title: William Hague

text: Hague contested Wentworth unsuccessfully in 1987, before being elected to Parliament at a by-election in 1989 as Member for the safe Conservative seat of Richmond, North Yorkshire, where he succeeded former Home Secretary Leon Brittan. Following his election he became the then-youngest Conservative MP and despite having only recently become an MP, Hague was invited to join the Government in 1990, serving as Parliamentary Private Secretary to the Chancellor of the Exchequer, Norman Lamont. After Lamont was sacked in 1993, Hague moved to the Department of Social Security (DSS) where he was Parliamentary Under-Secretary of State. The following year he was promoted as Minister of State in the DSS with responsibility for Social Security and Disabled People. His fast rise up through Government ranks was attributed to his intelligence and debating skills.


title: William Hague

text: Hague was appointed a Cabinet Minister in 1995 as Secretary of State for Wales; succeeding John Redwood, who had been castigated for being seen on TV apparently miming the Welsh national anthem at a conference; thus, Hague sought a Welsh Office civil servant, Ffion Jenkins, to teach him the words; they later married. He continued serving in Cabinet until the Conservatives were replaced by Labour at the 1997 general election.


title: William Hague

text: Following the 1997 general election defeat, Hague was elected Leader of the Conservative Party in succession to John Major, defeating more experienced figures such as Kenneth Clarke and Michael Howard.


title: William Hague

text: At the age of 36, Hague was tasked with rebuilding the Conservative Party (fresh from their worst general election result of the 20th century) by attempting to build a more modern image. £250,000 was spent on the "Listening to Britain" campaign to try to put the Conservatives back in touch with the public after losing power; he welcomed ideas about "compassionate conservatism" including from the then-Governor of Texas, later President George W. Bush.


title: William Hague

text: When he visited a theme park with his Chief of Staff and former local MP, Sebastian (now Lord) Coe, Hague took a ride on a log flume wearing a baseball cap emblazoned 'HAGUE'; Cecil Parkinson described the exercise as "juvenile".


title: William Hague

text: Hague steered the Conservatives to a successful result at the European parliamentary elections in June 1999, where the Conservatives gained 36 MEPs ahead of Labour's 29. Hague considered his opposition to the single European currency (the Euro) to be later vindicated by Labour Prime Minister Gordon Brown's adoption and subsequent approval of the policy.


title: William Hague

text: Hague's authority was challenged by the appointment of Michael Portillo as Shadow Chancellor in 2000. Portillo had been widely tipped to be the next Conservative Party Leader before dramatically losing his seat at the 1997 general election; he was elected as MP for Kensington and Chelsea at a by-election two years later. Soon after Portillo's return to Parliament, Conservative policy on two of Labour's flagship policies was reversed: the minimum wage and independence of the Bank of England. From then and until the 2001 general election Hague's supporters waged an increasingly bitter battle with Portillo's faction; such internecine war significantly contributed to the Conservatives' two subsequent election defeats.


title: William Hague

text: Hague was widely ridiculed for claiming he used to drink "14 pints of beer a day" as a teenager. His reputation suffered further damage when a 2001 poll for "The Daily Telegraph" found that 66% of voters considered him to be "a bit of a wally", and 70% of voters believed he would "say almost anything to win votes".


title: William Hague

text: Former Conservative Deputy Prime Minister Michael Heseltine, a prominent One-nation Conservative, was critical of Hague's Eurosceptic view that Britain was becoming a "foreign land", betraying in newspaper interviews that he was uncertain as to whether he could support a Hague-led Conservative Party.


title: William Hague

text: Hague's critics assiduously monitored his performance at Prime Minister's Questions each Wednesday in Parliament, having difficulty to find fault. During one particular exchange, while responding to the Queen's Speech of 2000, Hague attacked the Prime Minister's record:


title: William Hague

text: On the morning of Labour's second consecutive landslide victory at the 2001 general election, Hague stated: "we have not been able to persuade a majority, or anything approaching a majority, that we are yet the alternative government that they need." At that election the Conservative Party gained just one parliamentary seat more than at the 1997 general election; following this defeat, Hague resigned as party leader. Hague thus became the second twentieth century Conservative party leader not to become Prime Minister (after Austen Chamberlain) and the first ever to spend his entire tenure in Opposition.


title: William Hague

text: On the backbenches he occasionally spoke in the House of Commons on issues of the day. Between 1997 and 2002, he was the Chairman of the International Democrat Union. Hague's profile and personal popularity rose thereafter among both Conservative Party members and the wider public following his spell as Party Leader. He has written a biography of 18th-century Prime Minister Pitt the Younger (published in 2004), taught himself how to play the piano, and hosted the 25th anniversary programme for Radio 4 on the political television satire "Yes Minister" in 2005. In June 2007 he published his second book, a biography of the anti-slave trade campaigner William Wilberforce, shortlisted for the 2008 Orwell Prize for political writing.


title: William Hague

text: Hague's annual income was the highest in Parliament, with earnings of about £400,000 a year from directorships, consultancy, speeches and his parliamentary salary. His income was previously estimated at £1 million annually, but he dropped several commitments and in effect took a salary cut of some £600,000 on becoming Shadow Foreign Secretary in 2005.


title: William Hague

text: Together with former Prime Minister John Major, former Chancellor Kenneth Clarke, and Hague's successor Iain Duncan Smith, Hague served for a time on the Conservative Leadership Council, which was set up by Michael Howard upon his election unopposed as Leader of the Conservative Party in 2003.


title: William Hague

text: Following the 2005 general election, the Conservative Party Leader Michael Howard apparently offered Hague the post of Shadow Chancellor of the Exchequer, which he turned down citing that his business commitments would make it difficult for him to take on such a high-profile job.


title: William Hague

text: On 6 December 2005, David Cameron was elected Leader of the Conservative Party. Hague was offered and accepted the role of Shadow Foreign Secretary and Senior Member of the Shadow cabinet, effectively serving as Cameron's deputy (though not formally, unlike previous Deputy Conservative Leaders Willie Whitelaw, Peter Lilley and Michael Ancram). He had been widely tipped to return to the frontbench under either Cameron or leadership contest runner-up David Davis.


title: William Hague

text: On 30 January 2006, by Cameron's instructions, Hague travelled to Brussels for talks to pull Conservative Party MEPs out of the European People's Party–European Democrats Group (EPP-ED) in the European Parliament. ("The Daily Telegraph", 30 January 2006). Further, on 15 February 2006, Hague deputed, during David Cameron's paternity leave, at Prime Minister's Questions (PMQs). This appearance gave rise to jokes at the expense of Blair, that all three parties that day were being led by 'stand-ins', with the Liberal Democrats represented by Acting Leader Sir Menzies Campbell, the Labour Party by the departing Blair, and the Conservatives by Hague. Hague again deputised for Cameron for several sessions in 2006.


title: William Hague

text: Prime Minister Cameron's first appointment was Hague as Foreign Secretary. He was also accorded the honorary title of First Secretary of State. In his first overseas visit as British Foreign Secretary, Hague met US Secretary of State, Hillary Clinton, at Washington


title: William Hague

text: In August 2010, Hague set out a values-based foreign policy, stating that: "We cannot have a foreign policy without a conscience. Foreign policy is domestic policy written large. The values we live by at home do not stop at our shores. Human rights are not the only issue that informs the making of foreign policy, but they are indivisible from it, not least because the consequences of foreign policy failure are human".


title: William Hague

text: Hague further said that: "There will be no downgrading of human rights under this Government and no resiling from our commitments to aid and development". He continued saying that "Indeed I intend to improve and strengthen our human rights work. It is not in our character as a nation to have a foreign policy without a conscience, and neither is it in our interests". However, in March 2011, Hague was criticised by Cardinal Keith O'Brien for increasing financial aid to Pakistan despite persecution of its Christian minority: "To increase aid to the Pakistan Government when religious freedom is not upheld and those who speak up for religious freedom are gunned down is tantamount to an anti-Christian foreign policy".


title: William Hague

text: In September 2011, Hague told BBC Radio 4's "File on 4" investigation "Cyber Spies" into the legality of domestic cyber surveillance and the export of this technology from the UK to countries with questionable human rights records that the UK had a strong export licence system. The programme also obtained confirmation from the UK's Department for Business Innovation and Skills that cyber surveillance products that break, as opposed to create, encryption do not require export licences.


title: William Hague

text: In June 2012, he continued to stand in for David Cameron at PMQs when both the Prime Minister and Deputy Prime Minister Nick Clegg were out of the country.


title: William Hague

text: In January 2013, Hague visited New Zealand in his capacity as Foreign Secretary, holding talks with New Zealand government ministers, Murray McCully and David Shearer. In March 2013, Hague established the International Leaders Programme, designed to identify and develop partnerships among future global leaders.


title: William Hague

text: In early September 2010, newspapers including "The Daily Telegraph", "The Independent" and "Daily Mail" released stories about allegations surrounding Hague's friendship with 25-year-old Christopher Myers, a history graduate from Durham University, whom he employed as a parliamentary special adviser. A spokesperson stated that "Any suggestion that the Foreign Secretary's relationship with Chris Myers is anything other than a purely professional one is wholly inaccurate and unfounded."


title: William Hague

text: On 1 September 2010, Myers resigned from his appointment in light of that press speculation, which prompted Hague to issue a public statement, wherein he confirmed that he had "occasionally" shared a hotel room with Myers [for reasons of frugality by upbringing], but refuting the "utterly false" suggestions that he had ever been involved in a relationship with man. A spokesperson for Prime Minister David Cameron reported that he gave his "full support" over the media rumours. Figures from both within and without the Conservative Party criticised Hague for his personal response to the stories, with former Conservative leadership candidate, John Redwood, commenting that Hague had shown "poor judgement", and the Speaker's wife, Labour-supporting Sally Bercow, speculating that Hague had been given "duff PR advice", whilst a parliamentary and ministerial colleague, the Conservative MP, Alan Duncan, described the media coverage as "contemptible".


title: William Hague

text: Hague was criticised by Israeli leaders after meeting with Palestinians who demonstrated against Israel's barrier in the West Bank. He expressed solidarity with the idea of non-violence and listened to the accounts of left-wing and Palestinian activists. Israeli Opposition Leader Tzipi Livni condemned the statements and said:


title: William Hague

text: The security barrier has saved lives, and its construction was necessary. The barrier has separated Israel from Palestinian cities and completely changed the reality in Israel, where citizens were exposed to terror every day.


title: William Hague

text: In February 2011 security forces in the Bahrain dispersed thousands of anti-government protesters at Pearl Square in the centre of the capital, Manama. Hague informed the House of Commons that he had stressed the need for peaceful action in dealing with the protesters: "At least three people died in the operation, with hundreds more injured. We are greatly concerned about the deaths that have occurred. I have this morning spoken to the Foreign Minister of Bahrain and HM Ambassador spoke last night to the Bahraini Minister of the Interior. In both cases we stressed the need for peaceful action to address the concerns of protesters, the importance of respect for the right to peaceful protest and for freedom of expression".


title: William Hague

text: Hague told "Sky News" that the use of force by the Libyan authorities during the 2011 Libyan Civil War was "dreadful and horrifying" and called on the leader to respect people's human rights. A vicious crackdown led by special forces, foreign mercenaries and Muammar Gaddafi loyalists was launched in the country's second city Benghazi, which has been the focus of anti-regime protests. Hague stated to Dermot Murnaghan on "Sky": "I think we have to increase the international pressure and condemnation. The United Kingdom condemns what the Libyan Government has been doing and how they have responded to these protests, and we look to other countries to do the same".


title: William Hague

text: Following delays in extracting British citizens from Libya, a disastrous helicopter attempt to contact the protesters ending with eight British diplomats/SAS arrested and no aircraft carriers or Harriers to enforce a no-fly zone he was accused, by the Labour Opposition, of "losing his mojo" in March 2011.


title: William Hague

text: In March 2011, Hague said in a speech to business leaders that the examples being set in north Africa and the Middle East will ultimately transform the relationship between governments and their populations in the region. However following the row over whether Libyan leader Muammar Gaddafi was being targeted by coalition forces, the Foreign Secretary stated that the Libyan people must be free to determine their own future. Hague said: "It is not for us to choose the government of Libya—that is for the Libyan people themselves. But they have a far greater chance of making that choice now than they did on Saturday, when the opposition forces were on the verge of defeat."


title: William Hague

text: Hague has warned that autocratic leaders including Robert Mugabe, President of Zimbabwe, could be shaken and even toppled by a wave of popular uprisings rippling out from north Africa. said that recent revolts against authoritarian leaders in countries including Libya and Egypt will have a greater historic significance than the 9/11 attacks on the US or the recent financial crisis. He stopped short of threatening military intervention against other dictators, but warned that they will inevitably face "judgement" for oppressing their people and suppressing democracy. Repressive African régimes will also face challenges from their populations and from the international community, the Foreign Secretary said: "Demands for freedom will spread, and that undemocratic governments elsewhere should take heed." He added: "Governments that use violence to stop democratic development will not earn themselves respite forever. They will pay an increasingly high price for actions which they can no longer hide from the world with ease, and will find themselves on the wrong side of history."


title: William Hague

text: Hague, on his way to Qatar Summit in April 2011, called for intensified sanctions on the Libyan régime and for a clear statement that Gaddafi must go: "we have sent more ground strike aircraft in order to protect civilians. We do look to other countries to do the same, if necessary, over time". "We would like a continued increase in our (NATO's) capability to protect civilians in Libya", he added. Whether NATO ratcheted up operations depended on what happened on the ground, Hague said. "These air strikes are a response to movements of, or attacks from, régime forces so what happens will be dependent on that", he said. Whether the Americans could again be asked to step up their role would also "depend on the circumstances", he added.


title: William Hague

text: Hague, speaking on the protests in Syria said "Political reforms should be brought forward and implemented without delay". It is thought as many as 60 people have been killed by security forces in the country today (22 April 2011), making it the worst day for deaths since protests against President Bashar al-Assad began over a month ago, reported BBC News.


title: William Hague

text: Speaking on the Syrian civil war in August 2011 Hague said of military intervention: "It's not a remote possibility. Even if we were in favour [of UN-backed military action], which we are not because there's no call from the Arab League for intervention as in the case of Libya, there is no prospect of a legal, morally sanctioned military intervention. Hague added that it was a "frustrating situation" and that the "levers" at the international community's disposal were severely limited but said countries had to concentrate on other ways of influencing the Assad government. "We want to see stronger international pressure all round. Of course, to be effective that just can't be pressure from Western nations, that includes from Arab nations... and it includes from Turkey who has been very active in trying to persuade President Assad to reform instead of embarking on these appalling actions", he said. "I would also like to see a United Nations Security Council resolution to condemn this violence, to call for the release of political prisoners, to call for legitimate grievances to be responded to", he added.


title: William Hague

text: During 2012 the UK started training Syrian opposition activists in Istanbul on media, civil society and local government matters, and supplying non-lethal equipment such as satellite communications and computers.


title: William Hague

text: On 24 February 2012, Hague recognised the Syrian National Council as a "legitimate representative" of the country. Hague also said Bashar al-Assad's government had "forfeited the right to lead" by "miring itself in the blood of innocent people". Hague said: "Today we must show that we will not abandon the Syrian people in their darkest hour". He added that "Those responsible for the murder of entire families, the shelling of homes, the execution of detainees, the cleansing of political opponents and the torture and rape of women and children must be held to account", he said.


title: William Hague

text: In March 2012, Hague ordered the evacuation of all British diplomats from Syria and closed the UK embassy in Damascus because of mounting security threats. Hague told Parliament: "We have maintained an embassy in Damascus despite the violence to help us communicate with all parties in Syria and to provide insight into the situation on the ground". He added: "We now judge that the deterioration of the security situation in Damascus puts our embassy staff and premises at risk." Hague said that his decision "in no way reduces the UK's commitment to active diplomacy to maintain pressure on the Assad régime to end the violence". He went on to say that: "We will continue to work closely with other nations to co-ordinate diplomatic and economic pressure on the Syrian régime."


title: William Hague

text: On 1 April 2012, Hague met 74 other nations at a Friends of Syria Group conference in Istanbul, Turkey. Hague said the issue could return to the United Nations Security Council if current efforts to resolve the crisis fail. The government of President Assad has said it accepts a peace plan by the UN-Arab League envoy Kofi Annan, but there has been little evidence that it is prepared to end its crackdown on the opposition. Hague accused Assad of "stalling for time" and warned that if the issue does return to the Security Council, he may no longer be able to rely on the backing of Russia and China, who blocked a previous resolution calling for him to stand down. "There isn't an unlimited period of time for this, for the Kofi Annan process to work before many of the nations here want us to go back to the UN Security Council—some of them will call for arming the opposition if there isn't progress made," Hague told the BBC. He added that "What is now being put to them is a plan from Kofi Annan supported by the whole United Nations Security Council, and this is an important point, it's supported by Russia and by China as well as by the more obvious countries—the United States, the United Kingdom, France, the Arab League and so on".


title: William Hague

text: On 20 November 2012, Hague recognised the National Coalition for Syrian Revolutionary and Opposition Forces as the "sole legitimate representative" of the Syrian people, and a credible alternative to the current Syrian Government.


title: William Hague

text: On 29 August 2013, the British Parliament refused to ratify the British Government's plan to participate in military strikes against the Syrian Government in the wake of a chemical-weapons attack at Ghouta. Hague denied suggestions that he had threatened to resign over Prime Minister David Cameron's decision to go straight to a parliamentary vote. After the vote, Hague continued to urge other governments to take action against the Syrian Government, saying "If it is decided in the various parliaments of the world that no-one will stand up to the use of chemical weapons and take any action about that, that would be a very alarming moment in the affairs of the world". Ultimately a negotiated agreement was reached to eliminate Syria's chemical weapons.


title: William Hague

text: In June 2011, Hague dismissed Tony Blair's vision for an elected-head of the European Union by insisting that member states have more pressing priorities than further "constitutional tinkering". Hague made clear his view after Blair argued that a directly elected President of Europe, representing almost 400m people from 27 countries, would give the EU clear leadership and enormous authority. In an interview with "The Times", Blair set out the agenda that he thought a directly elected EU President should pursue, although he conceded, there was "no chance" of such a post being created "at the present time". Asked about the former Prime Minister's call for further European integration and the creation of an elected-President, Hague suggested that Blair may have been thinking of the role for himself. "I can't think who he had in mind", Hague joked, further adding on a serious note: "Elected presidents are for countries. The EU is not a country and it's not going to become a country, in my view, now or ever in the future. It is a group of countries working together".


title: William Hague

text: In June 2011, Hague said that Britain helped initiate "distasteful" peace talks with the Taliban in Afghanistan. Hague made the comments while on a three-day tour of the country to meet President Hamid Karzai and visited British troops. He told "The Sun" newspaper that Britain had led the way in persuading US President Barack Obama's administration that negotiation was the best potential solution to the conflict. Hague admitted that any deal might mean accepting "distasteful things" and could anger military veterans and relatives of the 374 British troops killed in Afghanistan. However, he said he believed that Britain as a whole was "realistic and practical" enough to accept that ending fighting and starting talks was the best way to safeguard national security. He told the newspaper: "An eventual settlement of these issues is the ultimate and most desirable way of safeguarding that national security." He added, "but reconciliation with people who have been in a military conflict can be very distasteful. In all these types of situations, you do have to face up to some distasteful things." The previous night US President Barack Obama told Americans that "the tide of war is receding" as he announced plans to withdraw 33,000 US troops from Afghanistan by September 2012.


title: William Hague

text: In September 2011, Hague said that the Euro is "a burning building with no exits" for some of the countries which adopted the currency. Hague first used the expression when he was Conservative Leader in 1998—and said in an interview with "The Spectator" he had been proved right: "It was folly to create this system. It will be written about for centuries as a kind of historical monument to collective folly. But it's there and we have to deal with it," he said. "I described the Euro as a burning building with no exits and so it has proved for some of the countries in it," he further said, adding "I might take the analogy too far but the Euro wasn't built with exits so it is very difficult to leave it".


title: William Hague

text: In February 2012, Hague warned in a BBC interview about Iran's "increasing willingness to contemplate" terrorism around the world. He cited the 2011 Iran assassination plot, an attempt to assassinate Adel al-Jubeir, the Saudi Ambassador to the United States, as well as alleged involvement in recent attacks in New Delhi, Georgia, and Bangkok. He said it showed "the danger Iran is currently presenting to the peace of the world".


title: William Hague

text: Hague spoke the Commons on 20 February about the nuclear program of Iran and said that if the Tehran régime managed to construct a viable weapon, its neighbours would be forced to build their own nuclear warheads too. He accused Iranian President Mahmoud Ahmadinejad of pursuing "confrontational policies" and described the country's enrichment of uranium in defiance of United Nations Security Council resolutions as "a crisis coming steadily down the track". "Our policy is that whilst we remain unswervingly committed to diplomacy, it is important to emphasise to Iran that all options are on the table", Hague told MPs.


title: William Hague

text: In March he condemned the way parliamentary elections were staged, claiming they were not "free and fair". He said the poll had been held against a backdrop of fear that meant the result would not reflect the will of the people. Hague said: "It has been clear for some time that these elections would not be free and fair. "The régime has presented the vote as a test of loyalty, rather than an opportunity for people freely to choose their own representatives. The climate of fear, created by the régime's crushing of opposition voices since 2009, persists."


title: William Hague

text: The 30th anniversary of the beginning of the 1982 Falklands War was on 2 April 2012. On 29 March, before the Lord Mayor of London's banquet guests, namely the entire foreign diplomatic corps of more than 100 ambassadors, including Alicia Castro (Argentinian Ambassador), Hague said the UK was keen to deepen its relationship with Latin America—and reiterated Britain's commitment to the Falklands. He said: "We are reversing Britain's decline in Latin America, where we are opening a new Embassy in El Salvador. This determination to deepen our relations with Latin America is coupled with our steadfast commitment to the right of self-determination of the people of the Falkland Islands".


title: William Hague

text: Tensions over the Falklands had risen in the weeks prior to the anniversary. In February, Hague said deployments of a British warship, HMS "Dauntless" and the Duke of Cambridge to the Falklands were "entirely routine". Hague said that Britain affirmed the Falklanders' self-determination and would seek to prevent Argentina from "raising the diplomatic temperature" over the issue. He further said: "(the events) are not so much celebrations as commemorations. I think Argentina will also be holding commemorations of those who died in the conflict. Since both countries will be doing that I don't think there is anything provocative about that."


title: William Hague

text: Hague set out Her Majesty's Government's plans, on 12 June 2012, for the reintroduction of self-government in the Turks and Caicos Islands, where direct rule of the Governor had been in place since the islands had been subject to corruption and maladministration under the previous autonomous administration.


title: William Hague

text: In August 2012, Hague declared that Julian Assange, the WikiLeaks organisation founder, would not be granted political asylum by the United Kingdom. Hague declared the UK's willingness to extradite Assange to the Swedish authorities who had requested his extradition; thus Swedish prosecutors, unwilling to break diplomatic protocol, have deferred from interrogating Assange at the Embassy of Ecuador, London.


title: William Hague

text: Hague confirmed the British Government's position – that it is lawfully obliged to extradite Julian Assange. "We're disappointed by the statement by Ecuador's Foreign Minister today that Ecuador has offered political asylum to Julian Assange. Under our Laws, with Mr. Assange having exhausted all options of appeal, the British authorities are under a binding obligation to extradite him to Sweden. We must carry out that obligation and of course we fully intend to do so," Hague confirmed.


title: William Hague

text: Following "The Guardian" newspaper outcry over a Foreign Office note sanctioned by Hague sent to the Ecuadorian Embassy—in which it raised the possibility of the revocation of their diplomatic status under the Diplomatic and Consular Premises Act 1987—the Foreign Secretary reaffirmed the UK remained "committed to a diplomatic solution" and played down any suggestion of a police raid of the Ecuadorian Embassy, stating "there is no threat here to storm an embassy".


title: William Hague

text: The former ambassador to Uzbekistan, Craig Murray, warned that using the 1987 Act to raid the Ecuadorian Embassy would be in "breach of the Vienna Convention of 1961". Russia warned Britain against violating fundamental diplomatic principles (Vienna Convention on Diplomatic Relations, and in particular the Article 22 spelling out the inviolability of diplomatic premises), which the Government of Ecuador invoked.


title: William Hague

text: Once Hague had formally declared his intention not to seek re-election as MP for Richmond at the forthcoming 2015 general election, he told David Cameron he would be standing down as Foreign Secretary. Cameron instigated a Cabinet reshuffle whereby Hague became Leader of the House of Commons. Hague remained as Cameron's ""de facto" political deputy", retained his membership of the National Security Council and played a lead role in reaching out to voters in the North of England in the run up to the general election.


title: William Hague

text: In a surprise motion on his last day in the House of Commons, Hague moved to make the election for Speaker in the next parliament a secret ballot, in what was seen as an effort to oust the incumbent John Bercow for lacking the neutrality expected of a Speaker of the House. Charles Walker, Conservative MP for Broxbourne, Chairman of the Procedure Committee and responsible for Speaker elections, stated that he had written a report about such an idea "years ago" and despite speaking with Hague and Michael Gove earlier that week, neither had told him of any such move. A visibly emotional Walker told the House, "I have been played as a fool. When I go home tonight, I will look in the mirror and see an honourable fool looking back at me. I would much rather be an honourable fool, in this and any other matter, than a clever man." Walker received a standing ovation, mainly from the Labour benches, whilst the Government lost its parliamentary motion by 228 to 202 votes. During the debate the future Father of the House, Gerald Kaufman, denounced Hague, saying: "Is the right hon. Gentleman aware that this grubby decision is what he personally will be remembered for? After a distinguished career in the House of Commons, both as a leader of a party and as a senior Cabinet Minister, he has now descended to squalor in the final days of the Parliament."


title: William Hague

text: He was succeeded as MP for Richmond (Yorks) by future Chancellor of the Exchequer and future Prime Minister Rishi Sunak.


title: William Hague

text: On 9 October 2015, Hague was created Baron Hague of Richmond, "of Richmond in the County of North Yorkshire".


title: William Hague

text: In August 2020, Hague endorsed Joe Biden for U.S. president over incumbent Donald Trump, arguing that a Biden victory was in the UK's interest.


title: William Hague

text: Hague and the then Duke of Cambridge identified, while the former was in post as Foreign Secretary, that the illegal wildlife trade (IWT) was among the most profitable criminal enterprises in the world, and in order to combat it formed in 2014 the Transport Task Force (TTF). The TTF seeks to identify and stop wildlife trafficking. They continue to work at this in 2020. The Financial Task Force was created in 2018 to help further the goal.


title: William Hague

text: In September 2020, Hague was appointed as chairman of the Royal Foundation, a charitable organisation operating under the auspices of the then Duke and Duchess of Cambridge, in succession to Sir Keith Mills who retired.


title: William Hague

text: Hague is an author of political biographies, and since his retirement from public life he has maintained a weekly column in first the "Daily Telegraph" and subsequently "The Times". Hague also writes the occasional book review and appears on TV shows and in radio presentations.


title: William Hague

text: Hague has been particularly sharp on the COVID–19 pandemic, writing as early as 10 February 2020 that "Coronavirus is a calamity for China. It cannot continue its dangerous wildlife practices any longer." Hague wrote on 2 March that: "The rise of coronavirus is a clear indication that the degrading of nature will come back to hit humans very hard." Hague returned to the subject on 13 April, when he said that the "world must act now on wildlife markets or run the risk of worse pandemics in future".


title: William Hague

text: Hague married Ffion Jenkins at the Chapel of St Mary Undercroft on 19 December 1997. Ffion Hague is now styled The Lady Hague of Richmond.


title: William Hague

text: Hague serves as a Vice-President of the Friends of the British Library, which provides funding support for the British Library to make new acquisitions. He is a Patron of the European Youth Parliament UK, an educational charity organisation that runs debating competitions and discussion forums across the UK and is President of the Britain–Australia Society. Hague practises judo, and has a keen interest in music, learning to play the piano, shortly after the 2001 general election. He is an enthusiast for the natural history and countryside of his native Yorkshire.


title: Knowing Bros

text: Knowing Bros (), also known as Men on a Mission or Ask Us Anything, is a South Korean television entertainment program produced by SM C&C that premiered on December 5, 2015, and currently airs on Saturdays at JTBC. Although the show has taken on different formats in some episodes, the program features a high school classroom setting with the fixed cast - the titular "brothers" - acting as same-aged students. Guests arrive posing as visiting students from another school and are interviewed by the cast in a talk show segment; the program's second half usually features a variety segment, with the cast and guests taking of activities such as games, talent show, or improvisational acting.


title: Knowing Bros

text: "Knowing Bros"' current fixed cast includes Kang Ho-dong, Lee Soo-geun, Kim Young-chul, Seo Jang-hoon, Kim Heechul, Min Kyunghoon, Lee Sangmin, and Lee Jin-ho; Hwang Chi-yeul and Kim Se-hwang were initial cast members in the program's first few episodes. Jang Sung-kyu and Shindong have appeared as recurring special guests by hosting the show's game segments. The show received several accolades during its run.


title: Knowing Bros

text: The current format (from episode 17) explores the high school concept where the cast members act like students in a classroom while guests come as newly transferred students. Most of the segments for this format portray activities that every student would do with their classmates. Subsequently, the program sometimes diversifies at the end with a segment outside of the school concept. This format has received praise from the viewers and led to a significant increase in ratings and popularity of the program.


title: Knowing Bros

text: A notable aspect of the show is the usage of the "banmal" (Hangul: 반말; "informal language") manner of speech used by everyone regardless of their age or seniority, thus disregarding the Korean language's strict seniority rules with spoken language and the "sunbae"–"hoobae" hierarchy in the Korean entertainment industry. The informal speech and casual setting also encourages the guests and cast to interact with each other as schoolmates often do, leading them to be comfortable enough to tease one another.


title: Knowing Bros

text: The episodes may also feature any of the following segments, usually within, in between, or following the above permanent segments:


title: Knowing Bros

text: The program's original cast consisted of Kang Ho-dong, Lee Soo-geun, Seo Jang-hoon, Kim Young-chul, Kim Hee-chul, Hwang Chi-yeul and Kim Se-hwang. Min Kyung-hoon was added to the cast formation on the second episode aired on December 12, 2015. Hwang Chi-yeul and Kim Se-hwang left the program after the seventh episode, which aired on January 16, 2016. Lee Sang-min joined the cast on March 5, 2016.


title: Knowing Bros

text: In December 2016, the program received a disciplinary action from South Korea's Broadcast Censorship Committee for using improper remarks, which included objectifying female guests on multiple occasions. In one episode, Min Kyung-hoon gives a handcrafted bra to a female guest (Jeon So-min). The committee also highlighted the use of improper language between cast members and the use of homophobic remarks. Besides a warning given by the committee, it is reported that the program has also received numerous complaints from viewers regarding the matter.


title: Analog-to-digital converter

text: In electronics, an analog-to-digital converter (ADC, A/D, or A-to-D) is a system that converts an analog signal, such as a sound picked up by a microphone or light entering a digital camera, into a digital signal. An ADC may also provide an isolated measurement such as an electronic device that converts an analog input voltage or current to a digital number representing the magnitude of the voltage or current. Typically the digital output is a two's complement binary number that is proportional to the input, but there are other possibilities.


title: Analog-to-digital converter

text: There are several ADC architectures. Due to the complexity and the need for precisely matched components, all but the most specialized ADCs are implemented as integrated circuits (ICs). These typically take the form of metal–oxide–semiconductor (MOS) mixed-signal integrated circuit chips that integrate both analog and digital circuits.


title: Analog-to-digital converter

text: A digital-to-analog converter (DAC) performs the reverse function; it converts a digital signal into an analog signal.


title: Analog-to-digital converter

text: An ADC converts a continuous-time and continuous-amplitude analog signal to a discrete-time and discrete-amplitude digital signal. The conversion involves quantization of the input, so it necessarily introduces a small amount of error or noise. Furthermore, instead of continuously performing the conversion, an ADC does the conversion periodically, sampling the input, limiting the allowable bandwidth of the input signal.


title: Analog-to-digital converter

text: The performance of an ADC is primarily characterized by its bandwidth and signal-to-noise ratio (SNR). The bandwidth of an ADC is characterized primarily by its sampling rate. The SNR of an ADC is influenced by many factors, including the resolution, linearity and accuracy (how well the quantization levels match the true analog signal), aliasing and jitter. The SNR of an ADC is often summarized in terms of its effective number of bits (ENOB), the number of bits of each measure it returns that are on average not noise. An ideal ADC has an ENOB equal to its resolution. ADCs are chosen to match the bandwidth and required SNR of the signal to be digitized. If an ADC operates at a sampling rate greater than twice the bandwidth of the signal, then per the Nyquist–Shannon sampling theorem, perfect reconstruction is possible. The presence of quantization error limits the SNR of even an ideal ADC. However, if the SNR of the ADC exceeds that of the input signal, its effects may be neglected resulting in an essentially perfect digital representation of the analog input signal.


title: Analog-to-digital converter

text: The resolution of the converter indicates the number of different, ie discrete, values it can produce over the allowed range of analog input values. Thus a particular resolution determines the magnitude of the quantization error and therefore determines the maximum possible signal-to-noise ratio for an ideal ADC without the use of oversampling. The input samples are usually stored electronically in binary form within the ADC, so the resolution is usually expressed as the audio bit depth. In consequence, the number of discrete values available is usually a power of two. For example, an ADC with a resolution of 8 bits can encode an analog input to one in 256 different levels (2 = 256). The values can represent the ranges from 0 to 255 (i.e. as unsigned integers) or from −128 to 127 (i.e. as signed integer), depending on the application.


title: Analog-to-digital converter

text: Resolution can also be defined electrically, and expressed in volts. The change in voltage required to guarantee a change in the output code level is called the least significant bit (LSB) voltage. The resolution "Q" of the ADC is equal to the LSB voltage. The voltage resolution of an ADC is equal to its overall voltage measurement range divided by the number of intervals:


title: Analog-to-digital converter

text: where "M" is the ADC's resolution in bits and "E" is the full scale voltage range (also called 'span'). "E" is given by


title: Analog-to-digital converter

text: In many cases, the useful resolution of a converter is limited by the signal-to-noise ratio (SNR) and other errors in the overall system expressed as an ENOB.


title: Analog-to-digital converter

text: Quantization error is introduced by the quantization inherent in an ideal ADC. It is a rounding error between the analog input voltage to the ADC and the output digitized value. The error is nonlinear and signal-dependent. In an ideal ADC, where the quantization error is uniformly distributed between −1/2 LSB and +1/2 LSB, and the signal has a uniform distribution covering all quantization levels, the signal-to-quantization-noise ratio (SQNR) is given by


title: Analog-to-digital converter

text: where Q is the number of quantization bits. For example, for a 16-bit ADC, the quantization error is 96.3 dB below the maximum level.


title: Analog-to-digital converter

text: Quantization error is distributed from DC to the Nyquist frequency. Consequently, if part of the ADC's bandwidth is not used, as is the case with oversampling, some of the quantization error will occur out-of-band, effectively improving the SQNR for the bandwidth in use. In an oversampled system, noise shaping can be used to further increase SQNR by forcing more quantization error out of band.


title: Analog-to-digital converter

text: In ADCs, performance can usually be improved using dither. This is a very small amount of random noise (e.g. white noise), which is added to the input before conversion. Its effect is to randomize the state of the LSB based on the signal. Rather than the signal simply getting cut off altogether at low levels, it extends the effective range of signals that the ADC can convert, at the expense of a slight increase in noise. Note that dither can only increase the resolution of a sampler. It cannot improve the linearity, and thus accuracy does not necessarily improve.


title: Analog-to-digital converter

text: Quantization distortion in an audio signal of very low level with respect to the bit depth of the ADC is correlated with the signal and sounds distorted and unpleasant. With dithering, the distortion is transformed into noise. The undistorted signal may be recovered accurately by averaging over time. Dithering is also used in integrating systems such as electricity meters. Since the values are added together, the dithering produces results that are more exact than the LSB of the analog-to-digital converter.


title: Analog-to-digital converter

text: Dither is often applied when quantizing photographic images to a fewer number of bits per pixel—the image becomes noisier but to the eye looks far more realistic than the quantized image, which otherwise becomes banded. This analogous process may help to visualize the effect of dither on an analog audio signal that is converted to digital.


title: Analog-to-digital converter

text: An ADC has several sources of errors. Quantization error and (assuming the ADC is intended to be linear) non-linearity are intrinsic to any analog-to-digital conversion. These errors are measured in a unit called the least significant bit (LSB). In the above example of an eight-bit ADC, an error of one LSB is 1/256 of the full signal range, or about 0.4%.


title: Analog-to-digital converter

text: All ADCs suffer from nonlinearity errors caused by their physical imperfections, causing their output to deviate from a linear function (or some other function, in the case of a deliberately nonlinear ADC) of their input. These errors can sometimes be mitigated by calibration, or prevented by testing. Important parameters for linearity are integral nonlinearity and differential nonlinearity. These nonlinearities introduce distortion that can reduce the signal-to-noise ratio performance of the ADC and thus reduce its effective resolution.


title: Analog-to-digital converter

text: When digitizing a sine wave formula_5, the use of a non-ideal sampling clock will result in some uncertainty in when samples are recorded. Provided that the actual sampling time uncertainty due to clock jitter is formula_6, the error caused by this phenomenon can be estimated as formula_7. This will result in additional recorded noise that will reduce the effective number of bits (ENOB) below that predicted by quantization error alone. The error is zero for DC, small at low frequencies, but significant with signals of high amplitude and high frequency. The effect of jitter on performance can be compared to quantization error: formula_8, where q is the number of ADC bits.


title: Analog-to-digital converter

text: Clock jitter is caused by phase noise. The resolution of ADCs with a digitization bandwidth between 1 MHz and 1 GHz is limited by jitter. For lower bandwidth conversions such as when sampling audio signals at 44.1 kHz, clock jitter has a less significant impact on performance.


title: Analog-to-digital converter

text: An analog signal is continuous in time and it is necessary to convert this to a flow of digital values. It is therefore required to define the rate at which new digital values are sampled from the analog signal. The rate of new values is called the "sampling rate" or "sampling frequency" of the converter. A continuously varying bandlimited signal can be sampled and then the original signal can be reproduced from the discrete-time values by a reconstruction filter. The Nyquist–Shannon sampling theorem implies that a faithful reproduction of the original signal is only possible if the sampling rate is higher than twice the highest frequency of the signal.


title: Analog-to-digital converter

text: Since a practical ADC cannot make an instantaneous conversion, the input value must necessarily be held constant during the time that the converter performs a conversion (called the "conversion time"). An input circuit called a sample and hold performs this task—in most cases by using a capacitor to store the analog voltage at the input, and using an electronic switch or gate to disconnect the capacitor from the input. Many ADC integrated circuits include the sample and hold subsystem internally.


title: Analog-to-digital converter

text: An ADC works by sampling the value of the input at discrete intervals in time. Provided that the input is sampled above the Nyquist rate, defined as twice the highest frequency of interest, then all frequencies in the signal can be reconstructed. If frequencies above half the Nyquist rate are sampled, they are incorrectly detected as lower frequencies, a process referred to as aliasing. Aliasing occurs because instantaneously sampling a function at two or fewer times per cycle results in missed cycles, and therefore the appearance of an incorrectly lower frequency. For example, a 2 kHz sine wave being sampled at 1.5 kHz would be reconstructed as a 500 Hz sine wave.


title: Analog-to-digital converter

text: To avoid aliasing, the input to an ADC must be low-pass filtered to remove frequencies above half the sampling rate. This filter is called an "anti-aliasing filter", and is essential for a practical ADC system that is applied to analog signals with higher frequency content. In applications where protection against aliasing is essential, oversampling may be used to greatly reduce or even eliminate it.


title: Analog-to-digital converter

text: Although aliasing in most systems is unwanted, it can be exploited to provide simultaneous down-mixing of a band-limited high-frequency signal (see undersampling and frequency mixer). The alias is effectively the lower heterodyne of the signal frequency and sampling frequency.


title: Analog-to-digital converter

text: For economy, signals are often sampled at the minimum rate required with the result that the quantization error introduced is white noise spread over the whole passband of the converter. If a signal is sampled at a rate much higher than the Nyquist rate and then digitally filtered to limit it to the signal bandwidth produces the following advantages:


title: Analog-to-digital converter

text: Oversampling is typically used in audio frequency ADCs where the required sampling rate (typically 44.1 or 48 kHz) is very low compared to the clock speed of typical transistor circuits (>1 MHz). In this case, the performance of the ADC can be greatly increased at little or no cost. Furthermore, as any aliased signals are also typically out of band, aliasing can often be completely eliminated using very low cost filters.


title: Analog-to-digital converter

text: The speed of an ADC varies by type. The Wilkinson ADC is limited by the clock rate which is processable by current digital circuits. For a successive-approximation ADC, the conversion time scales with the logarithm of the resolution, i.e. the number of bits. Flash ADCs are certainly the fastest type of the three; The conversion is basically performed in a single parallel step.


title: Analog-to-digital converter

text: There is a potential tradeoff between speed and precision. Flash ADCs have drifts and uncertainties associated with the comparator levels results in poor linearity. To a lesser extent, poor linearity can also be an issue for successive-approximation ADCs. Here, nonlinearity arises from accumulating errors from the subtraction processes. Wilkinson ADCs have the best linearity of the three.


title: Analog-to-digital converter

text: The sliding scale or randomizing method can be employed to greatly improve the linearity of any type of ADC, but especially flash and successive approximation types. For any ADC the mapping from input voltage to digital output value is not exactly a floor or ceiling function as it should be. Under normal conditions, a pulse of a particular amplitude is always converted to the same digital value. The problem lies in that the ranges of analog values for the digitized values are not all of the same widths, and the differential linearity decreases proportionally with the divergence from the average width. The sliding scale principle uses an averaging effect to overcome this phenomenon. A random, but known analog voltage is added to the sampled input voltage. It is then converted to digital form, and the equivalent digital amount is subtracted, thus restoring it to its original value. The advantage is that the conversion has taken place at a random point. The statistical distribution of the final levels is decided by a weighted average over a region of the range of the ADC. This in turn desensitizes it to the width of any specific level.


title: Analog-to-digital converter

text: A direct-conversion or flash ADC has a bank of comparators sampling the input signal in parallel, each firing for a specific voltage range. The comparator bank feeds a digital encoder logic circuit that generates a binary number on the output lines for each voltage range.


title: Analog-to-digital converter

text: ADCs of this type have a large die size and high power dissipation. They are often used for video, wideband communications, or other fast signals in optical and magnetic storage.


title: Analog-to-digital converter

text: The circuit consists of a resistive divider network, a set of op-amp comparators and a priority encoder. A small amount of hysteresis is built into the comparator to resolve any problems at voltage boundaries. At each node of the resistive divider, a comparison voltage is available. The purpose of the circuit is to compare the analog input voltage with each of the node voltages.


title: Analog-to-digital converter

text: The circuit has the advantage of high speed as the conversion takes place simultaneously rather than sequentially. Typical conversion time is 100 ns or less. Conversion time is limited only by the speed of the comparator and of the priority encoder. This type of ADC has the disadvantage that the number of comparators required almost doubles for each added bit. Also, the larger the value of n, the more complex is the priority encoder.


title: Analog-to-digital converter

text: A successive-approximation ADC uses a comparator and a binary search to successively narrow a range that contains the input voltage. At each successive step, the converter compares the input voltage to the output of an internal digital to analog converter which initially represents the midpoint of the allowed input voltage range. At each step in this process, the approximation is stored in a successive approximation register (SAR) and the output of the digital to analog converter is updated for a comparison over a narrower range.


title: Analog-to-digital converter

text: When the ramp starts, a timer starts counting. When the ramp voltage matches the input, a comparator fires, and the timer's value is recorded. Timed ramp converters can be implemented economically, however, the ramp time may be sensitive to temperature because the circuit generating the ramp is often a simple analog integrator. A more accurate converter uses a clocked counter driving a DAC. A special advantage of the ramp-compare system is that converting a second signal just requires another comparator and another register to store the timer value. To reduce sensitivity to input changes during conversion, a sample and hold can charge a capacitor with the instantaneous input voltage and the converter can time the time required to discharge with a constant current.


title: Analog-to-digital converter

text: The Wilkinson ADC was designed by Denys Wilkinson in 1950. The Wilkinson ADC is based on the comparison of an input voltage with that produced by a charging capacitor. The capacitor is allowed to charge until a comparator determines it matches the input voltage. Then, the capacitor is discharged linearly. The time required to discharge the capacitor is proportional to the amplitude of the input voltage. While the capacitor is discharging, pulses from a high-frequency oscillator clock are counted by a register. The number of clock pulses recorded in the register is also proportional to the input voltage.


title: Analog-to-digital converter

text: An integrating ADC (also dual-slope or multi-slope ADC) applies the unknown input voltage to the input of an integrator and allows the voltage to ramp for a fixed time period (the run-up period). Then a known reference voltage of opposite polarity is applied to the integrator and is allowed to ramp until the integrator output returns to zero (the run-down period). The input voltage is computed as a function of the reference voltage, the constant run-up time period, and the measured run-down time period. The run-down time measurement is usually made in units of the converter's clock, so longer integration times allow for higher resolutions. Likewise, the speed of the converter can be improved by sacrificing resolution. Converters of this type (or variations on the concept) are used in most digital voltmeters for their linearity and flexibility.


title: Analog-to-digital converter

text: A "delta-encoded" or "counter-ramp" ADC has an up-down counter that feeds a digital to analog converter (DAC). The input signal and the DAC both go to a comparator. The comparator controls the counter. The circuit uses negative feedback from the comparator to adjust the counter until the DAC's output matches the input signal and number is read from the counter. Delta converters have very wide ranges and high resolution, but the conversion time is dependent on the input signal behavior, though it will always have a guaranteed worst-case. Delta converters are often very good choices to read real-world signals as most signals from physical systems do not change abruptly. Some converters combine the delta and successive approximation approaches; this works especially well when high frequency components of the input signal are known to be small in magnitude.


title: Analog-to-digital converter

text: A "pipelined ADC" (also called "subranging quantizer") uses two or more conversion steps. First, a coarse conversion is done. In a second step, the difference to the input signal is determined with a digital to analog converter (DAC). This difference is then converted more precisely, and the results are combined in the last step. This can be considered a refinement of the successive-approximation ADC wherein the feedback reference signal consists of the interim conversion of a whole range of bits (for example, four bits) rather than just the next-most-significant bit. By combining the merits of the successive approximation and flash ADCs this type is fast, has a high resolution, and can be implemented efficiently.


title: Analog-to-digital converter

text: A sigma-delta ADC (also known as a delta-sigma ADC) oversamples the incoming signal by a large factor using a smaller number of bits than required are converted using a flash ADC and filters the desired signal band. The resulting signal, along with the error generated by the discrete levels of the flash, is fed back and subtracted from the input to the filter. This negative feedback has the effect of noise shaping the quantization error that it does not appear in the desired signal frequencies. A digital filter (decimation filter) follows the ADC which reduces the sampling rate, filters off unwanted noise signal and increases the resolution of the output.


title: Analog-to-digital converter

text: A time-interleaved ADC uses M parallel ADCs where each ADC samples data every M:th cycle of the effective sample clock. The result is that the sample rate is increased M times compared to what each individual ADC can manage. In practice, the individual differences between the M ADCs degrade the overall performance reducing the spurious-free dynamic range (SFDR). However, techniques exist to correct for these time-interleaving mismatch errors.


title: Analog-to-digital converter

text: An ADC with an intermediate FM stage first uses a voltage-to-frequency converter to produce an oscillating signal with a frequency proportional to the voltage of the input signal, and then uses a frequency counter to convert that frequency into a digital count proportional to the desired signal voltage. Longer integration times allow for higher resolutions. Likewise, the speed of the converter can be improved by sacrificing resolution. The two parts of the ADC may be widely separated, with the frequency signal passed through an opto-isolator or transmitted wirelessly. Some such ADCs use sine wave or square wave frequency modulation; others use pulse-frequency modulation. Such ADCs were once the most popular way to show a digital display of the status of a remote analog sensor.


title: Analog-to-digital converter

text: A Time-stretch analog-to-digital converter (TS-ADC) digitizes a very wide bandwidth analog signal, that cannot be digitized by a conventional electronic ADC, by time-stretching the signal prior to digitization. It commonly uses a photonic preprocessor to time-stretch the signal, which effectively slows the signal down in time and compresses its bandwidth. As a result, an electronic ADC, that would have been too slow to capture the original signal, can now capture this slowed-down signal. For continuous capture of the signal, the frontend also divides the signal into multiple segments in addition to time-stretching. Each segment is individually digitized by a separate electronic ADC. Finally, a digital signal processor rearranges the samples and removes any distortions added by the preprocessor to yield the binary data that is the digital representation of the original analog signal.


title: Analog-to-digital converter

text: In many cases, the most expensive part of an integrated circuit is the pins, because they make the package larger, and each pin has to be connected to the integrated circuit's silicon. To save pins, it is common for ADCs to send their data one bit at a time over a serial interface to the computer, with each bit coming out when a clock signal changes state. This saves quite a few pins on the ADC package, and in many cases, does not make the overall design any more complex.


title: Analog-to-digital converter

text: Commercial ADCs often have several inputs that feed the same converter, usually through an analog multiplexer. Different models of ADC may include sample and hold circuits, instrumentation amplifiers or differential inputs, where the quantity measured is the difference between two inputs.


title: Analog-to-digital converter

text: Analog-to-digital converters are integral to modern music reproduction technology and digital audio workstation-based sound recording. Music may be produced on computers using an analog recording and therefore analog-to-digital converters are needed to create the pulse-code modulation (PCM) data streams that go onto compact discs and digital music files. The current crop of analog-to-digital converters utilized in music can sample at rates up to 192 kilohertz. Many recording studios record in 24-bit/96 kHz pulse-code modulation (PCM) format and then downsample and dither the signal for Compact Disc Digital Audio production (44.1 kHz) or to 48 kHz for radio and television broadcast applications.


title: Analog-to-digital converter

text: ADCs are required in digital signal processing systems that process, store, or transport virtually any analog signal in digital form. TV tuner cards, for example, use fast video analog-to-digital converters. Slow on-chip 8-, 10-, 12-, or 16-bit analog-to-digital converters are common in microcontrollers. Digital storage oscilloscopes need very fast analog-to-digital converters, also crucial for software-defined radio and their new applications.


title: Analog-to-digital converter

text: Digital imaging systems commonly use analog-to-digital converters for digitizing pixels. Some radar systems use analog-to-digital converters to convert signal strength to digital values for subsequent signal processing. Many other in situ and remote sensing systems commonly use analogous technology.


title: Analog-to-digital converter

text: Many sensors in scientific instruments produce an analog signal; temperature, pressure, pH, light intensity etc. All these signals can be amplified and fed to an ADC to produce a digital representation.


title: Analog-to-digital converter

text: Some non-electronic or only partially electronic devices, such as rotary encoders, can also be considered ADCs. Typically the digital output of an ADC will be a two's complement binary number that is proportional to the input. An encoder might output a Gray code.


title: Analog-to-digital converter

text: Flat-panel displays are inherently digital and need an ADC to process an analog signal such as composite or VGA.


title: Analog-to-digital converter

text: Testing an analog-to-digital converter requires an analog input source and hardware to send control signals and capture digital data output. Some ADCs also require an accurate source of reference signal.


title: Rivers of Blood speech

text: The "Rivers of Blood" speech was made by British Member of Parliament (MP) Enoch Powell on 20 April 1968, to a meeting of the Conservative Political Centre in Birmingham, United Kingdom. His speech strongly criticised mass immigration, especially Commonwealth immigration to the United Kingdom and the proposed Race Relations Bill. It became known as the "Rivers of Blood" speech, although Powell always referred to it as "the Birmingham speech".


title: Rivers of Blood speech

text: The expression "rivers of blood" did not appear in the speech but is an allusion to a line from Virgil's "Aeneid" which he quoted: "as I look ahead, I am filled with foreboding; like the Roman, I seem to see 'the River Tiber foaming with much blood'."


title: Rivers of Blood speech

text: The speech caused a political storm, making Powell one of the most talked about and divisive politicians in the country, and leading to his controversial dismissal from the Shadow cabinet by Conservative Party leader Edward Heath. According to most accounts, the popularity of Powell's perspective on immigration may have been a decisive factor in the Conservatives' surprise victory in the 1970 general election, and he became one of the most persistent opponents of the subsequent Heath government.


title: Rivers of Blood speech

text: Powell, the Conservative MP for Wolverhampton South West and Shadow Secretary of State for Defence, was addressing the general meeting of the West Midlands Area Conservative Political Centre. The Labour government's 1968 Race Relations Bill was to have its second reading three days later, and the Conservative Opposition had tabled an amendment significantly weakening its provisions. The bill was a successor to the Race Relations Act 1965.


title: Rivers of Blood speech

text: The Birmingham-based television company ATV saw an advance copy of the speech on the Saturday morning, and its news editor ordered a television crew to go to the venue, where they filmed sections of the speech. Earlier in the week, Powell had said to his friend Clement "Clem" Jones, a journalist and then editor at the Wolverhampton "Express & Star", "I'm going to make a speech at the weekend and it's going to go up 'fizz' like a rocket; but whereas all rockets fall to the earth, this one is going to stay up."


title: Rivers of Blood speech

text: In preparing his speech, Powell had applied Clem Jones's advice that to make hard-hitting political speeches and short-circuit interference from his party organisation, his best timing was on Saturday afternoons, after delivering embargoed copies the previous Thursday or Friday to selected editors and political journalists of Sunday newspapers. This tactic could ensure coverage of the speech over three days through Saturday evening bulletins, then Sunday newspapers, thus the coverage would be picked up in Monday newspapers.


title: Rivers of Blood speech

text: In the speech Powell recounted a conversation with one of his constituents, a middle-aged working man, a few weeks earlier. Powell said that the man told him: "If I had the money to go, I wouldn't stay in this country... I have three children, all of them been through grammar school and two of them married now, with family. I shan't be satisfied till I have seen them all settled overseas." The man finished by saying to Powell: "In this country in 15 or 20 years' time the black man will have the whip hand over the white man".


title: Rivers of Blood speech

text: Powell quoted a letter he received from a woman in Northumberland, about an elderly woman living on a Wolverhampton street where she was the only white resident. The woman's husband and two sons had died in World War II and she had rented out the rooms in her house. Once immigrants had moved into the street in which she lived, her white lodgers left. Two black men had knocked on her door at 7:00 am to use her telephone to call their employers, but she refused, as she would have done to any other stranger knocking at her door at such an hour, and was subsequently verbally abused. The woman had asked her local authority for a rates reduction, but was told by a council officer to let out the rooms of her house. When the woman said the only tenants would be black, the council officer replied: "Racial prejudice won't get you anywhere in this country."


title: Rivers of Blood speech

text: Powell advocated voluntary re-emigration by "generous grants and assistance" and he mentioned that immigrants had asked him whether it was possible. He said that all citizens should be equal before the law, and that:


title: Rivers of Blood speech

text: He argued that journalists who urged the government to pass anti-discrimination laws were "of the same kidney and sometimes on the same newspapers which year after year in the 1930s tried to blind this country to the rising peril which confronted it". Powell described what he perceived to be the evolving position of the indigenous population:


title: Rivers of Blood speech

text: Powell warned that if the legislation proposed for the then–Race Relations Bill were to be passed it would bring about discrimination against the native population:


title: Rivers of Blood speech

text: Powell argued that he felt that although "many thousands" of immigrants wanted to integrate, he felt that the majority did not, and that some had vested interests in fostering racial and religious differences "with a view to the exercise of actual domination, first over fellow-immigrants and then over the rest of the population". Powell's peroration of the speech gave rise to its popular title. He quotes the Sibyl's prophecy in the epic poem "Aeneid", 6, 86–87, of "wars, terrible wars, / and the Tiber foaming with much blood".


title: Rivers of Blood speech

text: According to C. Howard Wheeldon, who was present at the meeting in which Powell gave the speech, "it is fascinating to note what little hostility emerged from the audience. To the best of my memory, only one person voiced any sign of annoyance." The day after the speech, Powell went to Sunday Communion at his local church, and when he emerged, there was a crowd of journalists, and a local plasterer (Sidney Miller) said to Powell: "Well done, sir. It needed to be said." Powell asked the assembled journalists: "Have I really caused such a furore?" At midday, Powell went on the BBC's "World This Weekend" to defend his speech, and he appeared later that day on ITN news.


title: Rivers of Blood speech

text: The Labour MP Edward Leadbitter said he would refer the speech to the Director of Public Prosecutions, and the Liberal Party leader Jeremy Thorpe spoke of a "prima facie" case against Powell for incitement. Lady Gaitskell called the speech "cowardly", and the West Indian cricketer Sir Learie Constantine condemned it.


title: Rivers of Blood speech

text: The leading Conservatives in the Shadow Cabinet were outraged by the speech. Iain Macleod, Edward Boyle, Quintin Hogg and Robert Carr all threatened to resign from the front bench unless Powell was sacked. Margaret Thatcher, who was then the Shadow Cabinet's Fuel and Power Spokesman, thought that some of Powell's speech was "strong meat", and said to the Conservative leader, Edward Heath when he telephoned her to inform her Powell was to be sacked: "I really thought that it was better to let things cool down for the present rather than heighten the crisis". Heath sacked Powell from his post as Shadow Defence Secretary, telling him on the telephone that Sunday evening (it was the last conversation they would have). Heath said of the speech in public that it was "racialist in tone and liable to exacerbate racial tensions". Conservative MPs on the right of the party—Duncan Sandys, Gerald Nabarro, Teddy Taylor—spoke against Powell's sacking. On 22 April 1968, Heath went on "Panorama", telling Robin Day: "I dismissed Mr Powell because I believed his speech was inflammatory and liable to damage race relations. I am determined to do everything I can to prevent racial problems developing into civil strife ... I don't believe the great majority of the British people share Mr Powell's way of putting his views in his speech."


title: Rivers of Blood speech

text: "The Times" newspaper declared it "an evil speech", stating "This is the first time that a serious British politician has appealed to racial hatred in this direct way in our postwar history." "The Times" went on to record incidents of racial attacks in the immediate aftermath of Powell's speech. One such incident, reported under the headline "Coloured family attacked", took place on 30 April 1968 in Wolverhampton itself: it involved a slashing incident with 14 white youths chanting "Powell" and "Why don't you go back to your own country?" at patrons of a West Indian christening party. One of the West Indian victims, Wade Crooks of Lower Villiers Street, was the child's grandfather. He had to have eight stitches over his left eye. He was reported as saying, "I have been here since 1955 and nothing like this has happened before. I am shattered." An opinion poll commissioned by the BBC television programme "Panorama" in December 1968 found that eight per cent of immigrants believed that they had been treated worse by white people since Powell's speech, 38 per cent would like to return to their country of origin if offered financial help, and 47 per cent supported immigration control, with 30 per cent opposed.


title: Rivers of Blood speech

text: The speech generated much correspondence to newspapers, most markedly with the "Express & Star" in Wolverhampton itself, whose local sorting office over the following week received 40,000 postcards and 8,000 letters addressed to its local newspaper. Clem Jones recalled:


title: Rivers of Blood speech

text: At the end of that week there were two simultaneous processions in Wolverhampton, one of Powell's supporters and another of opponents, who each brought petitions to Clem Jones outside his office, the two columns being kept apart by police.


title: Rivers of Blood speech

text: On 23 April 1968, the Race Relations Bill had its second reading in the House of Commons. Many MPs referred or alluded to Powell's speech. For Labour, Paul Rose, Maurice Orbach, Reginald Paget, Dingle Foot, Ivor Richard, and David Ennals were all critical. Among the Conservatives, Quintin Hogg and Nigel Fisher were critical, while Hugh Fraser, Ronald Bell, Dudley Smith, and Harold Gurden were sympathetic. Powell himself was present for the debate but did not speak.


title: Rivers of Blood speech

text: Earlier that day, 1,000 London dockers had gone on strike in protest of Powell's sacking and marched from the East End to the Palace of Westminster carrying placards with sayings such as "we want Enoch Powell!", "Enoch here, Enoch there, we want Enoch everywhere", "Don't knock Enoch" and "Back Britain, not Black Britain". Three hundred of them went into the palace, 100 to lobby the MP for Stepney, Peter Shore, and 200 to lobby the MP for Poplar, Ian Mikardo. Shore and Mikardo were shouted down and some dockers kicked Mikardo. Lady Gaitskell shouted: "You will have your remedy at the next election." The dockers replied: "We won't forget." The organiser of the strike, Harry Pearman, headed a delegation to meet Powell and said after: "I have just met Enoch Powell and it made me feel proud to be an Englishman. He told me that he felt that if this matter was swept under the rug he would lift the rug and do the same again. We are representatives of the working man. We are not racialists."


title: Rivers of Blood speech

text: On 24 April 600 dockers at St Katharine Docks voted to strike and numerous smaller factories across the country followed. Six hundred Smithfield meat porters struck and marched to Westminster and handed Powell a 92-page petition supporting him. Powell advised against strike action and asked them to write to Harold Wilson, Heath or their MP. However, strikes continued, reaching Tilbury by 25 April and he allegedly received his 30,000th letter supporting him, with 30 protesting against his speech. By 27 April, 4,500 dockers were on strike. On 28 April, 1,500 people marched to Downing Street chanting "Arrest Enoch Powell". Powell claimed to have received 43,000 letters and 700 telegrams supporting him by early May, with 800 letters and four telegrams against. On 2 May, the attorney-general, Sir Elwyn Jones, announced he would not prosecute Powell after consulting the Director of Public Prosecutions.


title: Rivers of Blood speech

text: The Gallup Organization took an opinion poll at the end of April and found that 74 per cent agreed with what Powell had said in his speech; 15 per cent disagreed. 69 per cent felt Heath was wrong to sack Powell and 20 per cent believed Heath was right. Before his speech Powell was favoured to replace Heath as Conservative leader by one per cent, with Reginald Maudling favoured by 20 per cent; after his speech 24 per cent favoured Powell and 18 per cent Maudling. 83 per cent now felt immigration should be restricted (75 per cent before the speech) and 65 per cent favoured anti-discrimination legislation. According to George L. Bernstein, the speech made the British people think that Powell "was the first British politician who was actually listening to them".


title: Rivers of Blood speech

text: Powell defended his speech on 4 May through an interview for the "Birmingham Post": "What I would take 'racialist' to mean is a person who believes in the inherent inferiority of one race of mankind to another, and who acts and speaks in that belief. So the answer to the question of whether I am a racialist is 'no'—unless, perhaps, it is to be a racialist in reverse. I regard many of the peoples in India as being superior in many respects—intellectually, for example, and in other respects—to Europeans. Perhaps that is over-correcting." On 5 May the Prime Minister, Harold Wilson, made his first public statement on race and immigration since Powell's speech. He told Labour supporters at a May Day rally in Birmingham Town Hall:


title: Rivers of Blood speech

text: I am not prepared to stand aside and see this country engulfed by the racial conflict which calculating orators or ignorant prejudice can create. Nor in the great world confrontation on race and colour, where this country must declare where it stands, am I prepared to be a neutral, whether that confrontation is in Birmingham or Bulawayo. In these issues there can be no neutrals and no escape from decision. For in the world of today, while political isolationism invites danger and economic isolationism invites bankruptcy, moral isolationism invites contempt.


title: Rivers of Blood speech

text: We are the party of human rights—the only party of human rights that will be speaking from this platform this month. (Loud applause.) The struggle against racialism is a worldwide fight. It is the dignity of man for which we are fighting. If what we assert is true for Birmingham, it is true for Bulawayo. If ever there were a condemnation of the values of the party which forms the Opposition it is the fact that the virus of Powellism has taken so firm a hold at every level.


title: Rivers of Blood speech

text: During the 1970 general election the majority of the Parliamentary Labour Party did not wish to "stir up the Powell issue". However, Labour MP Tony Benn said:


title: Rivers of Blood speech

text: According to most accounts, the popularity of Powell's perspective on immigration may have played a decisive contributory factor in the Conservatives' surprise victory in the 1970 general election, although Powell became one of the most persistent opponents of the subsequent Heath government. In "exhaustive research" on the election, the American pollster Douglas Schoen and University of Oxford academic R. W. Johnson believed it "beyond dispute" that Powell had attracted 2.5 million votes to the Conservatives, but nationally the Conservative vote had increased by only 1.7 million since 1966. In his own constituency at that election – his last in Wolverhampton – his majority of 26,220 and a 64.3 per cent share of the vote were then the highest of his career.


title: Rivers of Blood speech

text: Powell reflected on the speech in an interview in 1977 when the interviewer asked him, "nine years after the speech, are we still in your view on a kind of funeral pyre?":


title: Rivers of Blood speech

text: The interviewer then asked him, "what do you see as the likely prospect now? Still the 'River Tiber foaming with blood'?":


title: Rivers of Blood speech

text: Polls in the 1960s and 1970s showed that Powell's views were popular among the British population at the time. A Gallup poll, for example, showed that 75% of the population were sympathetic to Powell's views. An NOP poll showed that approximately 75% of the British population agreed with Powell's demand for non-white immigration to be halted completely, and about 60% agreed with his call for the repatriation of non-whites already resident in Britain.


title: Rivers of Blood speech

text: The Rivers of Blood speech has been blamed for leading to violent attacks against British Pakistanis and other British Asians, which became frequent after the speech in 1968; however, there is "little agreement on the extent to which Powell was responsible for racial attacks". These "Paki-bashing" attacks later peaked during the 1970s and 1980s.


title: Rivers of Blood speech

text: Powell was mentioned in early versions of the 1969 song "Get Back" by the Beatles. This early version of the song, known as the "No Pakistanis" version, parodied the anti-immigrant views of Enoch Powell.


title: Rivers of Blood speech

text: On 5 August 1976, Eric Clapton provoked an uproar and lingering controversy when he spoke out against increasing immigration during a concert in Birmingham. Visibly intoxicated, Clapton voiced his support of the controversial speech, and announced on stage that Britain was in danger of becoming a "black colony". Among other things, Clapton said "Keep Britain white!" which was at the time a National Front (NF) slogan.


title: Rivers of Blood speech

text: In November 2010, the actor and comedian Sanjeev Bhaskar recalled the fear which the speech instilled in Britons of Indian origin: "At the end of the 1960s, Enoch Powell was quite a frightening figure to us. He was the one person who represented an enforced ticket out, so we always had suitcases that were ready and packed. My parents held the notion that we may have to leave."


title: Rivers of Blood speech

text: Whilst a section of the white population appeared to warm to Powell over the speech, the author Mike Phillips recalls that it legitimised hostility, and even violence, towards black Britons like himself.


title: Rivers of Blood speech

text: In his book "The British Dream" (2013), David Goodhart claims that Powell's speech in effect "put back by more than a generation a robust debate about the successes and failures of immigration".


title: Rivers of Blood speech

text: After Powell delivered the speech, there were attempts to locate the Wolverhampton constituent whom Powell described as being victimised by non-white residents. The editor of the local Wolverhampton newspaper the "Express & Star", Clem Jones (a close friend of Powell who broke off relations with him over the controversy) claimed he was unable to identify the woman using the electoral register and other sources.


title: Rivers of Blood speech

text: Shortly after Powell's death, Kenneth Nock, a Wolverhampton solicitor, wrote to the "Express and Star" in April 1998 to claim that his firm had acted for the woman in question, but that he could not name her owing to rules concerning client confidentiality. In January 2007, the BBC Radio Four programme "Document" claimed to have uncovered the woman's identity. They said she was Druscilla Cotterill (1907–1978), the widow of Harry Cotterill, a battery quartermaster sergeant with the Royal Artillery who had been killed in World War II. (She was also the second cousin of Mark Cotterill, a figure in British far-right politics.) She lived in Brighton Place in Wolverhampton, which by the 1960s was dominated by immigrant families. In order to increase her income, she rented rooms to lodgers, but did not wish to rent rooms to West Indians and stopped taking in any lodgers when the Race Relations Act 1968 banned racial discrimination in housing. She locked up the spare rooms and lived only in two rooms of the house.


title: Rivers of Blood speech

text: In the United Kingdom, particularly in England, "Enoch [Powell] was right" is a phrase of political rhetoric, inviting comparison of aspects of current English society with the predictions made by Powell in the "Rivers of Blood" speech. The phrase implies criticism of racial quotas, immigration and multiculturalism. Badges, T-shirts and other items bearing the slogan have been produced at different times in the United Kingdom. Powell gained support from both right-wing and traditionally left-leaning, working-class voters for his anti-immigration stance.


title: Rivers of Blood speech

text: Powell gained the support of the far-right in Britain. Badges, T-shirts and fridge magnets emblazoned with the slogan "Enoch was right" are regularly seen at far-right demonstrations, according to VICE News. Powell also has a presence on social media, with an Enoch Powell page on Facebook run by the far-right Traditional Britain Group which amassed several thousands of likes, and similar pages which post "racist memes and "Daily Mail" stories" have been equally successful, such as British nationalist and anti-immigration Britain First's Facebook page.


title: Rivers of Blood speech

text: In "The Trial of Enoch Powell", a Channel 4 television broadcast in April 1998, on the thirtieth anniversary of his Rivers of Blood speech (and two months after his death), 64% of the studio audience voted that Powell was not a racist. Some in the Church of England, of which Powell had been a member, took a different view. Upon Powell's death, Barbados-born Wilfred Wood, then Bishop of Croydon, stated, "Enoch Powell gave a certificate of respectability to white racist views which otherwise decent people were ashamed to acknowledge".


title: Rivers of Blood speech

text: In March 2016, right-wing German writer Michael Stürmer wrote a retrospective pro-Powell piece in "Die Welt", opining that nobody else had been "punished so mercilessly" by fellow party members and media for their viewpoints.


title: Rivers of Blood speech

text: Trevor Phillips wrote in May 2016 "Rome may not yet be in flames, but I think I can smell the smouldering whilst we hum to the music of liberal self-delusion" by ignoring the effects of mass immigration. He explicitly compared his warning to Powell's: "He too summoned up echoes of Rome with his reference to Virgil's dire premonition of the River Tiber 'foaming with much blood'". From the damage the reaction to the speech did to Powell's career, Phillips wrote, "Everyone in British public life learnt the lesson: adopt any strategy possible to avoid saying anything about race, ethnicity (and latterly religion and belief) that is not anodyne and platitudinous".


title: Rivers of Blood speech

text: In October 2018, support for the speech was expressed by the Plymouth University Conservatives who referenced the phrase "Enoch was Right" on one of the apparel worn for a society gathering.


title: Rivers of Blood speech

text: In an interview for "Today" shortly after her departure from office as Prime Minister in 1990, Margaret Thatcher, said that Powell had "made a valid argument, if in sometimes regrettable terms".


title: Rivers of Blood speech

text: Thirty years after the speech, Edward Heath said that Powell's remarks on the "economic burden of immigration" had been "not without prescience".


title: Rivers of Blood speech

text: The Labour Party MP Michael Foot remarked to a reporter that it was "tragic" that this "outstanding personality" had been widely misunderstood as predicting actual bloodshed in Britain, when in fact he had used the "Aeneid" quotation merely to communicate his own sense of foreboding.


title: Rivers of Blood speech

text: In November 2007, Nigel Hastilow resigned as Conservative candidate for Halesowen and Rowley Regis after he wrote an article in the Wolverhampton "Express & Star" that included the statement: "Enoch, once MP for Wolverhampton South-West, was sacked from the Conservative front bench and marginalised politically for his 1968 'Rivers of Blood' speech, warning that uncontrolled immigration would change Britain irrevocably. He was right and immigration has changed the face of Britain dramatically."


title: Rivers of Blood speech

text: In January 2014, UK Independence Party leader Nigel Farage, after being told during an interview that a statement just read to him had come from Powell's speech, said: "Well what he was warning about was the large influx of people into an area, that change an area beyond recognition, there is tension – the basic principle is right." In June of that year, in response to the alleged Islamist Operation Trojan Horse, Conservative peer and former minister Norman Tebbit wrote in "The Daily Telegraph", "No one should have been surprised at what was going on in schools in Birmingham. It is precisely what I was talking about over 20 years ago and Enoch Powell was warning against long before that. We have imported far too many immigrants who have come here not to live in our society, but to replicate here the society of their homelands." Conservative MP Gerald Howarth said on the same issue, "Clearly, the arrival of so many people of non-Christian faith has presented a challenge, as so many of us, including the late Enoch Powell, warned decades ago."


title: Rivers of Blood speech

text: In April 2018, the leader of UKIP in Wales, Neil Hamilton, said that "the idea that Enoch Powell was some kind of uniquely racist villain is absolute nonsense". Hamilton said that Powell had been "proved right by events" in terms of social change if not violence. In response, the leader of Plaid Cymru, Leanne Wood, accused Hamilton of "keeping Powell's racist rhetoric going". Labour AM Hefin David described Hamilton's comments as "outrageous".


title: Rivers of Blood speech

text: The speech is the subject of a play, "What Shadows", written by Chris Hannan. The play was staged in Birmingham from 27 October to 12 November 2016, with Powell portrayed by Ian McDiarmid and Clem Jones by George Costigan.


title: Rivers of Blood speech

text: "The Speech", a novel by author Andrew Smith set in Wolverhampton during the ten days before and after the speech and featuring Powell as a character, was published in October 2016 by Urbane Publications.


title: Rivers of Blood speech

text: In April 2018 the BBC announced that Archive on 4 would transmit "50 Years On: Rivers of Blood", a programme marking the 50th anniversary of the speech. Ian McDiarmid would read the entire speech, the first time it would be broadcast on British radio, and it would be discussed and analysed. In the days before the broadcast, there was criticism from a number of commentators of the BBC's decision to broadcast the still-controversial speech.


title: Euclidean algorithm

text: In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his "Elements" (c. 300 BC).


title: Euclidean algorithm

text: It is an example of an "algorithm", a step-by-step procedure for performing a calculation according to well-defined rules,


title: Euclidean algorithm

text: and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations.


title: Euclidean algorithm

text: The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 252 − 105 = 147. Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that occurs, they are the GCD of the original two numbers. By reversing the steps or using the extended Euclidean algorithm, the GCD can be expressed as a linear combination of the two original numbers, that is the sum of the two numbers, each multiplied by an integer (for example, 21 = 5 × 105 + (−2) × 252). The fact that the GCD can always be expressed in this way is known as Bézout's identity.


title: Euclidean algorithm

text: The version of the Euclidean algorithm described above (and by Euclid) can take many subtraction steps to find the GCD when one of the given numbers is much bigger than the other. A more efficient version of the algorithm shortcuts these steps, instead replacing the larger of the two numbers by its remainder when divided by the smaller of the two (with this version, the algorithm stops when reaching a zero remainder). With this improvement, the algorithm never requires more steps than five times the number of digits (base 10) of the smaller integer. This was proven by Gabriel Lamé in 1844, and marks the beginning of computational complexity theory. Additional methods for improving the algorithm's efficiency were developed in the 20th century.


title: Euclidean algorithm

text: The Euclidean algorithm has many theoretical and practical applications. It is used for reducing fractions to their simplest form and for performing division in modular arithmetic. Computations using this algorithm form part of the cryptographic protocols that are used to secure internet communications, and in methods for breaking these cryptosystems by factoring large composite numbers. The Euclidean algorithm may be used to solve Diophantine equations, such as finding numbers that satisfy multiple congruences according to the Chinese remainder theorem, to construct continued fractions, and to find accurate rational approximations to real numbers. Finally, it can be used as a basic tool for proving theorems in number theory such as Lagrange's four-square theorem and the uniqueness of prime factorizations.


title: Euclidean algorithm

text: The original algorithm was described only for natural numbers and geometric lengths (real numbers), but the algorithm was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials of one variable. This led to modern abstract algebraic notions such as Euclidean domains.


title: Euclidean algorithm

text: The Euclidean algorithm calculates the greatest common divisor (GCD) of two natural numbers "a" and "b". The greatest common divisor "g" is the largest natural number that divides both "a" and "b" without leaving a remainder. Synonyms for GCD include "greatest common factor" (GCF), "highest common factor" (HCF), "highest common divisor" (HCD), and "greatest common measure" (GCM). The greatest common divisor is often written as gcd("a", "b") or, more simply, as ("a", "b"), although the latter notation is ambiguous, also used for concepts such as an ideal in the ring of integers, which is closely related to GCD.


title: Euclidean algorithm

text: If gcd("a", "b") = 1, then "a" and "b" are said to be coprime (or relatively prime). This property does not imply that "a" or "b" are themselves prime numbers. For example, 6 and 35 factor as 6 = 2 × 3 and 35 = 5 × 7, so they are not prime, but their prime factors are different, so 6 and 35 are coprime, with no common factors other than 1.


title: Euclidean algorithm

text: Let "g" = gcd("a", "b"). Since "a" and "b" are both multiples of "g", they can be written "a" = "mg" and "b" = "ng", and there is no larger number "G" > "g" for which this is true. The natural numbers "m" and "n" must be coprime, since any common factor could be factored out of "m" and "n" to make "g" greater. Thus, any other number "c" that divides both "a" and "b" must also divide "g". The greatest common divisor "g" of "a" and "b" is the unique (positive) common divisor of "a" and "b" that is divisible by any other common divisor "c".