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Boston Massacre
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For the baseball event nicknamed the "Boston Massacre", see Yankees – Red Sox rivalry.
For the 2013 terrorist attack, see Boston Marathon bombing.
Boston Massacre
Part of the American Revolution
This famous depiction of the event was engraved by Paul Revere (copied from an engraving by Henry Pelham), colored by Christian Remick, and printed by Benjamin Edes. The Old State House is depicted in the background.[1]
Location
Boston, Massachusetts, British America
(now State Street, Boston, Massachusetts, U.S.)
Coordinates
42°21′32″N 71°03′26″W / 42.35879°N 71.05717°W / 42.35879; -71.05717Coordinates: 42°21′32″N 71°03′26″W / 42.35879°N 71.05717°W / 42.35879; -71.05717
Date
March 5, 1770; 247 years ago (1770-03-05)
Attack type
Massacre
Weapons
Flintlock muskets
Deaths
5
Non-fatal injuries
6
Perpetrators
British Army infantrymen
This article is part of a series about the
American Revolution
1765–1783
Origins[show]
Sugar Act
Currency Act
Quartering Acts
Stamp Act 1765
Declaratory Act
No taxation without representation
Townshend Acts
Tea Act
Boston Massacre
Boston Tea Party
Intolerable Acts
Revolutionary War[show]
Continental Army
Continental Navy
Minutemen
Battles
Intelligence
Treaty of Paris
Costs
Declaration of Independence[show]
Continental Congress
Committee of Five
Lee Resolution
Articles of Confederation
United States Constitution[show]
Philadelphia Convention
Annapolis Convention
The Federalist Papers
Bill of Rights
A New Republic[show]
Republicanism
American Enlightenment
Liberalism
First Great Awakening
Role of Women
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The Boston Massacre, known as the Incident on King Street by the British,[2] was an incident on March 5, 1770, in which British Army soldiers shot and killed people while under attack by a mob. The incident was heavily publicized by leading Patriots, such as Paul Revere and Samuel Adams, to encourage rebellion against the British authorities.[3][4][5] British troops had been stationed in Boston, capital of the Province of Massachusetts Bay, since 1768 in order to protect and support crown-appointed colonial officials attempting to enforce unpopular Parliamentary legislation. Amid ongoing tense relations between the population and the soldiers, a mob formed around a British sentry, who was subjected to verbal abuse and harassment. He was eventually supported by eight additional soldiers, who were subjected to verbal threats and repeatedly hit by clubs, stones and snowballs. They fired into the crowd, without orders, instantly killing three people and wounding others. Two more people died later of wounds sustained in the incident.
The crowd eventually dispersed after Acting Governor Thomas Hutchinson promised an inquiry, but the crowd re-formed the next day, prompting the withdrawal of the troops to Castle Island. Eight soldiers, one officer, and four civilians were arrested and charged with murder. Defended by lawyer and future American president John Adams, six of the soldiers were acquitted, while the other two were convicted of manslaughter and given reduced sentences. The men found guilty of manslaughter were sentenced to branding on their hand. Depictions, reports, and propaganda about the event, notably the colored engraving produced by Paul Revere (shown at top-right), further heightened tensions throughout the Thirteen Colonies.
Contents
[hide]
1 Background
2 Incident
3 Aftermath
3.1 Investigation
3.2 Propaganda battle
3.3 Trial
4 Legacy
4.1 Contribution to American Revolution
4.2 Commemorations
5 See also
6 References
7 Sources
8 Further reading
9 External links
Background
Main article: American Revolution
View of the Old State House, Boston, Massachusetts, the seat of British colonial government from 1713 to 1776. The Boston Massacre took place in front of the balcony, and the site is now marked by a cobblestone circle in the square (photo 2009)
Massacre Site
Boston, the capital of the Province of Massachusetts Bay and an important shipping town, was a major center of resistance to unpopular acts of taxation by the British Parliament in the 1760s.[6] In 1768, the Townshend Acts were placed upon the colonists, by which a variety of common items that were manufactured in Britain and exported to the colonies were subjected to import tariffs. Colonists objected that the Townshend Acts were a violation of the natural, charter, and constitutional rights of British subjects in the colonies.[6] The Massachusetts House of Representatives began a campaign against the Townshend Acts by sending a petition to King George III asking for the repeal of the Townshend Revenue Act. The House also sent what became known as the Massachusetts Circular Letter to other colonial assemblies, asking them to join the resistance movement,[6] and called for a boycott of merchants importing the affected goods.[7]
In Great Britain, Lord Hillsborough, who had recently been appointed to the newly created office of Colonial Secretary, was alarmed by the actions of the Massachusetts House. In April 1768 he sent a letter to the colonial governors in America, instructing them to dissolve the colonial assemblies if they responded to the Massachusetts Circular Letter. He also ordered Massachusetts Governor Francis Bernard to direct the Massachusetts House to rescind the letter. The house refused to comply.[8]
Boston's chief customs officer, Charles Paxton, wrote to Hillsborough, asking for military support because "the Government is as much in the hands of the people as it was in the time of the Stamp Act."[9] Commodore Samuel Hood responded by sending the fifty-gun warship HMS Romney, which arrived in Boston Harbor in May 1768.[10] On June 10, 1768, customs officials seized Liberty, a sloop owned by leading Boston merchant John Hancock, on allegations that the ship had been involved in smuggling. Bostonians, already angry because the captain of Romney had been impressing local sailors, began to riot.[11] Customs officials fled to Castle William for protection.[12]
Daniel Calfe declares, that on Saturday evening the 3d of March, a camp-woman, wife to James McDeed, a grenadier of the 29th, came into his father's shop, and the people talking about the affrays at the ropewalks, and blaming the soldiers for the part they had acted in it, the woman said, "the soldiers were in the right;" adding, "that before Tuesday or Wednesday night they would wet their swords or bayonets in New England people's blood."
—Excerpt from A Short Narrative, suggesting the soldiers were contemplating violence against the colonists[13]
Given the unstable state of affairs in Massachusetts, Hillsborough instructed General Thomas Gage, Commander-in-Chief, North America, to send "such Force as You shall think necessary to Boston".[14] On October 1, 1768, the first of four British Army regiments began disembarking in Boston.[15] Two regiments were removed from Boston in 1769, but the 14th and the 29th Regiments of Foot remained.[16]
The Journal of Occurrences, an anonymously written series of newspaper articles, chronicled clashes between civilians and soldiers while troops were stationed in Boston, feeding tensions with its sometimes exaggerated accounts of the events. Tensions rose markedly after Christopher Seider, "a young lad about eleven Years of Age", was killed by a customs employee on February 22, 1770.[16] Seider's death was glorified in the Boston Gazette, and his funeral was described as one of the largest of the time in Boston. The killing and subsequent propaganda inflamed tensions, with gangs of colonists looking for soldiers to harass, and soldiers also on occasion looking for confrontation.[17]
Incident
This 19th-century lithograph is a variation of Revere's famous engraving. Produced soon before the American Civil War, this image emphasizes Crispus Attucks, who had by then become an important symbol for abolitionists. (John Bufford after William L. Champey, circa 1856)[18]
On the evening of March 5, Private Hugh White, a British soldier, stood on guard duty outside the Custom house on King Street, today known as State Street. A young wigmaker's apprentice named Edward Garrick called out to a British officer, Captain-Lieutenant John Goldfinch, that Goldfinch had not paid a bill due to Garrick's master. Goldfinch had in fact settled his account and ignored the insult.[19] Private White called out to Garrick that he should be more respectful of the officer. Garrick exchanged insults with Private White. Then, after Garrick started poking the officer in the chest with his finger, the officer left his post, challenged the boy, and struck him on the side of the head with his musket. As Garrick cried in pain, one of his companions, Bartholomew Broaders, began to argue with White. This attracted a larger crowd.[20] Henry Knox, a 19-year-old bookseller (who would later serve as a general in the revolution), came upon the scene and warned White, "if he fired he must die for it."[19]
An 1868 print by Alonzo Chappel, showing a more chaotic scene contrary to most earlier representations.
As the evening progressed, the crowd around Private White grew larger and more boisterous. Church bells were rung, which usually signified a fire, bringing more people out. Over fifty Bostonians pressed around White, led by a mixed-race runaway slave named Crispus Attucks, throwing objects at the sentry and challenging him to fire his weapon. White, who had taken up a somewhat safer position on the steps of the Custom House, sought assistance. Runners alerted the nearby barracks and Captain Thomas Preston, the officer of the watch.[21][22] According to his report, Preston dispatched a non-commissioned officer and six privates of the 29th Regiment of Foot, with fixed bayonets, to relieve White.[23][24] The soldiers Preston sent were Corporal William Wemms, Hugh Montgomery, John Carroll, William McCauley, William Warren, and Matthew Kilroy. Accompanied by Preston, they pushed their way through the crowd. En route, Henry Knox, again trying to reduce tensions, warned Preston, "For God's sake, take care of your men. If they fire, you must die." [clarification needed] Captain Preston responded "I am aware of it."[25] When they reached Private White on the custom house stairs, the soldiers loaded their muskets, and arrayed themselves in a semicircular formation. Preston shouted at the crowd, estimated to number between three and four hundred, to disperse.[26]
Boston Massacre grave marker in the Granary Burying Ground
The crowd continued to press around the soldiers, taunting them by yelling, "Fire!", by spitting at and throwing snowballs and other small objects at them.[27] Richard Palmes, a local innkeeper who was carrying a cudgel, came up to Preston and asked if the soldiers' weapons were loaded. Preston assured him they were, but that they would not fire unless he ordered it, and (according to his own deposition) that he was unlikely to do so, since he was standing in front of them. A thrown object then struck Private Montgomery, knocking him down and causing him to drop his musket. He recovered his weapon, and was thought to angrily shout "Damn you, fire!", then discharged it into the crowd although no command was given. Palmes swung his cudgel first at Montgomery, hitting his arm, and then at Preston. He narrowly missed Preston's head, striking him on the arm instead.[27]
There was a pause of uncertain length (eyewitness estimates ranged from several seconds to two minutes), after which the soldiers fired into the crowd. Rather than a disciplined volley (Preston gave no orders to fire), a ragged series of shots was fired, which hit eleven men.[28] Three Americans—ropemaker Samuel Gray, mariner James Caldwell, and Crispus Attucks—died instantly.[29] Samuel Maverick, an apprentice ivory turner of seventeen,[30] was struck by a ricocheting musket ball at the back of the crowd, and died a few hours later, in the early morning of the next day. An Irish immigrant, Patrick Carr, died two weeks later.[29] Christopher Monk, another apprentice, was one of those seriously wounded in the attack.[31] Although he recovered to some extent, he was crippled and eventually died in 1780, purportedly due to the injuries he had sustained in the attack a decade earlier.[32][33]
The crowd moved away from the immediate area of the custom house, but continued to grow in nearby streets.[34] Captain Preston immediately called out most of the 29th Regiment, which adopted defensive positions in front of the state house.[35] Acting Governor Thomas Hutchinson was summoned to the scene, and was forced by the movement of the crowd into the council chamber of the state house. From its balcony he was able to minimally restore order, promising there would be a fair inquiry into the shootings if the crowd dispersed.[36]
Aftermath
Investigation
Hutchinson immediately began investigating the affair, and by morning, Preston and the eight soldiers had been arrested.[37] In a meeting of the governor's council held late the morning after the shootings, Boston's selectmen asked Hutchinson to order the removal of troops from the city to Castle William on Castle Island,[36] while a town meeting at Faneuil Hall met to discuss the affair. The governor's council was at first opposed to ordering the troop withdrawal, with Hutchinson correctly claiming he did not have the authority to order the troops to move. Lieutenant Colonel William Dalrymple, commander of the troops, did not offer to move them.[38] The town meeting, however, became more restive when it learned of this. Under an imminent threat of further violence, the council changed its position, and unanimously ("under duress", according to Hutchinson's report) agreed to request the troops' removal.[39] Secretary of State Andrew Oliver reported that, had the troops not been removed, "that they would probably be destroyed by the people — should it be called rebellion, should it incur the loss of our charter, or be the consequence what it would."[40] This decision left the governor without effective means to police the town.[40] The 14th was transferred to Castle Island without incident about a week later, with the 29th following shortly after.[41] The first four victims were buried amid great ceremony on March 8; Patrick Carr, the fifth and final victim, died on March 14 and was buried with them on March 17.[42]
Mr. John Gillespie, in his deposition, (No. 104) declares that, as he was going to the south end of the town, to meet some friends at a public house, he met several people in the streets in parties [armed with clubs], to the number, as he thinks, of forty or fifty persons; and that while he was sitting with his friends there, several persons of his acquaintance came in to them at different times, and took notice of the numbers of persons they had seen in the street armed in the above manner. [...] About half an hour after eight the bells rung, which [Gillespie] and his company took to be for fire; but they were told by the landlord of the house that it was to collect the mob. Mr. Gillespie upon this resolved to go home, and in his way met numbers of people who were running past him, of whom many were armed with clubs and sticks, and some with other weapons. At the same time a number of people passed by him with two fire-engines, as if there had been a fire in the town. But they were soon told that there was no fire, but that the people were going to fight the soldiers, upon which they immediately quitted the fire-engines, and swore they would go to their assistance. All this happened before the soldiers near the custom-house fired their muskets, which was not till half an hour after nine o'clock; and it [shows] that the inhabitants had formed, and were preparing to execute, a design of attacking the soldiers on that evening.
—Excerpt from A Fair Account, suggesting the colonists planned the attack on the soldiers[43]
On March 27 the eight soldiers, Captain Preston, and four civilians who were in the Customs House and were alleged to have fired shots, were all indicted for murder.[44] Bostonians continued to be hostile to the troops and their dependents. General Gage, convinced the troops were doing more harm than good, ordered the 29th Regiment out of the province in May.[45] Governor Hutchinson took advantage of the ongoing high tensions to orchestrate delays of the trials until later in the year.[46]
Propaganda battle
In the days and weeks following the incident, a propaganda battle was waged between Boston's radicals and supporters of the government. Both sides published pamphlets that told strikingly different stories, which were principally published in London in a bid to influence opinion there. The Boston Gazette's version of events, for example, characterized the massacre as part of an ongoing scheme to "quell a Spirit of Liberty", and harped on the negative consequences of quartering troops in the city.[47]
Henry Pelham's engraving copied by Paul Revere
A young Boston artist, Henry Pelham, half-brother of the celebrated portrait painter John Singleton Copley, depicted the event. Silversmith and engraver Paul Revere closely copied Pelham's image, and is often credited as its originator. In order to further public outrage, the engraving contained several inflammatory details. Captain Preston is shown ordering his men to fire, and a musket is seen shooting out of the window of the customs office, which is labeled "Butcher's Hall."[48] Artist Christian Remick hand-colored some prints.[1] Some copies of the print show a man with two chest wounds and a somewhat darker face, matching descriptions of Attucks; others show no victim as a person of color. The image was published in the Boston Gazette, circulating widely, and became an effective piece of anti-British propaganda. The image of bright red "lobster backs" and wounded men with red blood was hung in farmhouses across New England.[49]
Anonymous pamphlets were published describing the event from significantly different perspectives. A Short Narrative of the Horrid Massacre, published under the auspices of the Boston town meeting, was principally written by James Bowdoin, a member of the governor's council and a vocal opponent of British colonial policy, along with Samuel Pemberton, and Joseph Warren.[50] It described the shooting and other lesser incidents that took place in the days before as unprovoked attacks on peaceful, law-abiding inhabitants, and was, according to historian Neal Langley York, probably the most influential description of the event.[51] The account it provided was drawn from more than 90 depositions taken after the event, and it included accusations that the soldiers sent by Captain Preston had been deployed with the intention of causing harm.[52] In the interest of minimizing impact on the jury pool, city leaders held back local distribution of the pamphlet, but sent copies to other colonies and to London, where they knew depositions collected by Governor Hutchinson were en route.[53] A second pamphlet, Additional Observations on the Short Narrative, furthered the attack on crown officials by complaining that customs officials (one of whom had left Boston to carry Hutchinson's gathered depositions to London) were abandoning their posts under the pretense that it was too dangerous for them to do their duties.[54]
The depositions that Hutchinson collected and sent to London were eventually published in a pamphlet entitled A Fair Account of the Late Unhappy Disturbance in Boston.[55] Drawn mainly from depositions by soldiers, its account of affairs sought to blame selfish Bostonians for denying the validity of Parliamentary laws. It also blamed the city's hoodlums and gangs for the lawlessness preceding the event, and claimed that they set up an ambush of the soldiers.[56] However, as it was not published until well after the first pamphlet had arrived in London, it ended up having a much smaller impact on the public debate there.[55]
Trial
John Adams defended the soldiers six of whom were acquitted
The Part I took in Defence of Cptn. Preston and the Soldiers, procured me Anxiety, and Obloquy enough. It was, however, one of the most gallant, generous, manly and disinterested Actions of my whole Life, and one of the best Pieces of Service I ever rendered my Country. Judgment of Death against those Soldiers would have been as foul a Stain upon this Country as the Executions of the Quakers or Witches, anciently. As the Evidence was, the Verdict of the Jury was exactly right. This however is no Reason why the Town should not call the Action of that Night a Massacre, nor is it any Argument in favour of the Governor or Minister, who caused them to be sent here. But it is the strongest Proofs of the Danger of Standing Armies.
—  John Adams, on the third anniversary of the massacre[57]
The government was determined to give the soldiers a fair trial so there could be no grounds for retaliation from the British and so moderates would not be alienated from the Patriot cause. After several lawyers with Loyalist leanings refused to defend him, Preston sent a request to John Adams, pleading for him to work on the case. Adams, who was already a leading Patriot and who was contemplating a run for public office, agreed to help, in the interest of ensuring a fair trial.[58] Adams was joined by Josiah Quincy II after the latter was assured that the Sons of Liberty would not oppose his appointment, and by Robert Auchmuty, a Loyalist.[59] They were assisted by Sampson Salter Blowers, whose chief duty was to investigate the jury pool, and Paul Revere, who drew a detailed map of the bodies to be used in the trial of the British soldiers held responsible.[60][61] Massachusetts Solicitor General Samuel Quincy and private attorney Robert Treat Paine, hired by the town of Boston, handled the prosecution.[62] Preston was tried separately in late October 1770. He was acquitted after the jury was convinced that he had not ordered the troops to fire.[63]
The trial of the eight soldiers opened on November 27, 1770.[64] Adams told the jury to look beyond the fact the soldiers were British. He argued that if the soldiers were endangered by the mob, which he called "a motley rabble of saucy boys, negroes, and molattoes, Irish teagues and outlandish jack tarrs [i.e. sailors]",[65] they had the legal right to fight back, and so were innocent. If they were provoked but not endangered, he argued, they were at most guilty of manslaughter.[66]
The jury agreed with Adams and acquitted six of the soldiers after two and one-half hours' deliberation. Two of the soldiers were found guilty of manslaughter because there was overwhelming evidence that they had fired directly into the crowd. The jury's decisions suggest that they believed the soldiers had felt threatened by the crowd, but should have delayed firing.[67] Patrick Carr, the fifth victim, corroborated this with deathbed testimony delivered to his doctor. The convicted soldiers were granted reduced sentences by invoking Benefit of clergy, which reduced their punishment from a death sentence to branding of the thumb in open court.[68]
The four civilians were tried on December 13.[69] The principal prosecution witness, a servant of one of the accused, made claims that were easily rebutted by defense witnesses. In the face of this weak testimony, as well as waning public interest, the prosecution allegedly failed to press its case very hard. The civilians were all acquitted, and the servant was eventually convicted of perjury, whipped, and banished from the province.[70]
Legacy
Contribution to American Revolution
The Boston Massacre is considered one of the most important events that turned colonial sentiment against King George III and British Parliamentary authority.
Howard Zinn argues that Boston was full of class anger. He reports that in 1763, the Boston Gazette published that "a few persons in power" were promoting political projects "for keeping the people poor in order to make them humble."[71]
John Adams wrote that the "foundation of American independence was laid" on March 5, 1770, and Samuel Adams and other Patriots used annual commemorations (Massacre Day) of the event to fulminate against British rule.[72] Christopher Monk, the boy who was wounded in the attack and died in 1780, was paraded before the crowds as a reminder of British hostility.[33]
Later events, such as the Boston Tea Party, further illustrated the crumbling relationship between Britain and its colonies. Although five years passed between the massacre and outright revolution, and direct connections between the massacre and the later war are (according to historian Neil Langley York) somewhat tenuous,[73] it is widely perceived as a significant event leading to the violent rebellion that followed.[74][75]
Commemorations
Massacre memorial on the Boston Common
See also: Massacre Day and Freedom Trail
The massacre was remembered in 1858 in a celebration organized by William Cooper Nell, an African American abolitionist who saw the death of Crispus Attucks as an opportunity to demonstrate the role of African Americans in the Revolutionary War.[76] Partly because of this activism, artwork commemorating the massacre was produced that changed the color of a victim's skin to black (which in Revere's original engraving was white) to emphasize Attucks' claimed martyrdom.[77] In 1888, a monument was erected on the Boston Common to the men killed in the massacre, and the five victims, along with Christopher Seider, were reinterred in a prominent grave in the Granary Burying Ground.[78]
The massacre is reenacted annually on March 5[79] under the auspices of the Bostonian Society.[80][81] The Old State House, the massacre site, and the Granary Burying Ground are all part of Boston's Freedom Trail, connecting sites important in the city's revolutionary-era history.[78]
See also
Boston portal
List of massacres in Massachusetts
Timeline of United States revolutionary history (1760–1789)
References
^ Jump up to: a b Fischer, Paul Revere's Ride, 24.
Jump up ^ Antal, "7 Leadership Lessons of the American Revolution", 40.
Jump up ^ The Boston Massacre
Jump up ^ The American Past: A Survey of American History; Joseph Conlin P. 133
Jump up ^ Historical Dictionary of American Propaganda. Martin J. Manning Page 33
^ Jump up to: a b c Knollenberg, Growth, p. 54.
Jump up ^ Ross and McCaughey, From Loyalist to Founding Father, p. 94.
Jump up ^ Knollenberg, Growth, p. 56.
Jump up ^ Triber, A True Republican, p. 66.
Jump up ^ Knollenberg, Growth, p. 63.
Jump up ^ Triber, A True Republican, p. 63.
Jump up ^ Ross and McCaughey, From Loyalist to Founding Father, p. 93.
Jump up ^ A Short Narrative, p. 17.
Jump up ^ Knollenberg, Growth, 75.
Jump up ^ Knollenberg, Growth, p. 76.
^ Jump up to: a b Knollenberg, Growth, pp. 76–78.
Jump up ^ Middlekauff, Glorious Cause, pp. 208–210.
Jump up ^ O'Connor, The Hub, p. 56.
^ Jump up to: a b Allison, The Boston Massacre, 11.
Jump up ^ Zobel, The Boston Massacre, pp. 185–86.
Jump up ^ Allison, The Boston Massacre, p. 12.
Jump up ^ Archer, As if an Enemy's Country, p. 190.
Jump up ^ Zobel, Boston Massacre, p. 194.
Jump up ^ Archer, As if an Enemy's Country, p. 191.
Jump up ^ Zobel, Boston Massacre, 195.
Jump up ^ Zobel, Boston Massacre, p. 196.
^ Jump up to: a b Zobel, Boston Massacre, p. 197.
Jump up ^ Zobel, Boston Massacre, 198–200.
^ Jump up to: a b A Short Narrative, 11.
Jump up ^ Zobel, Boston Massacre, 191.
Jump up ^ Archer, As If an Enemy's Country, 196.
Jump up ^ Allison, The Boston Massacre, x.
^ Jump up to: a b Miller, Origins of the American Revolution, p. 395.
Jump up ^ Zobel, Boston Massacre, p. 201.
Jump up ^ York, "Rival Truths", p. 61.
^ Jump up to: a b Bailyn, Ordeal, p. 158.
Jump up ^ Zobel, Boston Massacre, p. 205.
Jump up ^ York, "Rival Truths", p. 63.
Jump up ^ Bailyn, Ordeal, p. 159.
^ Jump up to: a b Bailyn, Ordeal, p. 161.
Jump up ^ York, "Rival Truths", p. 64.
Jump up ^ York, "Rival Truths", p. 66.
Jump up ^ A Fair Account, pp. 14–15
Jump up ^ York, "Rival Truths", pp. 59–60.
Jump up ^ Zobel, Boston Massacre, p. 228.
Jump up ^ Zobel, Boston Massacre, p. 229.
Jump up ^ York, "Rival Truths", p. 68.
Jump up ^ Triber, A True Republican, p. 80.
Jump up ^ Ross, Jane (April 1975). "Paul Revere – Patriot Engraver". Early American Life: 34–37.
Jump up ^ Walett, pp. 330–33.
Jump up ^ York, "Rival Truths", p. 70.
Jump up ^ York, "Rival Truths", p. 72.
Jump up ^ York, "Rival Truths", pp. 73–74.
Jump up ^ York, "Rival Truths", p. 77.
^ Jump up to: a b York, "Rival Truths", p. 74.
Jump up ^ York, "Rival Truths", p. 75.
Jump up ^ Adams and Butterfield, Diary, p. 79.
Jump up ^ Zobel, Boston Massacre, p. 220.
Jump up ^ Zobel, Boston Massacre, pp. 220–21.
Jump up ^ Zobel, Boston Massacre, p. 268.
Jump up ^ Cumming, The Fate of a Nation, p. 24.
Jump up ^ York, "Rival Truths", p. 81.
Jump up ^ Zobel, Boston Massacre, pp. 243–65.
Jump up ^ Zobel, Boston Massacre, p. 269.
Jump up ^ Zinn, A People's History of the United States, p. 67.
Jump up ^ Zobel, Boston Massacre, p. 291.
Jump up ^ Zobel, Boston Massacre, p. 294.
Jump up ^ Zobel, Boston Massacre, pp. 285–86, 298.
Jump up ^ York, "Rival Truths", p. 84.
Jump up ^ York, "Rival Truths", p. 85.
Jump up ^ Zinn 1980.
Jump up ^ Zobel, Boston Massacre, pp. 301–02.
Jump up ^ York, The Boston Massacre, pp. 46–47.
Jump up ^ Woods, Exploring American History, p. 149.
Jump up ^ Wheeler et al, Discovering the American Past, pp. 101–02.
Jump up ^ Nell et al, William Cooper Nell, 515.
Jump up ^ Fitz, Karsten (2005). "Commemorating Crispus Attucks: Visual Memory and the Representations of the Boston Massacre, 1770–1857". American Studies (Volume 50, No. 3): 463–484. JSTOR 41158169.
^ Jump up to: a b York, The Boston Massacre, p. 46.
Jump up ^ "The Boston Massacre". The Freedom Trail Foundation. Archived from the original on November 25, 2011. Retrieved November 21, 2011.
Jump up ^ "Massacre Reenactment Registration". Bostonian Society. Retrieved November 21, 2011.
Jump up ^ Young, p. 20
Sources
A Fair Account of the Late Unhappy Disturbance at Boston. London: B. White. 1770. OCLC 535966548. Original printing of the governor's account.
A Short Narrative of the Horrid Massacre. London: W. Bingley. 1770. OCLC 510892519. Original printing of the colonists' account.
Adams, John (1962). Butterfield, L. H., ed. Diary and Autobiography of John Adams, vol. 2. Cambridge, MA: The Belknap Press of Harvard University Press. OCLC 19993300.
Allison, Robert J. (2006). The Boston Massacre. Beverly, MA: Applewood Books. ISBN 978-1-933212-10-4. OCLC 66392877.
Archer, Richard (2010). As if an Enemy's Country: the British Occupation of Boston and the Origins of Revolution. Oxford, New York: Oxford University Press. ISBN 978-0-19-538247-1. OCLC 313664751.
Bailyn, Bernard (1974). The Ordeal of Thomas Hutchinson. Cambridge, MA: Harvard University Press. ISBN 978-0-674-64160-0. OCLC 6825524.
Cumming, William P.; Rankin, Hugh F. (1975). The Fate of a Nation: The American Revolution Through Contemporary Eyes. New York: Phaidon Press. ISBN 978-0-7148-1644-9. OCLC 1510269.
Fischer, David Hackett (1994). Paul Revere's Ride. New York: Oxford University Press. ISBN 978-0-19-508847-2. OCLC 263430392.
Knollenberg, Bernhard (1975). Growth of the American Revolution, 1766–1775. New York: Free Press. ISBN 978-0-02-917110-3. OCLC 1416300.
Middlekauff, Robert (2007). The Glorious Cause: The American Revolution, 1763–1789. New York and Oxford: Oxford University Press. ISBN 978-0-19-516247-9. OCLC 496757346.
Miller, John (1959). Origins of the American Revolution. Stanford, CA: Stanford University Press. ISBN 9780804705936. OCLC 180556929.
Nell, William Cooper; Wesley, Dorothy Porter; Uzelac, Constance Porter (2002). William Cooper Nell, Nineteenth-Century African American Abolitionist, Historian, Integrationist: Selected Writings from 1832–1874. Baltimore, MD: Black Classic Press. ISBN 978-1-57478-019-2. OCLC 50673509.
O'Connor, Thomas H (2001). The Hub: Boston Past and Present. Boston: Northeastern University Press. ISBN 978-1-55553-474-5. OCLC 248309644.
Ross, Betsy McCaughey; McCaughey, Elizabeth P. (1980). From Loyalist to Founding Father: the Political Odyssey of William Samuel Johnson. New York: Columbia University Press. ISBN 978-0-231-04506-3. OCLC 479827879.
Triber, Jayne (1998). A True Republican: The Life of Paul Revere. Amherst, MA: University of Massachusetts Press. ISBN 978-1-55849-139-7. OCLC 171052850.
Walett, Francis (September 1950). "James Bowdoin, Patriot Propagandist". The New England Quarterly (Volume 23, No. 3). JSTOR 361420.
Wheeler, William Bruce; Becker, Susan Becker; Glover, Lorri (2011). Discovering the American Past: A Look at the Evidence: To 1877. Belmont, CA: Cengage Learning. ISBN 978-0-495-79984-9. OCLC 664324291.
Woods, Thomas (2008). Exploring American History: From Colonial Times to 1877. New York: Associated University Press. ISBN 978-0-7614-7746-4. OCLC 302384920.
York, Neil Longley (2009). "Rival Truths, Political Accommodation, and the Boston 'Massacre'". Massachusetts Historical Review (Volume 11): 57–95. JSTOR 40345980.
York, Neil Longley (2010). The Boston Massacre: a History with Documents. New York: Taylor & Francis. ISBN 978-0-415-87348-2. OCLC 695944958.
Young, Alfred (Spring 2003). "Revolution in Boston? Eight Propositions for Public History on the Freedom Trail" (PDF). The Public Historian (Volume 25, No. 2): 17–41.
Zinn, Howard (1980). A People's History of the United States. New York: HarperCollins. ISBN 978-0-06-019448-2. OCLC 42420960.
Zobel, Hiller B (1970). The Boston Massacre. New York: W.W. Norton & Company. ISBN 978-0-393-31483-0. OCLC 243696768.
Further reading
Reid, John Phillip (1974). "A Lawyer Acquitted: John Adams and the Boston Massacre". American Journal of Legal History (Volume 18, No. 3): 189–207. ISSN 0002-9319.
Ritter, Kurt W (1977). "Confrontation as Moral Drama: the Boston Massacre in Rhetorical Perspective". Southern Speech Communication Journal (Volume 42, No. 1): 114–136. ISSN 0361-8269.
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Positional notation
From Wikipedia, the free encyclopedia
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Numeral systems
Hindu–Arabic numeral system
Western Arabic
Eastern Arabic
Bengali
Gurmukhi
Indian
Sinhala
Tamil
Balinese
Burmese
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Mongolian
Thai
East Asian
Chinese
Suzhou
Hokkien
Japanese
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Counting rods
Alphabetic
Abjad
Armenian
Āryabhaṭa
Cyrillic

Ge'ez
Georgian
Greek
Hebrew
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Former
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Attic
Babylonian
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Egyptian
Etruscan
Inuit
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Mayan
Muisca
Quipu
Prehistoric
Positional systems by base
2
3
4
5
6
8
10
12
16
20
60
Non-standard positional numeral systems
Bijective numeration (1)
Signed-digit representation (Balanced ternary)
factorial
negative
Complex-base system (2i)
Non-integer representation (φ)
mixed
List of numeral systems
v
t
e
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations (such as Roman numerals) for its use of the same symbol for the different orders of magnitude (for example, the "ones place", "tens place", "hundreds place"). This greatly simplified arithmetic, leading to the rapid spread of the notation across the world.
With the use of a radix point (decimal point in base-10), the notation can be extended to include fractions and the numeric expansions of real numbers.
The Babylonian numeral system, base-60, was the first positional system developed, and its influence is present today in the way time and angles are counted in tallies related to 60, like 60 minutes in an hour, 360 degrees in a circle. The Hindu–Arabic numeral system, base-10, is the most commonly used system in the world today for most calculations.
Contents
[hide]
1 History
1.1 History of positional fractions
2 Issues
3 Mathematics
3.1 Base of the numeral system
3.2 Notation
3.3 Exponentiation
3.4 Digits and numerals
3.5 Radix point
3.6 Sign
3.7 Base conversion
3.8 Terminating fractions
3.9 Infinite representations
4 Applications
4.1 Decimal system
4.2 Sexagesimal system
4.3 Computing
4.4 Other bases in human language
5 Non-standard positional numeral systems
6 Non-positional positions
7 See also
8 Notes
9 References
10 External links
History[edit]
Suanpan (the number represented in the picture is 6,302,715,408)
Today, the base-10 (decimal) system, which is likely motivated by counting with the ten fingers, is ubiquitous. Other bases have been used in the past however, and some continue to be used today. For example, the Babylonian numeral system, credited as the first positional numeral system, was base-60, but it lacked a real 0 value. Zero was indicated by a space between sexagesimal numerals. By 300 BC, a punctuation symbol (two slanted wedges) was co-opted as a placeholder in the same system. In a tablet unearthed at Kish (dating from about 700 BC), the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges.[1] The Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number. Thus numbers like 2 and 120 (2×60), 3 and 180 (3×60), 4 and 240 (4×60), looked the same because the larger numbers lacked a final sexagesimal placeholder. Only context could differentiate them.
The polymath Archimedes (ca. 287–212 BC) invented a decimal positional system in his Sand Reckoner which was based on 108[2] and later led the German mathematician Carl Friedrich Gauss to lament what heights science would have already reached in his days if Archimedes had fully realized the potential of his ingenious discovery.[3]
Before positional notation became standard, simple additive systems (sign-value notation) such as Roman numerals were used, and accountants in ancient Rome and during the Middle Ages used the abacus or stone counters to do arithmetic.[4]
The world's earliest positional decimal system
Upper row vertical form
Lower row horizontal form
Counting rods and most abacuses have been used to represent numbers in a positional numeral system. With counting rods or abacus to perform arithmetic operations, the writing of the starting, intermediate and final values of a calculation could easily be done with a simple additive system in each position or column. This approach required no memorization of tables (as does positional notation) and could produce practical results quickly. For four centuries (from the 13th to the 16th) there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the latter continues to be used in Japan and other Asian countries.
After the French Revolution (1789-1799), the new French government promoted the extension of the decimal system.[5] Some of those pro-decimal efforts—such as decimal time and the decimal calendar—were unsuccessful. Other French pro-decimal efforts—currency decimalisation and the metrication of weights and measures—spread widely out of France to almost the whole world.
History of positional fractions[edit]
Main article: Decimal
counting rod decimal fraction 1/7
According to Joseph Needham and Lam Lay Yong, decimal fractions were first developed and used by the Chinese in the 1st century BC, and then spread to the Middle East and from there to Europe.[6][7] The written Chinese decimal fractions were non-positional.[7] However, counting rod fractions were positional.[6]
Qin Jiushao in his book Mathematical Treatise in Nine Sections (1247) denoted 0.96644 by
寸
, meaning
寸
096644
[8]
J. Lennart Berggren notes that positional decimal fractions for the first time by Arab mathematician Abu'l-Hasan al-Uqlidisi as early as the 10th century.[9] The Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them.[10] The Persian mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century.[9]Al Khwarizmi introduced fraction to Islamic countries in the early 9th century, his fraction presentation was an exact copy of traditional Chinese mathematical fraction from Sunzi Suanjing.[6] This form of fraction with numerator on top and denominator at bottom without a horizontal bar was also used by 10th century Abu'l-Hasan al-Uqlidisi and 15th century Jamshīd al-Kāshī's work "Arithmetic Key".[6][11]
A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century.[12]
Issues[edit]
A key argument against the positional system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing (e.g.) 100 into 5100, or 100 into 1000. Modern cheques require a natural language spelling of an amount, as well as the decimal amount itself, to prevent such fraud. For the same reason the Chinese also use natural language numerals, for example 100 is written as 壹佰, which can never be forged into 壹仟(1000) or 伍仟壹佰(5100).
Many of the advantages claimed for the metric system could be realized by any consistent positional notation. Dozenal advocates say dozenal has several advantages over decimal, although the switching cost appears to be high.
Mathematics[edit]
This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (March 2013) (Learn how and when to remove this template message)
Base of the numeral system[edit]
In mathematical numeral systems the base or radix is usually the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is 10, because it uses the 10 digits from 0 through 9. When a number "hits" 9, the next number will not be another different symbol, but a "1" followed by a "0". In binary, the radix is 2, since after it hits "1", instead of "2" or another written symbol, it jumps straight to "10", followed by "11" and "100".
The highest symbol of a positional numeral system usually has the value one less than the value of the base of that numeral system. The standard positional numeral systems differ from one another only in the base they use.
The base is an integer that is greater than 1 (or less than negative 1), since a radix of zero would not have any digits, and a radix of 1 would only have the zero digit. Negative bases are rarely used. In a system with a negative radix, numbers may have many different possible representations.
(In certain non-standard positional numeral systems, including bijective numeration, the definition of the base or the allowed digits deviates from the above.)
In base-10 (decimal) positional notation, there are 10 decimal digits and the number
2506 = 2 × 10 3 + 5 × 10 2 + 0 × 10 1 + 6 × 10 0 {\displaystyle 2506=2\times 10^{3}+5\times 10^{2}+0\times 10^{1}+6\times 10^{0}} .
In base-16 (hexadecimal), there are 16 hexadecimal digits (0–9 and A–F) and the number
171 B = 1 × 16 3 + 7 × 16 2 + 1 × 16 1 + B × 16 0 {\displaystyle 171\mathrm {B} =1\times 16^{3}+7\times 16^{2}+1\times 16^{1}+\mathrm {B} \times 16^{0}} (where B represents the number eleven as a single symbol)
In general, in base-b, there are b digits and the number
a 3 a 2 a 1 a 0 = a 3 × b 3 + a 2 × b 2 + a 1 × b 1 + a 0 × b 0 {\displaystyle a_{3}a_{2}a_{1}a_{0}=a_{3}\times b^{3}+a_{2}\times b^{2}+a_{1}\times b^{1}+a_{0}\times b^{0}} (Note that a 3 a 2 a 1 a 0 {\displaystyle a_{3}a_{2}a_{1}a_{0}} represents a sequence of digits, not multiplication)
Notation[edit]
Sometimes the base number is written in subscript after the number represented. For example, 238 indicates that the number 23 is expressed in base 8 (and is therefore equivalent in value to the decimal number 19). This notation will be used in this article.
When describing base in mathematical notation, the letter b is generally used as a symbol for this concept, so, for a binary system, b equals 2. Another common way of expressing the base is writing it as a decimal subscript after the number that is being represented. 11110112 implies that the number 1111011 is a base-2 number, equal to 12310 (a decimal notation representation), 1738 (octal) and 7B16 (hexadecimal). In books and articles, when using initially the written abbreviations of number bases, the base is not subsequently printed: it is assumed that binary 1111011 is the same as 11110112.
The base b may also be indicated by the phrase "base-b". So binary numbers are "base-2"; octal numbers are "base-8"; decimal numbers are "base-10"; and so on.
To a given radix b the set of digits {0, 1, ..., b−2, b−1} is called the standard set of digits. Thus, binary numbers have digits {0, 1}; decimal numbers have digits {0, 1, 2, ..., 8, 9}; and so on. Therefore, the following are notational errors: 522, 22, 1A9. (In all cases, one or more digits is not in the set of allowed digits for the given base.)
Exponentiation[edit]
Positional numeral systems work using exponentiation of the base. A digit's value is the digit multiplied by the value of its place. Place values are the number of the base raised to the nth power, where n is the number of other digits between a given digit and the radix point. If a given digit is on the left hand side of the radix point (i.e. its value is an integer) then n is positive or zero; if the digit is on the right hand side of the radix point (i.e., its value is fractional) then n is negative.
As an example of usage, the number 465 in its respective base b (which must be at least base 7 because the highest digit in it is 6) is equal to:
4 × b 2 + 6 × b 1 + 5 × b 0 {\displaystyle 4\times b^{2}+6\times b^{1}+5\times b^{0}}
If the number 465 was in base-10, then it would equal:
4 × 10 2 + 6 × 10 1 + 5 × 10 0 = 4 × 100 + 6 × 10 + 5 × 1 = 465 {\displaystyle 4\times 10^{2}+6\times 10^{1}+5\times 10^{0}=4\times 100+6\times 10+5\times 1=465}
(46510 = 46510)
If however, the number were in base 7, then it would equal:
4 × 7 2 + 6 × 7 1 + 5 × 7 0 = 4 × 49 + 6 × 7 + 5 × 1 = 243 {\displaystyle 4\times 7^{2}+6\times 7^{1}+5\times 7^{0}=4\times 49+6\times 7+5\times 1=243}
(4657 = 24310)
10b = b for any base b, since 10b = 1×b1 + 0×b0. For example, 102 = 2; 103 = 3; 1016 = 1610. Note that the last "16" is indicated to be in base 10. The base makes no difference for one-digit numerals.
This concept can be demonstrated using a diagram. One object represents one unit. When the number of objects is equal to or greater than the base b, then a group of objects is created with b objects. When the number of these groups exceeds b, then a group of these groups of objects is created with b groups of b objects; and so on. Thus the same number in different bases will have different values:
241 in base 5:
   2 groups of 52 (25) 4 groups of 5 1 group of 1 ooooo ooooo ooooo ooooo ooooo ooooo ooooo ooooo + + o ooooo ooooo ooooo ooooo ooooo ooooo

241 in base 8:
   2 groups of 82 (64) 4 groups of 8 1 group of 1 oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo + + o oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo

The notation can be further augmented by allowing a leading minus sign. This allows the representation of negative numbers. For a given base, every representation corresponds to exactly one real number and every real number has at least one representation. The representations of rational numbers are those representations that are finite, use the bar notation, or end with an infinitely repeating cycle of digits.
Digits and numerals[edit]
A digit is what is used as a position in place-value notation, and a numeral is one or more digits. Today's most common digits are the decimal digits "0", "1", "2", "3", "4", "5", "6", "7", "8", and "9". The distinction between a digit and a numeral is most pronounced in the context of a number base.
A non-zero numeral with more than one digit position will mean a different number in a different number base, but in general, the digits will mean the same.[13] The base-8 numeral 238 contains two digits, "2" and "3", and with a base number (subscripted) "8", means 19. In our notation here, the subscript "8" of the numeral 238 is part of the numeral, but this may not always be the case. Imagine the numeral "23" as having an ambiguous base number. Then "23" could likely be any base, base-4 through base-60. In base-4 "23" means 11, and in base-60 it means the number 123. The numeral "23" then, in this case, corresponds to the set of numbers {11, 13, 15, 17, 19, 21, 23, ..., 121, 123} while its digits "2" and "3" always retain their original meaning: the "2" means "two of", and the "3" three.
In certain applications when a numeral with a fixed number of positions needs to represent a greater number, a higher number-base with more digits per position can be used. A three-digit, decimal numeral can represent only up to 999. But if the number-base is increased to 11, say, by adding the digit "A", then the same three positions, maximized to "AAA", can represent a number as great as 1330. We could increase the number base again and assign "B" to 11, and so on (but there is also a possible encryption between number and digit in the number-digit-numeral hierarchy). A three-digit numeral "ZZZ" in base-60 could mean 7005215999000000000♠215999. If we use the entire collection of our alphanumerics we could ultimately serve a base-62 numeral system, but we remove two digits, uppercase "I" and uppercase "O", to reduce confusion with digits "1" and "0".[14] We are left with a base-60, or sexagesimal numeral system utilizing 60 of the 62 standard alphanumerics. (But see Sexagesimal system below.) In general, the number of possible values that can be represented by a d {\displaystyle d} digit number in base r {\displaystyle r} is r d {\displaystyle r^{d}} .
The common numeral systems in computer science are binary (radix 2), octal (radix 8), and hexadecimal (radix 16). In binary only digits "0" and "1" are in the numerals. In the octal numerals, are the eight digits 0–7. Hex is 0–9 A–F, where the ten numerics retain their usual meaning, and the alphabetics correspond to values 10–15, for a total of sixteen digits. The numeral "10" is binary numeral "2", octal numeral "8", or hexadecimal numeral "16".
Radix point[edit]
Main article: Radix point
The notation can be extended into the negative exponents of the base b. Thereby the so-called radix point, mostly ».«, is used as separator of the positions with non-negative from those with negative exponent.
Numbers that are not integers use places beyond the radix point. For every position behind this point (and thus after the units digit), the exponent n of the power bn decreases by 1 and the power approaches 0. For example, the number 2.35 is equal to:
2 × 10 0 + 3 × 10 − 1 + 5 × 10 − 2 {\displaystyle 2\times 10^{0}+3\times 10^{-1}+5\times 10^{-2}}
Sign[edit]
Main article: Sign (mathematics)
If the base and all the digits in the set of digits are non-negative, negative numbers cannot be expressed. To overcome this, a minus sign, here »-«, is added to the numeral system. In the usual notation it is prepended to the string of digits representing the otherwise non-negative number.
Base conversion[edit]
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The conversion to a base b 2 {\displaystyle b_{2}} of an integer n represented in base b 1 {\displaystyle b_{1}} can be done by a succession of Euclidean divisions by b 2 : {\displaystyle b_{2}:} the right-most digit in base b 2 {\displaystyle b_{2}} is the remainder of the division of n by b 2 ; {\displaystyle b_{2};} the second right-most digit is the remainder of the division of the quotient by b 2 , {\displaystyle b_{2},} and so on. More precisely, the kth digit from the right is the remainder of the division by b 2 {\displaystyle b_{2}} of the (k−1)th quotient.
Terminating fractions[edit]
The numbers which have a finite representation form the semiring
N 0 b N 0 := { m b − ν ∣ m ∈ N 0 ∧ ν ∈ N 0 } . {\displaystyle {\frac {\mathbb {N} _{0}}{b^{\mathbb {N} _{0}}}}:=\left\{mb^{-\nu }\mid m\in \mathbb {N} _{0}\wedge \nu \in \mathbb {N} _{0}\right\}.}
More explicitly, if p 1 ν 1 ⋅ … ⋅ p n ν n := b {\displaystyle p_{1}^{\nu _{1}}\cdot \ldots \cdot p_{n}^{\nu _{n}}:=b} is a factorization of b {\displaystyle b} into the primes p 1 , … , p n ∈ P {\displaystyle p_{1},\ldots ,p_{n}\in \mathbb {P} } with exponents ν 1 , … , ν n ∈ N {\displaystyle \nu _{1},\ldots ,\nu _{n}\in \mathbb {N} } ,[15] then with the non-empty set of denominators S := { p 1 , … , p n } {\displaystyle S:=\{p_{1},\ldots ,p_{n}\}} we have
Z S := { x ∈ Q | ∃ μ i ∈ Z : x ∏ i = 1 n p i μ i ∈ Z } = b Z Z = ⟨ S ⟩ − 1 Z {\displaystyle \mathbb {Z} _{S}:=\left\{x\in \mathbb {Q} \left|\,\exists \mu _{i}\in \mathbb {Z} :x\prod _{i=1}^{n}{p_{i}}^{\mu _{i}}\in \mathbb {Z} \right.\right\}=b^{\mathbb {Z} }\,\mathbb {Z} ={\langle S\rangle }^{-1}\mathbb {Z} }
where ⟨ S ⟩ {\displaystyle \langle S\rangle } is the group generated by the p ∈ S {\displaystyle p\in S} and ⟨ S ⟩ − 1 Z {\displaystyle {\langle S\rangle }^{-1}\mathbb {Z} } is the so-called localization of Z {\displaystyle \mathbb {Z} } with respect to S {\displaystyle S} .
The denominator of an element of Z S {\displaystyle \mathbb {Z} _{S}} contains if reduced to lowest terms only prime factors out of S {\displaystyle S} . This ring of all terminating fractions to base b {\displaystyle b} is dense in the field of rational numbers Q {\displaystyle \mathbb {Q} } . Its completion for the usual (Archimedean) metric is the same as for Q {\displaystyle \mathbb {Q} } , namely the real numbers R {\displaystyle \mathbb {R} } . So, if S = { p } {\displaystyle S=\{p\}} then Z { p } {\displaystyle \mathbb {Z} _{\{p\}}} has not to be confused with Z ( p ) {\displaystyle \mathbb {Z} _{(p)}} , the discrete valuation ring for the prime p {\displaystyle p} , which is equal to Z T {\displaystyle \mathbb {Z} _{T}} with T = P ∖ { p } {\displaystyle T=\mathbb {P} \setminus \{p\}} .
If b {\displaystyle b} divides c {\displaystyle c} , we have b Z Z ⊆ c Z Z . {\displaystyle b^{\mathbb {Z} }\,\mathbb {Z} \subseteq c^{\mathbb {Z} }\,\mathbb {Z} .}
Infinite representations[edit]
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The representation of non-integers can be extended to allow an infinite string of digits beyond the point. For example, 1.12112111211112 ... base-3 represents the sum of the infinite series:
1 × 3 0 + {\displaystyle 1\times 3^{0\,\,\,}+{}}
1 × 3 − 1 + 2 × 3 − 2 + {\displaystyle 1\times 3^{-1\,\,}+2\times 3^{-2\,\,\,}+{}}
1 × 3 − 3 + 1 × 3 − 4 + 2 × 3 − 5 + {\displaystyle 1\times 3^{-3\,\,}+1\times 3^{-4\,\,\,}+2\times 3^{-5\,\,\,}+{}}
1 × 3 − 6 + 1 × 3 − 7 + 1 × 3 − 8 + 2 × 3 − 9 + {\displaystyle 1\times 3^{-6\,\,}+1\times 3^{-7\,\,\,}+1\times 3^{-8\,\,\,}+2\times 3^{-9\,\,\,}+{}}
1 × 3 − 10 + 1 × 3 − 11 + 1 × 3 − 12 + 1 × 3 − 13 + 2 × 3 − 14 + ⋯ {\displaystyle 1\times 3^{-10}+1\times 3^{-11}+1\times 3^{-12}+1\times 3^{-13}+2\times 3^{-14}+\cdots }
Since a complete infinite string of digits cannot be explicitly written, the trailing ellipsis (...) designates the omitted digits, which may or may not follow a pattern of some kind. One common pattern is when a finite sequence of digits repeats infinitely. This is designated by drawing a vinculum across the repeating block:
2.42 314 ¯ 5 = 2.42314314314314314 … 5 {\displaystyle 2.42{\overline {314}}_{5}=2.42314314314314314\dots _{5}}
For base-10 it is called a recurring decimal or repeating decimal.
An irrational number has an infinite non-repeating representation in all integer bases. Whether a rational number has a finite representation or requires an infinite repeating representation depends on the base. For example, one third can be represented by:
0.1 3 {\displaystyle 0.1_{3}}
0. 3 ¯ 10 = 0.3333333 … 10 {\displaystyle 0.{\overline {3}}_{10}=0.3333333\dots _{10}}
or, with the base implied:
0. 3 ¯ = 0.3333333 … {\displaystyle 0.{\overline {3}}=0.3333333\dots }
0. 01 ¯ 2 = 0.010101 … 2 {\displaystyle 0.{\overline {01}}_{2}=0.010101\dots _{2}}
0.2 6 {\displaystyle 0.2_{6}}
For integers p and q with gcd(p, q) = 1, the fraction p/q has a finite representation in base b if and only if each prime factor of q is also a prime factor of b.
For a given base, any number that can be represented by a finite number of digits (without using the bar notation) will have multiple representations, including one or two infinite representations:
1. A finite or infinite number of zeroes can be appended:
3.46 7 = 3.460 7 = 3.460000 7 = 3.46 0 ¯ 7 {\displaystyle 3.46_{7}=3.460_{7}=3.460000_{7}=3.46{\overline {0}}_{7}}
2. The last non-zero digit can be reduced by one and an infinite string of digits, each corresponding to one less than the base, are appended (or replace any following zero digits):
3.46 7 = 3.45 6 ¯ 7 {\displaystyle 3.46_{7}=3.45{\overline {6}}_{7}}
1 10 = 0. 9 ¯ 10 {\displaystyle 1_{10}=0.{\overline {9}}_{10}}
220 5 = 214. 4 ¯ 5 {\displaystyle 220_{5}=214.{\overline {4}}_{5}}
Applications[edit]
Decimal system[edit]
Main article: Decimal representation
In the decimal (base-10) Hindu–Arabic numeral system, each position starting from the right is a higher power of 10. The first position represents 100 (1), the second position 101 (10), the third position 102 (10 × 10 or 100), the fourth position 103 (10 × 10 × 10 or 1000), and so on.
Fractional values are indicated by a separator, which can vary in different locations. Usually this separator is a period or full stop, or a comma. Digits to the right of it are multiplied by 10 raised to a negative power or exponent. The first position to the right of the separator indicates 10−1 (0.1), the second position 10−2 (0.01), and so on for each successive position.
As an example, the number 2674 in a base-10 numeral system is:
(2 × 103) + (6 × 102) + (7 × 101) + (4 × 100)
or
(2 × 1000) + (6 × 100) + (7 × 10) + (4 × 1).
Sexagesimal system[edit]
The sexagesimal or base-60 system was used for the integral and fractional portions of Babylonian numerals and other mesopotamian systems, by Hellenistic astronomers using Greek numerals for the fractional portion only, and is still used for modern time and angles, but only for minutes and seconds. However, not all of these uses were positional.
Modern time separates each position by a colon or point. For example, the time might be 10:25:59 (10 hours 25 minutes 59 seconds). Angles use similar notation. For example, an angle might be 10°25'59" (10 degrees 25 minutes 59 seconds). In both cases, only minutes and seconds use sexagesimal notation—angular degrees can be larger than 59 (one rotation around a circle is 360°, two rotations are 720°, etc.), and both time and angles use decimal fractions of a second. This contrasts with the numbers used by Hellenistic and Renaissance astronomers, who used thirds, fourths, etc. for finer increments. Where we might write 10°25'59.392", they would have written 10°25′59″23‴31⁗12''''' or 10°25I59II23III31IV12V.
Using a digit set of digits with upper and lowercase letters allows short notation for sexagesimal numbers, e.g. 10:25:59 becomes 'ARz' (by omitting I and O, but not i and o), which is useful for use in URLs, etc., but it is not very intelligible to humans.
In the 1930s, Otto Neugebauer introduced a modern notational system for Babylonian and Hellenistic numbers that substitutes modern decimal notation from 0 to 59 in each position, while using a semicolon (;) to separate the integral and fractional portions of the number and using a comma (,) to separate the positions within each portion. For example, the mean synodic month used by both Babylonian and Hellenistic astronomers and still used in the Hebrew calendar is 29;31,50,8,20 days, and the angle used in the example above would be written 10;25,59,23,31,12 degrees.
Computing[edit]
In computing, the binary (base-2), octal (base-8) and hexadecimal (base-16) bases are most commonly used. Computers, at the most basic level, deal only with sequences of conventional zeroes and ones, thus it is easier in this sense to deal with powers of two. The hexadecimal system is used as "shorthand" for binary—every 4 binary digits (bits) relate to one and only one hexadecimal digit. In hexadecimal, the six digits after 9 are denoted by A, B, C, D, E, and F (and sometimes a, b, c, d, e, and f).
The octal numbering system is also used as another way to represent binary numbers. In this case the base is 8 and therefore only digits 0, 1, 2, 3, 4, 5, 6, and 7 are used. When converting from binary to octal every 3 bits relate to one and only one octal digit.
Hexadecimal, decimal, octal, and a wide variety of other bases have been used for binary-to-text encoding, implementations of arbitrary-precision arithmetic, and other applications.
For a list of bases and their applications, see list of numeral systems.
Other bases in human language[edit]
Base-12 systems (duodecimal or dozenal) have been popular because multiplication and division are easier than in base-10, with addition and subtraction being just as easy. Twelve is a useful base because it has many factors. It is the smallest common multiple of one, two, three, four and six. There is still a special word for "dozen" in English, and by analogy with the word for 102, hundred, commerce developed a word for 122, gross. The standard 12-hour clock and common use of 12 in English units emphasize the utility of the base. In addition, prior to its conversion to decimal, the old British currency Pound Sterling (GBP) partially used base-12; there were 12 pence (d) in a shilling (s), 20 shillings in a pound (£), and therefore 240 pence in a pound. Hence the term LSD or, more properly, £sd.
The Maya civilization and other civilizations of pre-Columbian Mesoamerica used base-20 (vigesimal), as did several North American tribes (two being in southern California). Evidence of base-20 counting systems is also found in the languages of central and western Africa.
Remnants of a Gaulish base-20 system also exist in French, as seen today in the names of the numbers from 60 through 99. For example, sixty-five is soixante-cinq (literally, "sixty [and] five"), while seventy-five is soixante-quinze (literally, "sixty [and] fifteen"). Furthermore, for any number between 80 and 99, the "tens-column" number is expressed as a multiple of twenty. For example, eighty-two is quatre-vingt-deux (literally, four twenty[s] [and] two), while ninety-two is quatre-vingt-douze (literally, four twenty[s] [and] twelve). In Old French, forty was expressed as two twenties and sixty was three twenties, so that fifty-three was expressed as two twenties [and] thirteen, and so on.
In English the same base-20 counting appears in the use of "scores". Although mostly historical it is occasionally used colloquially. Verse 10 of Pslam 90 in the King James Version of the Bible starts: "The days of our years are threescore years and ten; and if by reason of strength they be fourscore years, yet is their strength labour and sorrow". The Gettysburg Address starts: "Four score and seven years ago".
The Irish language also used base-20 in the past, twenty being fichid, forty dhá fhichid, sixty trí fhichid and eighty ceithre fhichid. A remnant of this system may be seen in the modern word for 40, daoichead.
The Welsh language continues to use a base-20 counting system, particularly for the age of people, dates and in common phrases. 15 is also important, with 16–19 being "one on 15", "two on 15" etc. 18 is normally "two nines". A decimal system is commonly used.
Danish numerals display a similar base-20 structure.
The Maori language of New Zealand also has evidence of an underlying base-20 system as seen in the terms Te Hokowhitu a Tu referring to a war party (literally "the seven 20s of Tu") and Tama-hokotahi, referring to a great warrior ("the one man equal to 20").
The binary system was used in the Egyptian Old Kingdom, 3000 BC to 2050 BC. It was cursive by rounding off rational numbers smaller than 1 to 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64, with a 1/64 term thrown away (the system was called the Eye of Horus).
A number of Australian Aboriginal languages employ binary or binary-like counting systems. For example, in Kala Lagaw Ya, the numbers one through six are urapon, ukasar, ukasar-urapon, ukasar-ukasar, ukasar-ukasar-urapon, ukasar-ukasar-ukasar.
North and Central American natives used base-4 (quaternary) to represent the four cardinal directions. Mesoamericans tended to add a second base-5 system to create a modified base-20 system.
A base-5 system (quinary) has been used in many cultures for counting. Plainly it is based on the number of digits on a human hand. It may also be regarded as a sub-base of other bases, such as base-10, base-20, and base-60.
A base-8 system (octal) was devised by the Yuki tribe of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight.[16] There is also linguistic evidence which suggests that the Bronze Age Proto-Indo Europeans (from whom most European and Indic languages descend) might have replaced a base-8 system (or a system which could only count up to 8) with a base-10 system. The evidence is that the word for 9, newm, is suggested by some to derive from the word for "new", newo-, suggesting that the number 9 had been recently invented and called the "new number".[17]
Many ancient counting systems use five as a primary base, almost surely coming from the number of fingers on a person's hand. Often these systems are supplemented with a secondary base, sometimes ten, sometimes twenty. In some African languages the word for five is the same as "hand" or "fist" (Dyola language of Guinea-Bissau, Banda language of Central Africa). Counting continues by adding 1, 2, 3, or 4 to combinations of 5, until the secondary base is reached. In the case of twenty, this word often means "man complete". This system is referred to as quinquavigesimal. It is found in many languages of the Sudan region.
The Telefol language, spoken in Papua New Guinea, is notable for possessing a base-27 numeral system.
Non-standard positional numeral systems[edit]
Main article: Non-standard positional numeral systems
Interesting properties exist when the base is not fixed or positive and when the digit symbol sets denote negative values. There are many more variations. These systems are of practical and theoretic value to computer scientists.
Balanced ternary uses a base of 3 but the digit set is {1,0,1} instead of {0,1,2}. The "1" has an equivalent value of −1. The negation of a number is easily formed by switching the on the 1s. This system can be used to solve the balance problem, which requires finding a minimal set of known counter-weights to determine an unknown weight. Weights of 1, 3, 9, ... 3n known units can be used to determine any unknown weight up to 1 + 3 + ... + 3n units. A weight can be used on either side of the balance or not at all. Weights used on the balance pan with the unknown weight are designated with 1, with 1 if used on the empty pan, and with 0 if not used. If an unknown weight W is balanced with 3 (31) on its pan and 1 and 27 (30 and 33) on the other, then its weight in decimal is 25 or 1011 in balanced base-3.
10113 = 1 × 33 + 0 × 32 − 1 × 31 + 1 × 30 = 25.
The factorial number system uses a varying radix, giving factorials as place values; they are related to Chinese remainder theorem and residue number system enumerations. This system effectively enumerates permutations. A derivative of this uses the Towers of Hanoi puzzle configuration as a counting system. The configuration of the towers can be put into 1-to-1 correspondence with the decimal count of the step at which the configuration occurs and vice versa.
Decimal equivalents
−3 −2 −1 0 1 2 3 4 5 6 7 8
Balanced base 3
10 11 1 0 1 11 10 11 111 110 111 101
Base −2
1101 10 11 0 1 110 111 100 101 11010 11011 11000
Factoroid
0 10 100 110 200 210 1000 1010 1100
Non-positional positions[edit]
Each position does not need to be positional itself. Babylonian sexagesimal numerals were positional, but in each position were groups of two kinds of wedges representing ones and tens (a narrow vertical wedge ( | ) and an open left pointing wedge (<))—up to 14 symbols per position (5 tens (<<<<<) and 9 ones ( ||||||||| ) grouped into one or two near squares containing up to three tiers of symbols, or a place holder (\\) for the lack of a position).[18] Hellenistic astronomers used one or two alphabetic Greek numerals for each position (one chosen from 5 letters representing 10–50 and/or one chosen from 9 letters representing 1–9, or a zero symbol).[19]
See also[edit]
Examples:
List of numeral systems
Category:Positional numeral systems
Related topics:
Numeral system
Hindu-Arabic numeral system
Non-standard positional numeral systems
Mixed radix
Algorism
Subtractive notation
Notes[edit]
Jump up ^ Kaplan, Robert. (2000). The Nothing That Is: A Natural History of Zero. Oxford: Oxford University Press.
Jump up ^ Greek numerals
Jump up ^ Menninger, Karl: Zahlwort und Ziffer. Eine Kulturgeschichte der Zahl, Vandenhoeck und Ruprecht, 3rd. ed., 1979, ISBN 3-525-40725-4, pp. 150-153
Jump up ^ Ifrah, page 187
Jump up ^ L. F. Menabrea. Translated by Ada Augusta, Countess of Lovelace. "Sketch of The Analytical Engine Invented by Charles Babbage". 1842.
^ Jump up to: a b c d Lam Lay Yong, "The Development of Hindu-Arabic and Traditional Chinese Arithmetic", Chinese Science, 1996 p38, Kurt Vogel notation
^ Jump up to: a b Joseph Needham (1959). "Decimal System". Science and Civilisation in China, Volume III, Mathematics and the Sciences of the Heavens and the Earth. Cambridge University Press.
Jump up ^ Jean-Claude Martzloff, A History of Chinese Mathematics, Springer 1997 ISBN 3-540-33782-2
^ Jump up to: a b Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 518. ISBN 978-0-691-11485-9.
Jump up ^ Gandz, S.: The invention of the decimal fractions and the application of the exponential calculus by Immanuel Bonﬁls of Tarascon (c. 1350), Isis 25 (1936), 16–45.
Jump up ^ Lam Lay Yong, "A Chinese Genesis, Rewriting the history of our numeral system", Archive for History of Exact Science 38: 101–108.
Jump up ^ B. L. van der Waerden (1985). A History of Algebra. From Khwarizmi to Emmy Noether. Berlin: Springer-Verlag.
Jump up ^ The digit will retain its meaning in other number bases, in general, because a higher number base would normally be a notational extension of the lower number base in any systematic organization. In the mathematical sciences there is virtually only one positional-notation numeral system for each base below 10, and this extends with few, if insignificant, variations on the choice of alphabetic digits for those bases above 10.
Jump up ^ We do not usually remove the lowercase digits "l" and lowercase "o", for in most fonts they are discernible from the digits "1" and "0".
Jump up ^ The exact size of the ν 1 , … , ν n {\displaystyle \nu _{1},\ldots ,\nu _{n}} does not matter. They only have to be ≥ 1.
Jump up ^ Barrow, John D. (1992), Pi in the sky: counting, thinking, and being, Clarendon Press, p. 38, ISBN 9780198539568 .
Jump up ^ (Mallory & Adams 1997) Encyclopedia of Indo-European Culture
Jump up ^ Ifrah, pages 326, 379
Jump up ^ Ifrah, pages 261-264
References[edit]
O'Connor, John; Robertson, Edmund (December 2000). "Babylonian Numerals". Retrieved 21 August 2010.
Kadvany, John (December 2007). "Positional Value and Linguistic Recursion". Journal of Indian Philosophy.
Knuth, Donald (1997). The art of Computer Programming. 2. Addison-Wesley. pp. 195–213. ISBN 0-201-89684-2.
Ifrah, George (2000). The Universal History of Numbers: From Prehistory to the Invention of the Computer. Wiley. ISBN 0-471-37568-3.
Kroeber, Alfred (1976) [1925]. Handbook of the Indians of California. Courier Dover Publications. p. 176. ISBN 9780486233680.
External links[edit]
Accurate Base Conversion
The Development of Hindu Arabic and Traditional Chinese Arithmetics
Implementation of Base Conversion at cut-the-knot
Learn to count other bases on your fingers
Online Arbitrary Precision Base Converter
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