially in his later years.  Heaviside studied at Camden House School until he was 16 and continued to study at home until he was 18.  He got a job as telegraph operator and spent some time working in Denmark. He noticed that operating speeds were 40% faster from England to Denmark than the other way round, and in 1871 he found a fault in the telegraph cable from England to Denmark.  In 1873 he got a copy of the first edition of Maxwell's book A Treatise on Electricity and Magnetism.  He was very influenced by this book and it helped him to produce many mathematical ideas.  At this time he was working in Newcastle.  In 1875 he went back to London to live with his parents.  There he spent a lot of time writing up his discoveries, but no one wanted to publish his papers because they were too difficult to understand.  He analysed alternating currents 15 years before they started to be used commercially.

When his parents moved to Paignton he moved with them.  After their deaths he lived at Newton Abbott and then, from 1908 until death, in Torquay.  He was unmarried.

Eccentric behaviour
In his old age Heaviside became more and more eccentric.  He was poor, and lived alone in a large house in Torquay that he could not afford to heat properly.  He liked it when friends visited, but many people were put off because he could often be rude.  He could not cook, and a friendly policeman regularly came to the house to bring him food. Because Heaviside was deaf and could not hear the knock on the door the policeman used to open the letterbox and blow his whistle loudly.  The house and garden was in a terrible mess.  He had lumps of granite in his house for furniture. He often put the letters W.O.R.M. after his name, just for a joke.  He took to painting his finger nails pink.  On at least one occasion he dyed his hair black and, while it was wet, he wore a teacosy on his head.

When, in 1925, he fell off a ladder and was taken by ambulance to a nursing home, it was the first time he had ever been in a motor vehicle.  He told the ambulance driver that he had enjoyed the ride.  He died at the nursing home in Torquay a month later and was buried in Paignton.

His work

Heaviside was like Sir Isaac Newton in that he worked alone and did not always publish the whole of his experiments.  He wrote a lot of ideas in his notebooks, but he did not think these ideas were ready for publication, yet they turned out to be very important.  For example, he suggested that there was an atmospheric conducting layer many kilometers above the earth's surface which could be used for sending long-distance radio-waves around the curvature of the earth.  In 1923 it was indeed discovered that there were complex, ionized layers (called the ''ionosphere").  The Kennelly-Heaviside layer was later named after him.  By 1950, 25 years after his death, there were more than 50 stations worldwide experimenting with radio communication in the upper atmosphere.

Heaviside made studies on how to transmit speech clearly in telephone communication.  He studied reflection in telephone circuits (these produced disturbing echoes). From 1886 he spent most of his time on the propagation of electromagnetic waves through space which had been developed by Hertz and on the motion of atoms which had been discovered by J.J. Thomson.

Honours

Heaviside was not interested in receiving honours.  Indeed, he never made any money from his discoveries and often found it difficult to pay the bills.  He was probably very pleased to be made a Fellow of the Royal Society in 1891.  He was awarded a £120 a year Civil List pension by the government.  In 1908 he was elected an Honorary Member of the Institution of Electrical Engineers, and in 1918 Honorary Membership of the Americal Institute of Electrical Engineers.  In 1921 he was the first person to be honoured with the Faraday Medal.

A famous quotation of Heaviside is:
"Shall I refuse my dinner because I do not fully understand the process of digestion"?

Related pages
Heaviside function
Magnetic reluctance
Dirac delta function
Coaxial cable
Hughes Medal

References

Searle G.F.C. 1987. Oliver Heaviside, the man.

Other websites

British electrical engineers
English mathematicians
English physicists
Fellows of the Royal Society
1850 births
1925 deaths
Mad scientists

Oliver Heaviside


In computer science, runtime or run time describes the operation of a computer program, the duration of its execution, from beginning to termination.

The term runtime can also refer to a virtual machine to manage a program written in a computer language while it is running.

Run time is sometimes used to mean  runtime library, a library of basic code that is used by a particular compiler but when used in this fashion, runtime library is more accurate.

A runtime environment is a virtual machine state which provides software services for processes or programs while a computer is running.

Runtime activities include loading and linking of the classes needed to execute a program, optional machine code generation and dynamic optimization of the program, and actual program execution.

For example, a program written in Java calls for services from the Java Runtime Environment by issuing commands from which the expected result is returned by the Java Runtime. By providing these services, the Java Runtime Environment is considered the runtime environment of the program. Both the program and the Java Runtime Environment request services from the operating system. The operating system kernel provides services for itself and all processes and software running under its control. The Operating System may be considered as providing a runtime environment for itself.

Related pages
 Compile time and compiling
 Interpreter (computing)
 Run-time system

References

 Jon Byous, Java technology: The early years. Sun Developer Network, no date [ca. 1998]. Retrieved April 22, 2005.
 James Gosling, A brief history of the Green project . Java.net, no date [ca. Q1/1998]. Retrieved April 29, 2007.
 James Gosling, Bill Joy, Guy Steele, and Gilad Bracha, The Java language specification, third edition. Addison-Wesley, 2005.  (see also online edition of the specification)
 Tim Lindholm and Frank Yellin. The Java Virtual Machine specification, second edition. Addison-Wesley, 1999.  (see also online edition of the specification).

Computer science

Runtime


Butterfly gardening is a growing school of gardening, that is aimed at creating an environment that attracts butterflies, as well as some moths. Butterfly gardening is often done to invite those butterflies and moths to lay eggs as well. In addition to the planting of flowers that feed butterflies, other methods of attracting them are building "butterfly houses", with sand for puddling, water to drink, and food items, like rotten fruit.

Butterflies
Gardening

Butterfly gardening


The Bosnie mafia (or Bosnie organisit creeme) is the body o illegal gangs an creeminal organisations operatin in Bosnie an Herzegovinae an athin the Bosnie diaspora. Bosnie organisit creeme figurs operate maistly in Europe, as well as in Canada, the Unitit States an (Allerton, in Bradford, Unitit Kinrick.) Bosnian organised-crime groups are involved in a wide range of activities, includin drog trokin, gamblin, cigarette smugglin, human trokin, prostitution, illicit antiquities trokin, extortion (aften unner the cover o ostensible security an insurance companies) an the airms trade. Smugglin revenues are atween €150 million an €300 million, equivalent tae Bosnie an Herzegovinae's annual state budget.

Organised-crime groups an activities
The Bosnian Mafia can be broadly divided into four categories – the Bosniak Mafia, the Bosnie Serb Mafia, the Bosnie Croat Mafia an mony local organisit-creeme groups. The Bosnie Mafia groups are involvit in a wide range o activities; ane o their mair notorious activities occurred in 2003 when they smuggled an sauld 250,000 Zastava M70 Yugoslav-made AK-47s tae Iraq.

Judicial prosecution an the struggle against creeminal netwirks
The federalist govrenment o Bosnie an Herzegovinae maks fechtin organisit creeme groups extremely difficult. Bosnie an Herzegovinae consists o twa lairgely unthirldt entities, the Federation o Bosnie an Herzegovinae an Republika Srpska – baith o which hae their awn caipital, parliament, preses an polis force. Due tae the poor cooperation atween law-enforcement agencies in the twa regions, creeminals can aften evade prosecution simply bi muivin frae ane tae the ither. Mafia groups frae neebourin kintras (pairticularly Serbie an Croatie) aften base their operations in Sarajevo, due tae its proximity tae Bosnie an Herzegovinae.

Housomeivver, thare hae been some efforts tae increase collaboration atween the twa regions. In 2006 the European Union established the State Investigation an Protection Agency. SIPA is a Bosnie federal polis agency with authority in both regions, an is unner the direct admeenistration of the Ministry of Security of Bosnia an Herzegovinae (BIH MIS).

See an aw
Human trokin in Bosnie an Herzegovinae
Bosnie an Herzegovinae
Bosnie diaspora
Organisit creeme

References

Crime in Bosnie an Herzegovinae

Bosnie mafia


Early Christianity is known as the Christianity of around the three centuries (1st, 2nd, 3rd, early 4th) between the time of the Crucifixion of Jesus (c.26-36) and  the First Council of Nicaea in 325.

Since the 19th century, historians have learned more about the early Christian community. Books written early in the time period, such as the Didache (in second-millennium copies) and the Gospel of Thomas (in two manuscripts dated as early as about 200 and 340) as well as parts from the Jewish-Christian Gospels have been found again in the last 200 years

History
At the beginning of Christian history, the church was in Jerusalem, in an "upper room" perhaps where the Cenacle is today, and leaders included James, Peter, and John. The major source for the Apostolic Age (c.30-c.100) is the Acts of the Apostles. Following the Great Commission, the missionary actions of the Apostles, including Paul of Tarsus who said the title of "Apostle to the Gentiles", spread Christianity to cities in the Hellenistic world, such as Alexandria and Antioch, and also to Rome and even beyond the Roman Empire. The term "Christian" was first applied to members of the church at Antioch according to . The New Testament includes letters written by Paul to churches, such as those in Thessalonica and Corinth, during the years 50-62, see also Seven Churches of Asia. Early Christians continued the Jewish practice of respect for the Jewish Scriptures, using the Septuagint translation of Hellenistic Judaism that was in general use among Greek-speakers, or the Targums in use among Aramaic-speakers, but added to it their own writings.

References

Early Christianity


Hardcore punk (or hardcore) is a style of punk music that usually sounds louder, faster and more angry than earlier punk rock music from the 1970s. Hardcore began in the late 1970s. Some of the important bands of the genre are Bad Brains, Black Flag, D.O.A, Minor Threat and Cro-Mags. Hardcore songs often have more political lyrics (words) than songs from 1970s punk.

No one is quite sure where the term "hardcore punk" came from. The band D.O.A. may have helped to make the term more popular with the title of their album Hardcore '81 in 1981.

The first hardcore punk song was "Out of Vogue" by the Santa Ana band The Middle Class. This band started to play a loud type of punk with shouted words. At the same time, the Stimulators, a NYC thrash band, performed their song M.A.C.H.I.N.E. which contains all the elements of hardcore punk. The Stimulators played shows with the Bad Brains, who soon created the song Pay to  Cum, which was the fastest punk song ever played at the time.

People in hardcore punk bands and people who listened to hardcore punk wore clothes that were of two types. Some people wore clothes and hairstyles that were strange and a bit scary, like Mohawk hairstyles and leather jackets with metal studs. Other people wore regular clothes like black pants and T-shirts.

At hardcore punk concerts, people did a type of dancing called moshing. This type of dance involves people pushing each other and jumping off the stage.

In the early 1980s, hardcore bands from England began to get noticed, such as Discharge, The Exploited, and The Varukers. Discharge played with a sound that included heavy metal guitar styles.

Hardcore led to the creation of the straight edge movement. People who are straight edge do not use alcohol, tobacco, and other drugs. Straight edge led to two new groups, called hardline and youth crew. Hardcore musicians also created independent record labels in the 1980s.

Many bands had a "Do it Yourself" (DIY) attitude. Bands would record their own albums using simple four-track recorders, and the sell the records themselves at shows or through the mail. Hardcore influenced the creation of new styles of music, including alternative rock, grunge, alternative metal, metalcore, thrash metal and post-hardcore.

Politics

Many early hardcore punk bands had songs about liberal politics such as anarchism and socialism. They spoke against political leaders, such as American president Ronald Reagan and British prime minister Margaret Thatcher. While George W. Bush was President of the United States, a number of hardcore bands criticized him.
A small number of hardcore musicians have expressed right wing views, including the band Antiseen, whose guitarist Joe Young ran for office as a Libertarian in North Carolina. Former Misfits singer Michale Graves supported George W. Bush.

How it sounds
Hardcore band singers often sing by shouting the words of the song. Hardcore songs often sound like the singer is angry.

Hardcore bands use electric guitars, electric bass guitar, drums, and a singer. The electric guitars in hardcore bands are distorted, noisy, and fuzzy because the guitarists are using electronic boxes called distortion pedals. Some hardcore guitarists have changed the tuning of their guitar strings so they play notes that are lower than those a normal guitar can play.

The drums in hardcore music play loud and fast, often characterized by use of the double-bass pedal, particularly notable in the breakdowns.

References

Hardcore punk


Philistus (; c. 432 – 356 BC), son of Archomenidas, was a Greek historian from Sicily, Magna Graecia.

Life
Philistus was born in Syracuse around the time the Peloponnesian War began. He was a faithful supporter of the elder Dionysius, and commander of the citadel. Cicero, who had a high opinion of his work, called him the miniature Thucydides (pusillus Thucydides). He was admitted by the Alexandrian critics into the canon of historiographers, and his work was highly valued by Alexander the Great.

Philistus was quite wealthy. Before his rise to power, Dionysius criticized the generals of Syracuse on their poor performance against the Carthaginians. When the assembly fined Dionysius for his seditious language, Philistus stepped in to pay Dionysius' fines, with the promise to continue paying for them so long as the council saw fit to issue them. It was through Philistus' support that Dionysius was able to rise to power and control Syracuse.

In 386 BC, without consulting the tyrant, Philistus married the daughter of Dionysius' brother Leptines. Dionysius feared that Leptines and Philistus would use the marriage as the basis of an alliance against him, and exiled both men from Syracuse. Philistus settled at Thurii but afterwards moved to Adria, where he remained until the death of Dionysius in 367 BC.  During his stay at Adria, Philistus occupied himself with the composition of his history of Sicily in eleven books. The first part comprised the history of the island from the earliest times to the capture of Agrigentum by the Carthaginians (406 BC); the second, the history of the elder and the younger Dionysius (down to 363 BC). From this point the work was carried on by Athanas.

Philistus was then recalled by the younger Dionysius, whom he persuaded to dismiss Plato and Dion. When Dion set sail from Zacynthus with the object of liberating Syracuse in 356 BC, Philistus was entrusted by Dionysius II with the command of the fleet. After a lengthy sea battle, his ship was surrounded. Philistus took his own life to avoid retribution from his countrymen.

Notes

References

430s BC births
356 BC deaths
Classical-era Greek historians
Ancient Syracusans
5th-century BC Syracusans
4th-century BC Syracusans
4th-century BC historians
Historians from Magna Graecia
Suicides in Ancient Greece

Philistus


Mediastinoscopy is a procedure that enables visualization of the contents of the mediastinum, usually for the purpose of obtaining a biopsy. Mediastinoscopy is often used for staging of lymph nodes of lung cancer or for diagnosing other conditions affecting structures in the mediastinum such as sarcoidosis or lymphoma.

Mediastinoscopy involves making an incision approximately 1 cm above the suprasternal notch of the sternum, or breast bone. Dissection is carried out down to the pretracheal space and down to the carina. A scope (mediastinoscope) is then advanced into the created tunnel which provides a view of the mediastinum. The scope may provide direct visualization or may be attached to a video monitor.

Mediastinoscopy provides access to mediastinal lymph node levels 2, 4, and 7.

Current use
Historically, mediastinoscopy has been the gold standard for the staging of lung cancer. However, with advances in minimally invasive procedures and imaging, mediastinoscopy usage has declined significantly from 2006 to 2010.

Extended mediastinoscopy
Extended mediastinoscopy is a technique which allows access to the pre-aortic (station 6) and aortopulmonary window (station 5) lymph nodes.

Parasternal mediastinotomy
Parasternal mediastinotomy, aka, a Chamberlain procedure, is the standard approach to access lymph nodes at stations 5 and 6.

References

Thoracic surgical procedures
Endoscopy
Pulmonary thoracic surgery

Mediastinoscopy


Alfred George Gardyne de Chastelain, DSO, OBE (1906–1974) was a British-Canadian businessman, soldier, and secret agent, noted for his actions during World War II. He was the father of Canadian General John de Chastelain.

Early life and stay in Romania

Born in London in February 1906, of Anglo-Scottish parents and of Huguenot background, he was educated in Scotland and the University of London, from which he received a First Class degree in Engineering. On graduation, he moved to Romania and worked for Unirea (a British Petroleum branch) in Bucharest, rising to a managerial position towards the end of the 1930s. While in Bucharest, he married Marion Elizabeth Walsh, the daughter of Jack Walsh, an American accountant with Standard Oil of New Jersey in Romania.

Wartime activities with the SOE
On the outbreak of war with Nazi Germany, de Chastelain was commissioned into the Artists' Rifles and became a member of Special Operations Executive (SOE), with which organisation he conducted sabotage operations in the Balkans and served in the North African campaign.

In late 1943, with the rank of lieutenant-colonel, he led a team of two other SOE members (Ivor Porter and Silviu Meţianu) by parachute into Romania (Operation Autonomous), to test the possibility of the country's surrender to the Allies as the Axis was losing battles on the Eastern Front (see Romania during World War II). Captured by the Romanian Gendarmerie soon after their landing near Plosca, Teleorman County, he and his team were taken into custody as prisoners of war and held in a Bucharest apartment.

They were all released in August 1944, when Conducător Ion Antonescu was overthrown and King Michael I assumed full powers (see King Michael Coup). Decorated, de Chastelain was demobilised after the war, joining a unit of the Special Air Service in the Territorial Army.

Life in the post-war era
After the start of the Cold War, Soviet authorities alleged that he was keeping contacts with Iuliu Maniu, the leader of the National Peasants' Party; the latter had opposed both Antonescu's regime and the Soviet occupation of Romania. During Maniu's trial for treason (1947), the Minister of the Interior, Teohari Georgescu, was handed a report which indicated Maniu's alleged contacts with de Chastelain as proof that the politician was a British spy.

Alongside a group of partners, de Chastelain started "Griffin Enterprises" (1945), a business centered in London and Kent, dealing in film processing, import and exports and other activities. In 1954, he left that business to accept a post as Vice-President of the Canadian branch of an American oilwell drilling and services company, H. J. Eastman, in Calgary, Alberta.  Living for several years in Canada and then in Australia, he returned to Canada in the early 1970s and died in Calgary at the age of 68.

A book giving an account of the SOE Operation in Romania was written by one member of the team, Ivor Porter, later a British Ambassador (Operation Autonomous: With SOE in Wartime Rumania, Chatto and Windus, 1989).

He was a founding member of the Special Forces Club in Knightsbridge, and of the "Food and Wine Society" in Calgary.

Notes

References
 Alesandru Duţu, Florica Dobre, Andrei Siperco, "Pagini dintr-o istorie nescrisă: 1941-1945. Prizonieri de război în România" (Pages from an Unwritten History: 1941-1945. Prisoners of War in Romania"), in Magazin Istoric, March 1997
Tatiana Pokivailova, Ioan Chiper, "Procesul Maniu în arhivele de la Moscova" ("The Maniu Trial in the Moscow Archives"), in Magazin Istoric, April 2002

1906 births
1974 deaths
Alumni of the University of London
Anglo-Scots
Artists' Rifles officers
British Special Operations Executive personnel
Special Air Service officers
British emigrants to Canada
British Army personnel of World War II
20th-century Canadian businesspeople
Companions of the Distinguished Service Order
Officers of the Order of the British Empire
Businesspeople from London
World War II prisoners of war held by Romania
Romania in World War II
Scottish Canadians
BP people
20th-century English businesspeople

Alfred Gardyne de Chastelain


Sarchi is a city in Costa Rica. It is famous for its art and hand-crafted furniture. Sarchi is also known for its colorful hand carts that the people there have been making for centuries.

Sarchi has several small communities.  One of the most famous is "Los Bajos del Toro". It is  northeast from the center of Sarchi. Although the road is very old, travelers  will be able to see one of the best views in Costa Rica. "Los Bajos" has a climate that is very humid and rainy.

Sarchi is a community made up of artists and craftsmen. It is well known for its ox carts.  Most of the ox carts in Costa Rica are created in Sarchi and are sold throughout the country and to tourists.  They are usually painted in very bright colors and patterns. In the past, these ox carts were used to carry coffee to the markets.

There are many shops in the town that sell hand-made wooden furniture and souvenirs. Some of the many items that are sold are: tables, rocking chairs, beds, jewelry, sculptures, and paintings.

Cities in Costa Rica

Sarchí, Costa Rica


A Teddy boy was a young man who dressed in a style popular in 1950s Britain. The word 'Teddy' comes from the clothes, which were a version of clothes from the Edwardian period, about 1910. That period is named after Edward VII (19011910).

Teddy boys relate to early rock and roll music. People often thought of them as rebels. The teddy boy also meant the first British culture of young people. For the first time, teenagers who were in a working class could buy clothes, a bicycle or motorcycle and entertainment. Teddyboys wore clothes that might shock older people.

Style
The style featured a long and jacket with one row of buttons and narrow pants, "brothel creeper" shoes with thick crepe soles and pencil-thin ties. Duck-tail hairstyles were the norm. The movement was working class, and rather dandyfied. It was quite short-lived, and gave way to the mods and rockers of the early 1960s.

References

British culture
Fashion
1950s

Teddy boy


A valve is a device that regulates the flow of fluids.

Valve may also refer to:
 Valve Corporation, a computer software/games company responsible for the best-selling Half-Life games
 Valve hall, a building which contains the valves of the static inverters of an HVDC plant
 Mercury arc valve, a type of electrical rectifier which converts alternating current into direct current
 Poppet valve, a valve consisting of a hole and a tapered plug on the end of a shaft
 Heart valve, valves in the heart that maintain the unidirectional flow of blood
 Vein valve, valves in veins and other fluid cords in body
 Valve Records, a record label
 Valve Recordings, a record label
 Thermionic valve, an electronic component otherwise known as a vacuum tube
 Valve, a paired clasping process on the male genitalia of a moth
 Valve is one of eight overlapping shells on a chiton mollusc
 Valve Sound System

Valve (disambiguation)


Naval Station Pearl Harbor is a United States naval base on the island of Oahu, Hawaii. In 2010, as part of the recommendations of the Base Realignment and Closure (BRAC) commission, the naval station was consolidated with the  United States Air Force's Hickam Air Force Base to form Joint Base Pearl Harbor–Hickam. Since 1940, Pearl Harbor has been the headquarters of the United States Pacific Fleet.

Overview
Naval Station Pearl Harbor provides berthing and shore side support to surface ships and submarines, as well as maintenance and training. Pearl Harbor can accommodate the largest ships in the fleet, to include dry dock services, and is now home to over 160 commands. Housing, personnel, and family support are also provided and are an integral part of the shore side activities, which encompasses both permanent and transient personnel.

Because Pearl Harbor is the only intermediate maintenance facility for submarines in the Middle Pacific, it serves as host to a large number of visiting submariners.

The Naval Computer and Telecommunications Area Master Station, Pacific (NCTAMS PAC), Wahiawa, Hawaii is the world's largest communication station. The headquarters site of this shore command is located in the central section of the island of Oahu, approximately three miles north of Wahiawa.

History

1899–1941
Following the annexation of Hawaii, Pearl Harbor was refitted to allow for more navy ships. In May 1899, Commander John F. Merry was made naval representative with authority to transact business for the Navy Department and its Bureaus. He immediately assumed control of the Coal Depot and its equipment. To supplement his facilities, he was assigned the Navy tug  and two coal barges. Inquiries that commenced in June culminated in the establishment of the "Naval Station, Honolulu" on 17 November 1899. On 2 February 1900, this title was changed to "Naval Station, Hawaii".

The creation of the Naval Station allowed the Navy Department to explore territorial outposts. In October 1899,  and Iroquois made extensive surveys and sounding of the waterways to Midway and Guam. One of the reasons for these explorations was to select a possible cable route to Luzon.

A coal famine and an outbreak of the bubonic plague were the only two incidents that hindered the Commandant from fulfilling his duties. Because of the severe coal shortage in September 1899, the Commandant sold coal to the Oahu Railway and Land Company and the Inter-Island Steam Navigation Company, Ltd. Although this indicated the affinity of economic ties with the Navy, it was to a certain extent counteracted by the quarantine of the naval establishment from December 1899 to February 1900, because of the bubonic plague. Approximately 61 deaths were recorded in Honolulu for this period. Work was consequently delayed on nascent Navy projects in Honolulu Harbor.

From 1900 to 1908, the Navy devoted its time to improving the facilities of the  that constituted the naval reservation in Honolulu. Under the Appropriation Act of 3 March 1901, this tract of land was improved with the erection of additional sheds and housing. Improvements included a machine shop, smithery and foundry, Commandant's house and stables, cottage for the watchman, fencing, 10-ton wharf crane, and water-pipe system. Pearl Harbor was dredged and the channel enlarged to accommodate larger ships. On 28 May 1903, the first battleship, , entered the harbor for coal and water. However, when the vessels of the Asiatic station visited Honolulu in January 1904, Rear Admiral Silas Terry complained that they were inadequately accommodated with dockage and water.

Under the above Appropriation Act, Congress approved the acquisition of lands for the development of a naval station at Pearl Harbor and the improvement of the channel to the Lochs. The Commandant, under the direction of the Bureau of Equipment, attempted to obtain options on lands surrounding Pearl Harbor that were recommended for naval use. This endeavor was unsuccessful when the owners of the property refused to accept what was deemed to be a fair price. Condemnation proceedings, under the Hawaiian law of eminent domain, were begun on 6 July 1901. The land acquired by this suit included the present Navy Yard, Kauhua Island, and a strip on the southeast coast of Ford Island. The work of dredging the coral reef that blocked Pearl Harbor progressed rapidly enough to allow the gunboat  to proceed to the upper part of Main Loch in January 1905.

One of the early concerns of the growing station was that the Army would make claims on its property. Because of their facilities, as wharves, cranes, artesian wells, and coal supplies, many requests were made by the Army for their use. By February 1901, the Army had made application for the privilege of establishing on Navy docks movable cranes for handling coal and other stores, a saluting battery and a flag staff on the naval reservation, and an artesian well of its own. All these requests were rejected by the Bureau of Equipment on the theory that, once granted, they "will practically constitute a permanent foothold on the property, and end in dividing it between the two Departments, or in the entire exclusion of the Navy Department on the ground of military expediency as established by frequency of use." However, the Army Depot Quartermaster at Honolulu contracted for the sinking of an artesian well on the Naval Station with the Commandant's approval, who, in turn, acted on a recommendation of the Bureau of Yards and Docks. The flow of water obtained amounted to over 1.5 million gallons per day, sufficient for all purposes of the Army and Navy. The Bureau of Equipment felt that its word of caution was justified when the Depot Quartermaster in 1902 let it be known that any water used by the Navy from the artesian well was "only given by courtesy of the Army".

Despite the warnings of the Bureau of Equipment, the War Department, the Department of Labor and Commerce, and the Department of Agriculture had secured permission to settle on the naval reservation. By 1906, the Commandant believed that it was necessary for the Bureau of Yards and Docks to develop a policy on the future of the station. The docks were being used to a greater extent by the Army transports than by Navy ships, and the Army was actually attempting to get possession of Quarantine Wharf (which was built by the Territorial Government on the Naval Reservation, with the understanding that it could be taken over at any time by the Navy Department upon the payment of its appraised value). In 1903, the Department of Labor and Commerce received about  for an Immigration Station. The Department of Agriculture had, in the meanwhile, secured part of the site intended for a hospital as an experimental station. The Commandant felt that, if the station was going to develop beyond a mere coaling depot, these territorial encroachments on the part of other departments should be stopped, particularly when they were enjoying the benefits of naval appropriations. "On the other hand," he wrote, "if it is the intention to improve Pearl Harbor and eventually abandon this station every effort should be made to begin work there as soon as possible... I am informed that important commercial interests will make a strong effort next year to have Pearl Harbor improved, and I think that will be an opportune time for the Navy Department to make efforts in the same direction."

In 1908, the Pearl Harbor Naval Shipyard was established. The period from 1908 to 1919 was one of steady and continuous growth of the Naval Station, Pearl Harbor, with the exception of the discouraging collapse of the drydock in 1913. The Act of 13 May 1908 authorized the enlargement and dredging of the Pearl Harbor channel and lochs "to admit the largest ships," the building of shops and supply houses for the Navy Yard, and the construction of a drydock. Work on the dock started on 21 September 1909. In April 1910, the barquentine Amaranth became the fourth deep-sea, cargo-carrying vessel to venture into the newly dredged harbor, having been preceded by the three-masted schooner W.H. Marston on 8 March, and the schooner Ariel and bark Marston a few days later. Amaranth delivered materials for construction of the dry dock facility. Work progressed satisfactorily on all projects, except the drydock. After much wrangling with Congress to secure an appropriation of over three million dollars for its construction, the drydock was wrecked by "underground pressure". "On February 17, 1913, the entire drydock structure rumbled, rocked, and caved in." The drydock was ceremonially opened to flooding on 21 August 1919, by Mrs. Josephus Daniels, wife of the Secretary of the Navy. In 1917, Ford Island in the middle of Pearl Harbor was purchased for joint Army and Navy use in the development of military aviation in the Pacific.

As the Imperial Japanese military pressed its war in China, concern over Japan's intentions caused the U.S. to begin taking defensive measures. On 1 February 1933, the U.S. Navy staged a mock attack on the base at Pearl Harbor as part of a preparedness exercise. The attack "succeeded" and the defense was deemed a "failure".

Sunday 7 December 1941

The attack on Pearl Harbor by the Empire of Japan on Sunday 7 December 1941 brought the United States into World War II.

Aircraft and midget submarines of the Imperial Japanese Navy under the command of Admiral Chuichi Nagumo began bombing Honolulu at the U.S. naval base. Through earlier code breaking activity, the Americans had determined that an attack was likely to occur. However, while the Americans failed to discover Japan's target location, it was believed that the Philippines was the most likely target. At 06:05 on 7 December, the six Japanese carriers launched a first wave of 183 aircraft composed mainly of dive bombers, horizontal bombers and fighters.

The Japanese struck American ships and military installations at 07:51. The first wave attacked airfields of Ford Island. At 08:30, a second wave of 170 Japanese aircraft, mostly torpedo bombers, attacked the fleet anchored in Pearl Harbor. The battleship  was hit with an armor-piercing bomb which penetrated the forward ammunition compartment, sinking it within seconds, killing 1,177 crew members.

The overall death toll was 2,467 people: 2,403 American citizens—2,335 members of the U.S. military and 68 civilians—and 64 members of the Imperial Japanese Navy. Five U.S. battleships were sunk and the other remaining three were badly damaged. Overall, nine ships of the U.S. fleet were sunk and 21 ships were severely damaged. Three of the 21 would be irreparable. 188 U.S. aircraft were completely destroyed and 159 others were damaged. Japan lost 29 out of the 353 aircraft they attacked with. The attack on Pearl Harbor was the largest attack committed by foreign nationals on American soil in peacetime and in general prior to the September 11 attacks.

The first shots fired were from the destroyer  on a midget submarine that surfaced outside of Pearl Harbor; Ward sank the midget sub at approximately 06:55, about an hour before the attack on Pearl Harbor.

West Loch Explosion, 1944

Just after 3 p.m. on Sunday 21 May 1944 an explosion in a staging area for Landing Ships, Tank (LSTs) and other amphibious assault ships in West Loch leading to a fire which quickly spread among the ships being prepared for Operation Forager, the invasion of the Japanese-held Mariana Islands. Over the next 24 hours, six LSTs sank, 163 naval personnel died and 396 were injured.

A subsequent Naval Board of Inquiry never determined the exact cause of the disaster. But it concluded the initial explosion was caused when a mortar round aboard  detonated during an unloading operation because it was either dropped or went off when gasoline vapours ignited. The incident – together with the Port Chicago disaster two months later – led to major changes in weapon handling practices within the United States Navy.

Post-World War II years
Over the years, Pearl Harbor remained a main base for the US Pacific Fleet after World War II along with Naval Base San Diego. In 2010, the Navy and the Air Force merged their two nearby bases; Pearl Harbor joined with Hickam Air Force Base to create Joint Base Pearl Harbor–Hickam.

On 29 January 1964 the Navy base itself was recognized as a National Historic Landmark district and with the National Register of Historic Places since 1976. Within its bounds, it contains several other National Historic Landmarks associated with the attack on Pearl Harbor, including the Arizona, , and . As an active Navy base, many of the historic buildings that contributed to the NHL designation are under threat of demolition and rebuilding.

From 1932 to 1983 the most recognizable structure on the submarine base was the 100-foot-tall Escape Training Tank. A similar tower on Ford Island was converted to an airport control tower during World War II and still stands. Generations of submariners learned to escape in up to 80 feet of water using buoyant ascent, and were trained in the use of the Momsen lung or Steinke hood. The Steinke hood was replaced by the Submarine Escape Immersion Equipment in the 2000s.

Bloch Arena located on the base is still in use today and hosted the collegiate Pearl Harbor Basketball Invitational and was home to the Honolulu Chiefs of the American Basketball League. The address is 224 A Avenue, Honolulu, HI 96818.

On 4 December 2019, while the nuclear submarine USS Columbia was in dry dock at Pearl Harbor, 22 year-old Machinist Mate Auxiliary Fireman Gabriel Romero shot and killed two naval station civilian employees and injured another before shooting and killing himself. Romero was one of Columbias crew and been assigned to the armed Topside Roving Patrol aboard the vessel; both weapons involved in the shooting were issued from the Columbia. Later investigation by the Navy determined that Romero was "likely unfit" for service on a submarine and that officers aboard the Columbia had failed to recognize signs of Romero's degrading mental state.

Environmental contamination
On 14 October 1992, the US EPA added the Pearl Harbor Naval Complex as a superfund to the so-called National Priorities List.

Vessels

Surface ships presently homeported
s (9)

Submarines presently homeported

s (11)

s (6)

As part of the 2006 Quadrennial Defense Review, the Navy announced in early 2006 that it would shift 60% of its attack submarines to the Pacific by 2010.

USS Missouri ("Mighty Mo" or "Big Mo'''") is a United States Navy , and was the fourth ship of the U.S. Navy to be named in honor of the U.S. state of Missouri. Missouri was the last battleship built by the United States, and was the site of the surrender of the Empire of Japan which ended World War II.Missouri was ordered in 1940 and commissioned in June 1944. In the Pacific Theater of World War II she fought in the battles of Iwo Jima and Okinawa and shelled the Japanese home islands, and she fought in the Korean War from 1950 to 1953. She was decommissioned in 1955 into the United States Navy reserve fleets (the "Mothball Fleet"), but reactivated and modernized in 1984 as part of the 600-ship Navy plan, and provided fire support during Operation Desert Storm in January/February 1991.Missouri received a total of 11 battle stars for service in World War II, Korea, and the Persian Gulf, and was finally decommissioned on 31 March 1992, but remained on the Naval Vessel Register until her name was struck in January 1995. In 1998, she was donated to the USS Missouri'' Memorial Association and became a museum ship at Pearl Harbor, Hawaii.

See also
United States Navy submarine bases
World War II United States Merchant Navy
Naval Base Hawaii
Naval Submarine Base Pearl Harbor

References

External links

Naval Stations of the United States Navy
P
Installations of the United States Navy in Hawaii
National Historic Landmarks in Hawaii
Oahu
Superfund sites in Hawaii
United States Navy submarine bases
World War II on the National Register of Historic Places in Hawaii
1899 establishments in Hawaii
Historic districts on the National Register of Historic Places in Hawaii
National Register of Historic Places in Honolulu County, Hawaii
Military installations established in 1899
Pearl Harbor

Naval Station Pearl Harbor


John R. Massaro (born May 22, 1930) is a retired United States Marine who served as the 8th Sergeant Major of the Marine Corps from 1977 to 1979.

Early life
Massaro was born in Cleveland, Ohio on May 22, 1930. He graduated from Orrville High School in Orrville, Ohio, in 1948 and enlisted in the United States Marine Corps on August 6, 1948. He underwent recruit training at Marine Corps Recruit Depot Parris Island, South Carolina. He is a member of the Church of Jesus Christ of Latter-day Saints.

Military career
Following recruit training, Massaro was assigned to the 1st Battalion 6th Marines as a rifleman, before transferring to Marine Corps Base Camp Pendleton. He remained there until March 1950. From Camp Pendleton, he was assigned to the Marine Barracks, U.S. Navy Supply Depot, Clearfield, Utah, as a security guard. That was followed by the first of three tours as a drill instructor at the Marine Corps Recruit Depot San Diego. He served on the drill field until April 1952, when he rejoined the 1st Marine Division in Korea. He was decorated for actions in combat during that tour.

From Korea, Massaro returned to San Diego for duty as a drill instructor and later as an instructor at the Drill Instructor School. In August 1955, he was a company gunnery sergeant with the 1st Marine Division's Reconnaissance Company and later the 1st Force Reconnaissance Company, remaining there until November 1959. While serving as the company Gunnery Sergeant, he acquired the now famous spirited cry, "Oorah!". He there passed it on to the drill instructor students and they, in turn, passed it on to their recruits.

Massaro was then transferred to Headquarters and Headquarters Squadron, 1st Marine Aircraft Wing at Marine Corps Air Station Iwakuni, Japan. He returned to the 1st Force Reconnaissance Company for duty as company Gunnery Sergeant and later as Company First Sergeant. In June 1963, he returned to San Diego for a third tour as a drill instructor, serving as Chief Drill Instructor.

Massaro then transferred to the Inspector-Instructor Staff of the 4th Force Reconnaissance Company in San Bernardino, California. In 1967, the Sergeant Major joined the 3rd Engineer Battalion in South Vietnam. He was initially assigned as a company First Sergeant and later became the battalion sergeant major.

Massaro next reported to the Marine Corps Recruiting Station in San Francisco, California, and served as Recruiting Station Sergeant Major until August 1972. He then returned overseas and served as group Sergeant Major of Marine Aircraft Group 36 on Okinawa. Following that tour, he reported to Headquarters Marine Corps where he served as Sergeant Major to the Inspector General of the Marine Corps.

In May 1976, Massaro became Sergeant Major of the 1st Marine Division at Camp Pendleton, California, and remained in that billet until his selection as Sergeant Major of the Marine Corps. Sergeant Major John R. Massaro assumed the post of 8th Sergeant Major of the Marine Corps on 1 April 1977.

Awards and honors
Massaro's military decorations include:

References

External links

1930 births
Living people
Military personnel from Cleveland
United States Marine Corps personnel of the Korean War
United States Marine Corps personnel of the Vietnam War
United States Marines
Sergeants Major of the Marine Corps
Latter Day Saints from Ohio

John R. Massaro


Tall Ship Chronicles is a television series produced in Canada in 2001 and 2002. It followed the training of Canadian journalist and actor Andrew Younghusband on an 18-month sail training voyage around the World, on the barque Picton Castle.

Originally, a new episode was aired approximately once per month. Some of the people in the show are the ship's professional crew, while many are trainees who joined the ship to travel or learn about tall ship sailing. The number of crew when the ship began its voyage from Nova Scotia, Canada, was approximately four dozen. Some of the trainees had only booked one leg of the voyage while others had signed on for the entire 18 months. Some left early because they fell in love while on board — or because of personality conflicts — while others decided to stay on board longer than they'd initially planned. The show follows the interpersonal relations between many of the people on board while also showing a bit about the various islands the ship visits during the voyage.

This was the Picton Castle's second sail-training voyage around the World. The vessel contained a supply of text books in her hold, which she distributed to a number of isolated communities in the South Pacific, including The Cook Islands, Pitcairn Island, Tonga, French Polynesia, Samoa, and Fiji. The show first aired in Canada and has subsequently been aired in various European markets.

External links

 The sailing training vessel Picton Castle

Tall ships
2000s Canadian reality television series
2001 Canadian television series debuts
2002 Canadian television series endings

Tall Ship Chronicles


The word "rounding" for a numerical value means replacing it by another value that is approximately equal but has a shorter, simpler, or more explicit form. For example, US$23.74 could be rounded to US$24, the fraction 312/937 could be rounded to 1/3, and the expression  could be rounded to 1.41.

Rounding is often done on purpose to obtain a value that is easier to write and handle than the original. It may also be done to indicate the accuracy of a computed number; for example, a quantity that was computed as 123,456, but is known to be accurate only to within a few hundred units, is better stated as "about 123,500".

On the other hand, rounding can introduce some round-off error as a result. Rounding is almost unavoidable in many computations, especially when dividing two numbers in integer or doing fixed-point arithmetic; when computing mathematical functions such as square roots, logarithms, and sines; or when using a floating point representation with a fixed number of significant digits. In a sequence of calculations, these rounding errors generally accumulate, and in certain "ill-conditioned" cases, they may make the result meaningless.

Accurate rounding of transcendental mathematical functions is difficult, because the number of extra digits that need to be calculated to resolve whether to round up or down cannot be known in advance. This problem is known as "the table-maker's dilemma" (below).

Rounding has many similarities to the quantization that occurs when physical quantities must be encoded by numbers or digital signals.

Types of rounding

Typical rounding problems can include:
 Approximating an irrational number by a fraction. For example, π by 22/7.
 Approximating a fraction with periodic decimal expansion by a finite decimal fraction. For example, 5/3 by 1.6667.
 Replacing a rational number by a fraction with smaller numerator and denominator. For example, 3122/9417 by 1/3.
 Replacing a fractional decimal number by one with fewer digits. For example, 2.1784 dollars by 2.18 dollars.
 Replacing a decimal integer by an integer with more trailing zeros. For example. 23,217 people by 23,200 people.
 Replacing a value by a multiple of a specified amount. For example. 27.2 seconds by 30 seconds (a multiple of 15).

Rounding to a specified increment
The most common type of rounding is to round to an integer; or, more generally, to an integer multiple of some increment—such as rounding to whole tenths of seconds, hundredths of a dollar, to whole multiples of 1/2 or 1/8 inch, to whole dozens or thousands, etc..

In general, rounding a number x to a multiple of some specified increment m entails the following steps:

Divide x by m, let the result be y;
Round y to an integer value, call it q;
Multiply q by m to obtain the rounded value z.

For example, rounding x = 2.1784 dollars to whole cents (that is, to a multiple of 0.01) entails computing y = x/m = 2.1784/0.01 = 217.84, then rounding y to the integer q = 218, and finally computing z = q×m = 218×0.01 = 2.18.

When rounding to a predetermined number of significant digits, the increment m depends on the magnitude of the number to be rounded (or of the rounded result).

The increment m is normally a finite fraction in whatever numeral system that is used to represent the numbers. For display to humans, that usually means the decimal numeral system (that is, m is an integer times a power of 10, like 1/1000 or 25/100). For intermediate values stored in digital computers, it often means the binary numeral system (m is an integer times a power of 2).

The abstract single-argument "round()" function that returns an integer from an arbitrary real value has at least a dozen distinct concrete definitions presented in the rounding to integer section. The abstract two-argument "round()" function is formally defined here, but in many cases it is used with the implicit value m = 1 for the increment and then reduces to the equivalent abstract single-argument function, with also the same dozen distinct concrete definitions.

Rounding to integer
The most basic form of rounding is to replace an arbitrary number by an integer. All the following rounding modes are concrete implementations of the abstract single-argument "round()" function presented and used in the previous sections.

There are many ways of rounding a number y to an integer q.  The most common ones are
 Round down (or take the floor, or round towards minus infinity):  q is the largest integer that does not exceed y.

 Round up (or take the ceiling,  or round towards plus infinity):  q is the smallest integer that is not less than y.

 Round towards zero (or truncate, or round away from infinity): q is the integer part of y, without its fraction digits.

 Round away from zero (or round towards infinity): if y is an integer, q is y; else q is the integer that is closest to 0 and is such that y is between 0 and q.

 Round to nearest: q is the integer that is closest to y. This is sometimes written as  (see below for tie-breaking rules).

The first four methods are called directed rounding, as the displacements from the original number y to the rounded value q are all directed towards or away from the same limiting value (0, +∞, or −∞).

If y is positive, round-down is the same as round-towards-zero, and round-up is the same as round-away-from-zero. If y is negative, round-down is the same as round-away-from-zero, and round-up is the same as round-towards-zero. In any case, if y is integer, q is just y. The following table illustrates these rounding methods:

Where many calculations are done in sequence, the choice of rounding method can have a very significant effect on the result. A famous instance involved a new index set up by the Vancouver Stock Exchange in 1982. It was initially set at 1000.000, and after 22 months had fallen to about 520 — whereas stock prices had generally increased in the period. The problem was caused by the index being recalculated thousands of times daily, and always being rounded down to 3 decimal places, in such a way that the rounding errors accumulated. Recalculating with better rounding gave an index value of 1098.892 at the end of the same period.

Tie-breaking
Rounding a number y to the nearest integer requires some tie-breaking rule for those cases when y is exactly half-way between two integers — that is, when the fraction part of y is exactly 0.5.

Round half up
The following  tie-breaking rule, called round half up (or round half towards plus infinity), is widely used in many disciplines. That is, half-way values y are always rounded up.

 If the fraction of y is exactly 0.5, then q = y + 0.5.

For example, by this rule the value 23.5 gets rounded to 24, but −23.5 gets rounded to −23.

This is one of two rules generally taught in US elementary mathematics classes.

If it were not for the 0.5 fractions, the roundoff errors introduced by the round to nearest method would be quite symmetric: for every fraction that gets rounded up (such as 0.268), there is a complementary fraction (namely, 0.732) that gets rounded down, by the same amount. When rounding a large set of numbers with random fractional parts, these rounding errors would statistically compensate each other, and the expected (average) value of the rounded numbers would be equal to the expected value of the original numbers.

However, the round half up tie-breaking rule is not symmetric, as the fractions that are exactly 0.5 always get rounded up. This asymmetry introduces a positive bias in the roundoff errors. For example, if the fraction of y consists of three random decimal digits, then the expected value of q will be 0.0005 higher than the expected value of y. For this reason, round-to-nearest with the round half up rule is also (ambiguously) known as asymmetric rounding.

One reason for rounding up at 0.5 is that only one digit need be examined. When seeing 17.50000..., for example, the first three figures, 17.5, determines that the figure would be rounded up to 18. If the opposite rule were used (round half down), then all the zero decimal places would need to be examined to determine if the value is exactly 17.5.

Round half down
One may also use round half down (or round half towards minus infinity) as opposed to the more common round half up (the round half up method is a common convention, but is nothing more than a convention).

 If the fraction of y is exactly 0.5, then q = y − 0.5.

For example, 23.5 gets rounded to 23, and −23.5 gets rounded to −24.

The round half down tie-breaking rule is not symmetric, as the fractions that are exactly 0.5 always get rounded down. This asymmetry introduces a negative bias in the roundoff errors. For example, if the fraction of y consists of three random decimal digits, then the expected value of q will be 0.0005 lower than the expected value of y. For this reason, round-to-nearest with the round half down rule is also (ambiguously) known as asymmetric rounding.

Round half away from zero
The other tie-breaking method commonly taught and used is the round half away from zero (or round half towards infinity), namely:

 If the fraction of y is exactly 0.5, then q = y + 0.5 if y is positive, and q = y − 0.5 if y is negative.

For example, 23.5 gets rounded to 24, and −23.5 gets rounded to −24.

This method treats positive and negative values symmetrically, and therefore is free of overall bias if the original numbers are positive or negative with equal probability. However, this rule will still introduce a positive bias for positive numbers, and a negative bias for the negative ones.

It is often used for currency conversions and price roundings (when the amount is first converted into the smallest significant subdivision of the currency, such as cents of a euro) as it is easy to explain by just considering the first fractional digit, independently of supplementary precision digits or sign of the amount (for strict equivalence between the paying and recipient of the amount).

Round half towards zero
One may also round half towards zero (or round half away from infinity) as opposed to the more common round half away from zero (the round half away from zero method is a common convention, but is nothing more than a convention).

 If the fraction of y is exactly 0.5, then q = y − 0.5 if y is positive, and q = y + 0.5 if y is negative.

For example, 23.5 gets rounded to 23, and −23.5  gets rounded to −23.

This method also treats positive and negative values symmetrically, and therefore is free of overall bias if the original numbers are positive or negative with equal probability. However, this rule will still introduce a negative bias for positive numbers, and a positive bias for the negative ones.

Round half to even
A tie-breaking rule that is even less biased is round half to even, namely

 If the fraction of y is 0.5, then q is the even integer nearest to y.

Thus, for example, +23.5 becomes +24, +22.5 becomes +22, −22.5 becomes −22, and −23.5 becomes −24.

This method also treats positive and negative values symmetrically, and therefore is free of overall bias if the original numbers are positive or negative with equal probability. In addition, for most reasonable distributions of y values, the expected (average) value of the rounded numbers is essentially the same as that of the original numbers, even if the latter are all positive (or all negative). However, this rule will still introduce a positive bias for even numbers (including zero), and a negative bias for the odd ones.

This variant of the round-to-nearest method is also called unbiased rounding (ambiguously, and a bit abusively), convergent rounding, statistician's rounding, Dutch rounding, Gaussian rounding, or bankers' rounding. This is widely used in bookkeeping.

This is the default rounding mode used in IEEE 754 computing functions and operators.

Round half to odd
Another tie-breaking rule that is very similar to round half to even, namely

 If the fraction of y is 0.5, then q is the odd integer nearest to y.

Thus, for example, +22.5 becomes +23, +21.5 becomes +21, −21.5 becomes −21, and −22.5 becomes −23.

This method also treats positive and negative values symmetrically, and therefore is free of overall bias if the original numbers are positive or negative with equal probability. In addition, for most reasonable distributions of y values, the expected (average) value of the rounded numbers is essentially the same as that of the original numbers, even if the latter are all positive (or all negative). However, this rule will still introduce a negative bias for even numbers (including zero), and a positive bias for the odd ones.

This variant is almost never used in most computations, except in situations where one wants to avoid rounding 0.5 or −0.5 to zero, or to avoid increasing the scale of numbers represented as floating point (with limited ranges for the scaling exponent), so that a non infinite number would round to infinite, or that a small denormal value would round to a normal non-zero value (these could occur with the round half to even mode). Effectively, this mode prefers preserving the existing scale of tie numbers, avoiding out of range results when possible.

Stochastic rounding
Another unbiased tie-breaking method is stochastic rounding:

 If the fractional part of y is .5, choose q randomly among y + 0.5 and y − 0.5, with equal probability.

Like round-half-to-even, this rule is essentially free of overall bias; but it is also fair among even and odd q values. On the other hand, it introduces a random component into the result; performing the same computation twice on the same data may yield two different results. Also, it is open to unconscious bias if humans (rather than computers or devices of chance) are "randomly" deciding in which direction to round.

Alternating tie-breaking
One method, more obscure than most, is round half alternatingly.

 If the fractional part is 0.5, alternate round up and round down: for the first occurrence of a 0.5 fractional part, round up; for the second occurrence, round down; so on so forth.

This suppresses the random component of the result, if occurrences of 0.5 fractional parts can be effectively numbered. But it can still introduce a positive or negative bias according to the direction of rounding assigned to the first occurrence, if the total number of occurrences is odd.

Simple dithering
In some contexts, all the rounding methods above may be unsatisfactory. For example, suppose that y is an accurate measurement of an audio signal, which is being rounded to an integer q in order to reduce the storage or transmission costs. If y changes slowly with time, any of the rounding method above will result in q being completely constant for long intervals, separated by sudden jumps of ±1. When the q signal is played back, these steps will be heard as a very disagreeable noise, and any variations of the original signal between two integer values will be completely lost.

One way to avoid this problem is to round each value y upwards with probability equal to its fraction, and round it downwards with the complement of that probability. For example, the number 23.17 would be rounded up to 24 with probability 0.17, and down to 23 with probability 1 - 0.17 = 0.83. (This is equivalent to rounding down y + s, where s is a random number uniformly distributed between 0 and 1.) With this special rounding, known as dithering, the sudden steps get replaced by a less objectionable noise, and even small variations in the original signal will be preserved to some extent.  Like the stochastic approach to tie-breaking, dithering has no bias: if all fraction values are equally likely, rounding up by a certain amount is as likely as rounding down by that same amount; and the same is true for the sum of several rounded numbers. On the other hand, dithering introduces a random component in the result, much greater than that of stochastic tie-breaking.

More precisely, the roundoff error for each dithered number will be a uniformly distributed random variable with mean value of zero, but with a standard deviation , which is better than the 1/2 standard deviation with the simple predictive methods, but slightly higher than with the simpler stochastic method. However, the sum of n rounded numbers will be a random variable with expected error zero, but with standard deviation  (the total remaining noise) which diverges semi-quadratically and may become easily perceptible, even if the standard deviation of the roundoff error per sample will be  which slowly converges semi-quadratically to zero. So, this random distribution may still be too high for some applications that are rounding a lot of data.

Multidimensional dithering
This variant of the simple dithering method still rounds values with probability equal to its fraction. However, instead of using a random distribution for rounding isolated samples, the roundoff error occurring at each rounded sample is totalled for the next surrounding elements to sample or compute; this accumulated value is then added to the value of these next sampled or computed values to round, so that the modified values will take into account this difference using a predictive model (such as Floyd–Steinberg dithering).

The modified values are then rounded with any one of the above rounding methods, the best ones being with stochastic or dithering methods: in this last case, the sum of n rounded numbers will still be a random variable with expected error zero but with an excellent constant standard deviation of , instead of diverging semi-quadratically when dithering isolated samples; and the overall average roundoff error deviation per rounded sample will be  that will converge hyperbolically to zero, faster than with the semi-hyperbolic convergence when dithering isolated samples.

In practice, when rounding large sets of sampled data (such as audio, image and video rendering), the accumulation of roundoff errors is most frequently used with a simple predictive rounding of the modified values (such as rounding towards zero), because it will still preserve the hyperbolic convergence towards zero of the overall mean roundoff error bias and of its standard deviation. This enhancement is frequently used in image and audio processing (notably for accurate rescaling and antialiasing operations, where the simple probabilistic dithering of isolated values may still produce perceptible noise, sometimes even worse than the moiré effects occurring with simple non-probabilistic rounding methods applied to isolated samples).

The effective propagation of accumulated roundoff errors may depend on the discrete dimension of the sampled data to round: when sampling bidimensional images, including colored images (that add the discrete dimension of color planes), or tridimensional videos (that add a discrete time dimension), or on polyphonic audio data (using time and channel discrete dimensions), it may still be preferable to propagate this error into a preferred direction, or equally into several orthogonal dimensions, such as vertically vs. horizontally for bidimensional images, or into parallel color channels at the same position and/or timestamp, and depending on other properties of these orthogonal discrete dimensions (according to a perception model). In those cases, several roundoff error accumulators may be used (at least one for each discrete dimension), or a (n-1)-dimension vector (or matrix) of accumulators.

In some of these cases, the discrete dimensions of the data to sample and round may be treated non orthogonally: for example, when working with colored images, the trichromatic color planes data in each physical dimension (height, width and optionally time) could be remapped using a perceptive color model, so that the roundoff error accumulators will be designed to preserve lightness with a higher probability than hue or saturation, instead of propagating errors into each orthogonal color plane independently; and in stereophonic audio data the two rounded data channels (left and right) may be rounded together to preserve their mean value in priority to their effective difference that will absorb most of the remaining roundoff errors, in a balanced way around zero.

Rounding to simple fractions
In some contexts it is desirable to round a given number x to a "neat" fraction — that is, the nearest fraction z = m/n
whose numerator m and denominator n do not exceed a given maximum. This problem is fairly distinct from that of rounding a value to a fixed number of decimal or binary digits, or to a multiple of a given unit m. This problem is related to Farey sequences, the Stern-Brocot tree, and continued fractions.

Scaled rounding
This type of rounding, which is also named rounding to a logarithmic scale, is a variant of Rounding to a specified increment but with an increment that is modified depending on the scale and magnitude of the result. Concretely, the intent is to limit the number of significant digits, rounding the value so that non-significant digits will be dropped. This type of rounding occurs implicitly with numbers computed with floating-point values with limited precision (such as IEEE-754 float and double types), but it may be used more generally to round any real values with any positive number of significant digits and any strictly positive real base.

For example it can be used in engineering graphics for representing data with a logarithmic scale with variable steps (for example wave lengths, whose base is not necessarily an integer measure), or in statistical data to define classes of real values within intervals of exponentially growing widths (but the most common use is with integer bases such as 10 or 2).

This type of rounding is based on a logarithmic scale defined by a fixed non-zero real scaling factor s (in most frequent cases this factor is s=1) and a fixed positive base b>1 (not necessarily an integer and most often different from the scaling factor), and a fixed integer number n>0 of significant digits in that base (which will determine the value of the increment to use for rounding, along with the computed effective scale of the rounded number).

The primary argument number (as well as the resulting rounded number) is first represented in exponential notation x = s·a·m·bc, such that the sign s is either +1 or −1, the absolute mantissa a is restricted to the half-open positive interval [1/b,1), and the exponent c is any (positive or negative) integer. In that representation, all significant digits are in the fractional part of the absolute mantissa whose integer part is always zero.

If the source number (or rounded number) is 0, the absolute mantisssa a is defined as 0, the exponent c is fixed to an arbitrary value (0 in most conventions, but some floating-point representations cannot use a null absolute mantissa but reserve a specific maximum negative value for the exponent c to represent the number 0 itself), and the sign s may be arbitrarily chosen between −1 or +1 (it is generally set to +1 for simple zero, or it is set to the same sign as the argument in the rounded value if the number representation allows to differentiate positive and negative zeroes, even if they finally represent the same numeric value 0).

A scaled exponential representation as x = a·s·bc may also be used equivalently, with a signed mantissa a either equal to zero or within one of the two half-open intervals (−1,−1/b] and [+1/b,+1), and this will be the case in the algorithm below.

The steps to compute this scaled rounding are generally similar to the following:
 if x equals zero, simply return x; otherwise:
 convert x into the scaled exponential representation, with a signed mantissa:
 let x' be the unscaled value of x, by dividing it by the scaling factor s: ;
 let the scaling exponent c be one plus the base-b logarithm of the absolute value of x', rounded down to an integer (towards minus infinity): ;
 let the signed mantissa a be product of x' divided by b to the power c:
 compute the rounded value in this representation:
 let c' be the initial scaling exponent c of x':
 let m be the increment for rounding the mantissa a according to the number of significant digits to keep:
 let a' be the signed mantissa a rounded according to this increment m and the selected rounding mode:
 if the absolute value of a' is not lower than b, then decrement n (multiply the increment m by b), increment the scaling exponent c', divide the signed mantissa a by b, and restart the rounding of the new signed mantissa a into a' with the same formula; this step may be avoided only if the abtract "round()" function is always rounding a towards 0 (i.e. when it is a simple truncation), but is necessary if it may be rounding a towards infinity, because the rounded mantissa may have a higher scaling exponent in this case, leaving an extra digit of precision.
 return the rounded value: .

For the abstract "round()" function, this type of rounding can use any one of the rounding to integer modes described more completely in the next section, but it is most frequently the round to nearest mode (with tie-breaking rules also described more completely below).

For example:
 the scaled rounding of 1.234 with scaling factor 1 in base 10 and 3 significant digits (maximum relative precision=1/1000), when using any round to nearest mode, will return 1.23;
 similar scaled rounding of 1.236 will return 1.24;
 similar scaled rounding of 21.236 will return 21.2;
 similar scaled rounding of 321.236 will return 321;
 the scaled rounding of 1.234 scaling factor 1 in base 10 and 3 significant digits (maximum relative precision=1/1000), when using the round down mode, will return 1.23;
 similar scaled rounding of 1.236 will also return 1.23;
 the scaled rounding of  with scaling factor  in base 2 and 3 significant digits (maximum relative precision=1/8), when using the round down mode, will return ;
 similar scaled rounding of  will return ;
 similar scaled rounding of  will return .
 similar scaled rounding of  will also return .
 similar scaled rounding of  will return .

Round to available value

Finished lumber, writing paper, capacitors, and many other products are usually sold in only a few standard sizes.

Many design procedures describe how to calculate an approximate value, and then "round" to some standard size using phrases such as "round down to nearest standard value", "round up to nearest standard value", or "round to nearest standard value".

When a set of preferred values is equally spaced on a logarithmic scale,
Choosing the closest preferred value to any given value can be seen as a kind of scaled rounding.
Such "rounded" values can be directly calculated.

Floating-point rounding
In floating-point arithmetic, rounding aims to turn a given value x into a value z with a specified number of significant digits. In other words, z should be a multiple of a number m that depends on the magnitude of z. The number m is a power of the base (usually 2 or 10) of the floating-point form.

Apart from this detail, all the variants of rounding discussed above apply to the rounding of floating-point numbers as well. The algorithm for such rounding is presented in the Scaled rounding section above, but with a constant scaling factor s=1, and an integer base b>1.

For results where the rounded result would overflow the result for a directed rounding is either the appropriate signed infinity, or the highest representable positive finite number (or the lowest representable negative finite number if x is negative), depending on the direction of rounding. The result of an overflow for the usual case of round to even is always the appropriate infinity.

In addition, if the rounded result would underflow, i.e. if the exponent would exceed the lowest representable integer value, the effective result may be either zero (possibly signed if the representation can maintain a distinction of signs for zeroes), or the smallest representable positive finite number (or the highest representable negative finite number if x is negative), possibly a denormal positive or negative number (if the mantissa is storing all its significant digits, in which case the most significant digit may still be stored in a lower position by setting the highest stored digits to zero, and this stored mantissa does not drop the most significant digit, something that is possible when base b=2 because the most significant digit is always 1 in that base), depending on the direction of rounding. The result of an underflow for the usual case of round to even is always the appropriate zero.

Double rounding

Rounding a number twice in succession to different precisions, with the latter precision being coarser, is not guaranteed to give the same result as rounding once to the final precision except in the case of directed rounding. For instance rounding 9.46 to one decimal gives 9.5, and then 10 when rounding to integer using rounding half to even, but would give 9 when rounded to integer directly.

Some computer languages and the IEEE 754-2008 standard dictate that in straightforward calculations, the result should not be rounded twice. This has been a particular problem with Java as it is designed to be run identically on different machines, special programming tricks have had to be used to achieve this with x87 floating point.
The Java language was changed to allow different results where the difference does not matter and require a "strictfp" qualifier to be used when the results have to conform accurately.

Exact computation with rounded arithmetic
It is possible to use rounded arithmetic to evaluate the exact value of a function with a discrete domain and range. For example, if we know that an integer n is a perfect square, we can compute its square root by converting n to a floating-point value x, computing the approximate square root y of x with floating point, and then rounding y to the nearest integer q. If n is not too big, the floating-point roundoff error in y will be less than 0.5, so the rounded value q will be the exact square root of n. In most modern computers, this method may be much faster than computing the square root of n by an all-integer algorithm.

The table-maker's dilemma

William Kahan coined the term "The Table-Maker's Dilemma" for the unknown cost of rounding transcendental functions:
 "Nobody knows how much it would cost to compute  y^w  correctly rounded for every two floating-point arguments at which it does not over/underflow. Instead, reputable math libraries compute elementary transcendental functions mostly within slightly more than half an ulp and almost always well within one ulp. Why can't Y^W be rounded within half an ulp like SQRT? Because nobody knows how much computation it would cost... No general way exists to predict how many extra digits will have to be carried to compute a transcendental expression and round it correctly to some preassigned number of digits. Even the fact (if true) that a finite number of extra digits will ultimately suffice may be a deep theorem."

The IEEE floating point standard guarantees that add, subtract, multiply, divide, square root, and floating point remainder will give the correctly rounded result of the infinite precision operation. However, no such guarantee is given for more complex functions and they are typically only accurate to within the last bit at best.

Using the Gelfond–Schneider theorem and Lindemann–Weierstrass theorem, many of the standard elementary functions can be proved to return transcendental results when given rational non-zero arguments; therefore it is always possible to correctly round such functions. However determining a limit for a given precision on how accurate results needs to be computed before a correctly rounded result can be guaranteed may demand a lot of computation time.

There are some packages around now that offer full accuracy. The MPFR package gives correctly rounded arbitrary precision results. IBM has written a package for fast and accurate IEEE elementary functions and in the future the standard libraries may offer such precision.

It is possible to devise well-defined computable numbers which it may never be possible to correctly round no matter how many digits are calculated. For instance, if Goldbach's conjecture is true but unprovable, then it is impossible to correctly round down 0.5 + 10-n where n is the first even number greater than 4 which is not the sum of two primes, or 0.5 if there is no such number. This can however be approximated to any given precision even if the conjecture is unprovable.

History
The concept of rounding is very old, perhaps older even than the concept of division. Some ancient clay tablets found in Mesopotamia contain tables with rounded values of reciprocals and square roots in base 60.
Rounded approximations to π, the length of the year, and the length of the month are also ancient.

The Round-to-even method has served as the ASTM (E-29) standard since 1940. The origin of the terms unbiased rounding and statistician's rounding are fairly self-explanatory. In the 1906 4th edition of Probability and Theory of Errors  Robert Simpson Woodward called this "the computer's rule" indicating that it was then in common use by human computers who calculated mathematical tables. Churchill Eisenhart's 1947 paper "Effects of Rounding or Grouping Data" (in Selected Techniques of Statistical Analysis, McGrawHill, 1947, Eisenhart, Hastay, and Wallis, editors) indicated that the practice was already "well established" in data analysis.

The origin of the term "bankers' rounding" remains more obscure. If this rounding method was ever a standard in banking, the evidence has proved extremely difficult to find. To the contrary, section 2 of the European Commission report The Introduction of the Euro and the Rounding of Currency Amounts  suggests that there had previously been no standard approach to rounding in banking; and it specifies that "half-way" amounts should be rounded up.

Until the 1980s, the rounding method used in floating-point computer arithmetic was usually fixed by the hardware, poorly documented, inconsistent, and different for each brand and model of computer. This situation changed after the IEEE 754 floating point standard was adopted by most computer manufacturers. The standard allows the user to choose among several rounding modes, and in each case, specifies precisely how the results should be rounded. These features made numerical computations more predictable and machine-independent, and made possible the efficient and consistent implementation of interval arithmetic.

Rounding functions in programming languages
Most programming languages provide functions or special syntax to round fractional numbers in various ways. The earliest numeric languages, such as FORTRAN and C, would provide only one method, usually truncation (towards zero). This default method could be implied in certain contexts, such as when assigning a fractional number to an integer variable, or using a fractional number as an index of an array. Other kinds of rounding had to be programmed explicitly; for example, rounding a positive number to the nearest integer could be implemented by adding 0.5 and truncating.

In the last decades, however, the syntax and/or the standard libraries of most languages have commonly provided at least the four basic rounding functions (up/ceiling, down/floor, to nearest, and towards zero). The tie-breaking method may vary depending the language and version, and/or may be selectable by the programmer. Several languages follow the lead of the IEEE-754 floating-point standard, and define these functions as taking a double precision float argument and returning the result of the same type, which then may be converted to an integer if necessary.  Since the IEEE double precision format has 52 fraction bits, this approach may avoid spurious overflows in languages have 32-bit integers. Some languages, such as PHP, provide functions that round a value to a specified number of decimal digits, e.g. from 4321.5678 to 4321.57 or 4300. In addition, many languages provide a "printf" or similar string formatting function, which allows one to convert a fractional number to a string, rounded to a user-specified number of decimal places (the precision). On the other hand, truncation (round to zero) is still the default rounding method used by many languages, especially for the division of two integer values.

On the opposite, CSS and SVG do not define any specific maximum precision for numbers and measurements, that are treated and exposed in their Document Object Model and in their Interface-description-language interface as strings as if they had infinite precision, and do not discriminate between integers and floating point values; however, the implementations of these languages will typically convert these numbers into IEEE-754 double floating points before exposing the computed digits with a limited precision (notably within standard Javascript or ECMAScript interface bindings).

Other rounding standards

Some disciplines or institutions have issued standards or directives for rounding.

U.S. Weather Observations

In a guideline issued in mid-1966, the U.S. Office of the Federal Coordinator for Meteorology determined that weather data should be rounded to the nearest  round number, with the "round half up" tie-breaking rule. For example, 1.5 rounded to integer should become 2, and −1.5 should become −1. Prior to that date, the tie-breaking rule was "round half away from zero".

Negative zero in meteorology
Some meteorologists may write "-0" to indicate a temperature between 0.0 and -0.5 degrees (exclusive) that was rounded to integer. This notation is used when the negative sign is considered important, no matter how small is the magnitude; for example, when rounding temperatures in the Celsius scale, where below zero indicates freezing.

Related pages
 Arithmetic precision

Other websites

An introduction to different rounding algorithms  that is accessible to a general audience but especially useful to those studying computer science and electronics.
 How To Implement Custom Rounding Procedures by Microsoft

References

Mathematics

Rounding


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