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I wonder what is the tone of this question. Is it a sincere and innocent request for information? Or some sort of cynical rhetoric suggestion that the answer should be “very little”?I will give credit that it is a sincere “Ask” and answer accordingly. So here is a list of the inventions of just 10% of Great Britain - the Inventions of Scotland. If I included the inventions from the other 90% of Great Britain, I would swamp Quora!Road transport innovationsMacadamised roads (the basis for, but not specifically, tarmac): John Loudon McAdam (1756–1836)[3]The pedal bicycle: Attributed to both Kirkpatrick Macmillan (1813–1878)[2] and Thomas McCall (1834–1904)The pneumatic tyre: Robert William Thomson and John Boyd Dunlop (1822–1873)[9]The overhead valve engine: David Dunbar Buick (1854–1929)[10]Civil engineering innovationsTubular steel: Sir William Fairbairn (1789–1874)[11]The Falkirk wheel: Initial designs by Nicoll Russell Studios, Architects, RMJM, Architects and engineers Binnie Black and Veatch (Opened 2002)[12][13]The patent slip for docking vessels: Thomas Morton (1781–1832)[14][15]The Drummond Light: Thomas Drummond (1797–1840)[16]Canal design: Thomas Telford (1757–1834)[17]Dock design improvements: John Rennie (1761–1821)[18]Crane design improvements: James Bremner (1784–1856)[19]"Trac Rail Transposer", a machine to lay rail track patented in 2005, used by Network Rail in the United Kingdom and the New York Subway in the United States.[20][21][22]Aviation innovationsAircraft design: Frank Barnwell (1910) Establishing the fundamentals of aircraft design at the University of Glasgow.[23]Power innovationsCondensing steam engine improvements: James Watt (1736–1819)[1]Thermodynamic cycle: William John Macquorn Rankine (1820–1872)[24]Coal-gas lighting: William Murdoch (1754–1839)[25]The Stirling heat engine: Rev. Robert Stirling (1790–1878)[26]Carbon brushes for dynamos: George Forbes (1849–1936)[27]The Clerk cycle gas engine: Sir Dugald Clerk (1854–1932)[28]The wave-powered electricity generator: by South African Engineer Stephen Salter in 1977[29]The Pelamis Wave Energy Converter ("red sea snake" wave energy device): Richard Yemm, 1998[30]Shipbuilding innovationsEurope's first passenger steamboat: Henry Bell (1767–1830)[31]The first iron–hulled steamship: Sir William Fairbairn (1789–1874)[32]The first practical screw propeller: Robert Wilson (1803–1882)[citation needed]Marine engine innovations: James Howden (1832–1913)[33]John Elder & Charles Randolph (Marine Compound expansion engine)[33]Military innovationsLieutenant-General Sir David Henderson two areas: Field intelligence. Argued for the establishment of the Intelligence Corps. Wrote Field Intelligence: Its Principles and Practice (1904) and Reconnaissance (1907) on the tactical intelligence of modern warfare during World War I.[34]Special forces: Founded by Sir David Stirling, the SAS was created in World War II in the North Africa campaign to go behind enemy lines to destroy and disrupt the enemy. Since then it has been regarded as the most famous and influential special forces that has inspired other countries to form their own special forces too.Intelligence: Allan Pinkerton developed the still relevant intelligence techniques of "shadowing" (surveillance) and "assuming a role" (undercover work) in his time as head of the Union Intelligence Service.Heavy industry innovationsCoal mining extraction in the sea on an artificial island by Sir George Bruce of Carnock (1575). Regarded as one of the industrial wonders of the late medieval period.[35]Making cast steel from wrought iron: David Mushet (1772–1847)[36]Wrought iron sash bars for glass houses: John C. Loudon (1783–1865)[37]The hot blast oven: James Beaumont Neilson (1792–1865)[38]The steam hammer: James Nasmyth (1808–1890)[39]Wire rope: Robert Stirling Newall (1812–1889)[40]Steam engine improvements: William Mcnaught (1831–1881)[41]The Fairlie, a narrow gauge, double-bogie railway engine: Robert Francis Fairlie (1831–1885)[42]Cordite - Sir James Dewar, Sir Frederick Abel (1889)[43]Agricultural innovationsThreshing machine improvements: James Meikle (c.1690-c.1780) & Andrew Meikle (1719–1811)[44]Hollow pipe drainage: Sir Hew Dalrymple, Lord Drummore (1700–1753)[45]The Scotch plough: James Anderson of Hermiston (1739–1808)[46]Deanstonisation soil-drainage system: James Smith (1789–1850)[47]The mechanical reaping machine: Rev. Patrick Bell (1799–1869)[48]The Fresno scraper: James Porteous (1848–1922)[49]The Tuley tree shelter: Graham Tuley in 1979[50]Communication innovationsPrint stereotyping: William Ged (1690–1749)[51]Roller printing: Thomas Bell (patented 1783)[52]The adhesive postage stamp and the postmark: claimed by James Chalmers (1782–1853)[53]The Waverley pen nib innovations thereof: Duncan Cameron (1825–1901) The popular "Waverley" was unique in design with a narrow waist and an upturned tip designed to make the ink flow more smoothly on the paper.[54]Universal Standard Time: Sir Sandford Fleming (1827–1915)[55]Light signalling between ships: Admiral Philip H. Colomb (1831–1899)[56]The underlying principles of Radio - James Clerk Maxwell (1831–1879)[57]The Kinetoscope, a motion picture camera: devised in 1889 by William Kennedy Dickson (1860-1935)[58]The teleprinter: Frederick G. Creed (1871–1957)[59]The British Broadcasting Corporation BBC: John Reith, 1st Baron Reith (1922) its founder, first general manager and Director-general of the British Broadcasting Corporation[60]Radar: A significant contribution made by Robert Watson-Watt (1892–1973) alongside Englishman Henry Tizard (1885-1959) and others[61]The automated teller machine and Personal Identification Number system - James Goodfellow (born 1937)[62]Publishing firstsThe first edition of the Encyclopædia Britannica (1768–81)[63]The first English textbook on surgery(1597)[64]The first modern pharmacopaedia, William Cullen (1776). The book became 'Europe's principal text on the classification and treatment of disease'. His ideas survive in the terms nervous energy and neuroses (a word that Cullen coined).[65]The first postcards and picture postcards in the UK[66]The first eBook from a UK administration (March 2012). Scottish Government publishes 'Your Scotland, Your Referendum'.[67][citation needed]The educational foundation of Ophthalmology: Stewart Duke-Elder in his ground breaking work including ‘Textbook of Ophthalmology and fifteen volumes of System of Ophthalmology’[68]Culture and the artsScottish National Portrait Gallery, designed by Sir Robert Rowand Anderson (1889): the world's first purpose-built portrait gallery.[69]Fictional charactersSherlock Holmes, by Sir Arthur Conan DoylePeter Pan, by J.M. Barrie, born in Kirriemuir, AngusLong John Silver and Jekyll and Hyde, by Robert Louis StevensonJohn Bull: by John Arbuthnot although seen as a national personification of the United Kingdom in general, and England in particular,[70] the character of John Bull was invented by Arbuthnot in 1712[71]James Bond was given a Scottish background by Ian Fleming, himself of Scottish descent, after he was impressed by Sean Connery's performance.Scientific innovationsLogarithms: John Napier (1550–1617)[72]Modern Economics founded by Adam Smith (1776) 'The father of modern economics'[73] with the publication of The Wealth of Nations.[74][75]Modern Sociology: Adam Ferguson (1767) ‘The Father of Modern Sociology’ with his work An Essay on the History of Civil Society[76]Hypnotism: James Braid (1795–1860) the Father of Hypnotherapy[77]Tropical medicine: Sir Patrick Manson known as the father of Tropical Medicine[78]Modern Geology: James Hutton ‘The Founder of Modern Geology’[79][80][81]The theory of Uniformitarianism: James Hutton (1788): a fundamental principle of Geology the features of the geologic time takes millions of years.[82]The theory of electromagnetism: James Clerk Maxwell (1831–1879)[83]The discovery of the Composition of Saturn's Rings James Clerk Maxwell (1859): determined the rings of Saturn were composed of numerous small particles, all independently orbiting the planet. At the time it was generally thought the rings were solid. The Maxwell Ringlet and Maxwell Gap were named in his honor.[84]The Maxwell–Boltzmann distribution by James Clerk Maxwell (1860): the basis of the kinetic theory of gases, that speeds of molecules in a gas will change at different temperatures. The original theory first hypothesised by Maxwell and confirmed later in conjunction with Ludwig Boltzmann.[85]Popularising the decimal point: John Napier (1550–1617)[86]The first theory of the Higgs boson by English born [87] Peter Higgs particle-physics theorist at the University of Edinburgh (1964)[88]The Gregorian telescope: James Gregory (1638–1675)[89]The discovery of Proxima Centauri, the closest known star to the Sun, by Robert Innes (1861–1933)[90]One of the earliest measurements of distance to the Alpha Centauri star system, the closest such system outside of the Solar System, by Thomas Henderson (1798–1844)[91]The discovery of Centaurus A, a well-known starburst galaxy in the constellation of Centaurus, by James Dunlop (1793–1848)[92]The discovery of the Horsehead Nebula in the constellation of Orion, by Williamina Fleming (1857–1911)[93]The world's first oil refinery and a process of extracting paraffin from coal laying the foundations for the modern oil industry: James Young (1811–1883)[94]The identification of the minerals yttrialite, thorogummite, aguilarite and nivenite: by William Niven (1889)[95]The concept of latent heat by French-born Joseph Black (1728–1799)[96]Discovering the properties of Carbon dioxide by French-born Joseph Black (1728–1799)The concept of Heat capacity by French-born Joseph Black (1728–1799)The pyroscope, atmometer and aethrioscope scientific instruments: Sir John Leslie (1766–1832)[97]Identifying the nucleus in living cells: Robert Brown (1773–1858)[98]An early form of the Incandescent light bulb: James Bowman Lindsay (1799-1862)[99]Colloid chemistry: Thomas Graham (1805–1869)[100]The kelvin SI unit of temperature by Irishman William Thomson, Lord Kelvin (1824–1907)[101]Devising the diagramatic system of representing chemical bonds: Alexander Crum Brown (1838–1922)[102]Criminal fingerprinting: Henry Faulds (1843–1930)[103]The noble gases: Sir William Ramsay (1852–1916)[104]The cloud chamber recording of atoms: Charles Thomson Rees Wilson (1869–1959)[105][106]The discovery of the Wave of Translation, leading to the modern general theory of solitons by John Scott Russell (1808-1882)[107]Statistical graphics: William Playfair founder of the first statistical line charts, bar charts, and pie charts in (1786) and (1801) known as a scientific ‘milestone’ in statistical graphs and data visualization[108][109]The Arithmetic mean density of the Earth: Nevil Maskelyne conducted the Schiehallion experiment conducted at the Scottish mountain of Schiehallion, Perthshire 1774[110]The first isolation of methylated sugars, trimethyl and tetramethyl glucose: James Irvine[111][112]Discovery of the Japp–Klingemann reaction: to synthesize hydrazones from β-keto-acids (or β-keto-esters) and aryl diazonium salts 1887[113]Pioneering work on nutrition and poverty: John Boyd Orr (1880–1971)[114]Ferrocene synthetic substances: Peter Ludwig Pauson in 1955[115]The first cloned mammal (Dolly the Sheep): Was conducted in The Roslin Institute research centre in 1996 by English scientists Ian Wilmut (born 1944) and Keith Campbell (1954–2012).[116]The seismometer innovations thereof: James David Forbes[117]Metaflex fabric innovations thereof: University of St. Andrews (2010) application of the first manufacturing fabrics that manipulate light in bending it around a subject. Before this such light manipulating atoms were fixed on flat hard surfaces. The team at St Andrews are the first to develop the concept to fabric.[118]Tractor beam innovations thereof: St. Andrews University (2013) the world's first to succeed in creating a functioning Tractor beam that pulls objects on a microscopic level[119][120]Macaulayite: Dr. Jeff Wilson of the Macaulay Institute, Aberdeen.[121]Discovery of Catacol whitebeam by Scottish Natural Heritage and the Royal Botanic Garden Edinburgh (1990s): a rare tree endemic and unique to the Isle of Arran in south west Scotland. The trees were confirmed as a distinct species by DNA testing.[122]The first positive displacement liquid flowmeter, the reciprocating piston meter by Thomas Kennedy Snr.[123]Sports innovationsMain article: Sport in ScotlandScots have been instrumental in the invention and early development of several sports:Australian rules football Scots were prominent with many innovations in the early evolution of the game, including the establishment of the Essendon Football Club by the McCracken family from Ayrshire[124][125][126][127][128]several modern athletics events, i.e. shot put[129] and the hammer throw,[129] derive from Highland Games and earlier 12th century Scotland[129]Curling[130]Gaelic handball The modern game of handball is first recorded in Scotland in 1427, when koKing James I an ardent handball player had his men block up a cellar window in his palace courtyard that was interfering with his game.[131]Cycling, invention of the pedal-cycle[132]Golf (see Golf in Scotland)Ice Hockey, invented by the Scots regiments in Atlantic Canada by playing Shinty on frozen lakes.Shinty The history of Shinty as a non-standardised sport pre-dates Scotland the Nation. The rules were standardised in the 19th century by Archibald Chisholm[133]Rugby sevens: Ned Haig and David Sanderson (1883)[134]The Dugout was invented by Aberdeen FC Coach Donald Colmanin the 1920sThe world's first Robot Olympics which took place in Glasgow in 1990.Medical innovationsPioneering the use of surgical anaesthesia with Chloroform: Firstly in 1842 by Robert Mortimer Glover then extended for use on humans by Sir James Young Simpson (1811–1870)[135] Initial use of chloroform in dentistry by Francis Brodie ImlachThe saline drip by Dr Thomas Latta of Leith in 1831/32The hypodermic syringe: Alexander Wood (1817–1884)[136]Transplant rejection: Professor Thomas Gibson (1940s) the first medical doctor to understand the relationship between donor graft tissue and host tissue rejection and tissue transplantation by his work on aviation burns victims during World War II.[137]First diagnostic applications of an ultrasound scanner: Ian Donald (1910–1987)[138]Discovery of hypnotism (November 1841): James Braid (1795–1860)[139]Identifying the mosquito as the carrier of malaria: Sir Ronald Ross (1857–1932)[140]Identifying the cause of brucellosis: Sir David Bruce (1855–1931)[141]Discovering the vaccine for typhoid fever: Sir William B. Leishman (1865–1926)[142]Discovery of Staphylococcus: Sir Alexander Ogston (1880)[143]Discovering the Human papillomavirus vaccine Ian Frazer (2006): the second cancer preventing vaccine, and the world's first vaccine designed to prevent a cancer[144]Discovering insulin: John J R Macleod (1876–1935) with others[8] The discovery led him to be awarded the 1923 Nobel prize in Medicine.[145]Penicillin: Sir Alexander Fleming (1881–1955)[7]General anaesthetic - Pioneered by Scotsman James Young Simpson and Englishman John Snow[146]The establishment of standardized Ophthalmology University College London: Stewart Duke-Elder a pioneering Ophthalmologist[68]The first hospital Radiation therapy unit John Macintyre (1902): to assist in the diagnosis and treatment of injuries and illness at Glasgow Royal Infirmary[147]Pioneering of X-ray cinematography by John Macintyre (1896): the first moving real time X-ray image and the first KUB X-ray diagnostic image of a kidney stone in situ[147][148][149]The Haldane effect a property of hemoglobin first described by John Scott Haldane (1907)[150]Oxygen Therapy John Scott Haldane (1922): with the publication of ‘The Theraputic Administration of Oxygen Therapy’ beginning the modern era of Oxygen therapy[151]Ambulight PDT: light-emitting sticking plaster used in photodynamic therapy (PDT) for treating non-melanoma skin cancer. Developed by Ambicare Dundee's Ninewells Hospital and St Andrews University. (2010)[152]Discovering an effective tuberculosis treatment: Sir John Crofton in the 1950s[153]Primary creator of the artificial kidney (Professor Kenneth Lowe - Later Queen's physician in Scotland)[154]Developing the first beta-blocker drugs: Sir James W. Black in 1964[155] The discovery revolutionized the medical management of angina[156] and is considered to be one of the most important contributions to clinical medicine and pharmacology of the 20th century.[157] In 1988 he was awarded the Nobel Prize in Medicine.Developing modern asthma therapy based both on bronchodilation (salbutamol) and anti-inflammatory steroids (beclomethasone dipropionate) : Sir David Jack in 1972Glasgow coma scale: Graham Teasdale and Bryan J. Jennett (1974)[158]Glasgow Outcome Scale Bryan J. Jennett & Sir Michael Bond (1975): is a scale so that patients with brain injuries, such as cerebral traumas[159]Glasgow Anxiety Scale J.Mindham and C.A Espie (2003)[160]Glasgow Depression Scale Fiona Cuthill (2003): the first accurate self-report scale to measure the levels of depression in people with learning disabilities[161]ECG [Electrocardiography]: Alexander Muirhead. First recording of a human ECG (1869)[162][163]The first Decompression tables John Scott Haldane (1908): to calculate the safe return of deep-sea divers to surface atmospheric pressure[164]Surface Enhanced Raman Scattering (SERS): Strathclyde University (2014) A laser and nanoparticle test to detect Meningitis or multiple pathogenic agents at the same time.[165]Household innovationsThe television: John Logie Baird (1923)The refrigerator: William Cullen (1748)[166]The first electric bread toaster: Alan MacMasters (1893)The flush toilet: Alexander Cumming (1775)[167]The vacuum flask: Sir James Dewar (1847–1932)[168]The first distiller to triple distill Irish whiskey:[169]John Jameson (Whisky distiller)The piano footpedal: John Broadwood (1732–1812)[170]The first automated can-filling machine John West (1809–1888)[171]The waterproof macintosh: Charles Macintosh (1766–1843)[172]The kaleidoscope: Sir David Brewster (1781–1868)[173]Keiller's marmalade Janet Keiller (1797) - The first recipe of rind suspended marmalade or Dundee marmalade produced in Dundee.The modern lawnmower: Alexander Shanks (1801–1845)[174]The Lucifer friction match: Sir Isaac Holden (1807–1897)[175]The self filling pen: Robert Thomson (1822–1873)[176]Cotton-reel thread: J & J Clark of Paisley[177]Lime cordial: Lauchlan Rose in 1867Bovril beef extract: John Lawson Johnston in 1874[178]The electric clock: Alexander Bain (1840)[179]Chemical Telegraph (Automatic Telegraphy) Alexander Bain (1846) In England Bain's telegraph was used on the wires of the Electric Telegraph Company to a limited extent, and in 1850 it was used in America.[180]Barr's Irn Bru, refreshing soft drink produced by Barr's in Cumbernauld Scotland and exported to all around the world, The drink is so widely popular that in Scotland outsells both American colas Coca-Cola and Pepsi. And ranks 3rd most popular drink in the UK with Coca-Cola and Pepsi taking the first two spots.[181]Weapons innovationsThe carronade cannon: Robert Melville (1723–1809)[182]The Ferguson rifle: Patrick Ferguson in 1770 or 1776[183]The Lee bolt system as used in the Lee–Metford and Lee–Enfield series rifles: James Paris Lee[184]The Ghillie suit[185]The percussion cap: invented by Scottish Presbyterian clergyman Alexander Forsyth[186]Miscellaneous innovationsBoys' Brigade[187]Bank of England devised by William PatersonBank of France devised by John LawThe industrialisation and modernisation of Japan by Thomas Blake Glover[188]Colour photography: the first known permanent colour photograph was taken by James Clerk Maxwell (1831–1879)[189]Buick Motor Company by David Dunbar Buick[190]New York Herald newspaper by James Gordon Bennett, Sr.[190]Pinkerton National Detective Agency by Allan Pinkerton[190]Forbes magazine by B. C. Forbes[190]The establishment of a standardized botanical institute: Isaac Bayley Balfour major reform, development of botanical science, the concept of garden infrastructure therein improving scientific facilities[191]London School of Hygiene & Tropical Medicine: founded by Sir Patrick Manson in 1899[78]SERIES-B by JAC Vapour - first UK designed and engineered electronic cigarette[192]See alsoList of British innovations and discoveriesList of domesticated Scottish breedsHomecoming Scotland 2009References"BBC - History - James Watt". Retrieved 2008-12-31."BBC - History - Kirkpatrick Macmillan". Retrieved 2008-12-31."Encyclopædia Britannica: John Loudon Mcadam (British inventor)". Retrieved 2010-06-13."Scottish Science Hall of Fame - Alexander Graham Bell (1847-1922)". Retrieved 2010-02-20."BBC - History - John Logie Baird". Retrieved 2008-12-31.The World's First High Definition Colour Television System. McLean, p. 196."Nobelprize.org: Sir Alexander Fleming - Biography". Retrieved 2008-12-31."Nobelprize.org: John Macleod - Biography". Retrieved 2008-12-31."Robert William Thomson, Scotland's forgotten inventor". Retrieved 2010-06-13.Pelfrey, William (2006). Billy, Alfred, and General Motors: The Story of Two Unique Men, a Legendary Company, and a Remarkable Time in American History. AMACOM. ISBN 978-0-8144-2961-7."Gazetteer for Scotland: Overview of Sir William Fairbairn". Retrieved 2010-06-14."Falkirk Wheel & Visitor Centre". Retrieved 2015-11-30."SKF Evolution online". Retrieved 2010-06-13."Clydesite Magazine: The Real Inventor of the Patent Slip". Retrieved 2010-06-13.The Edinburgh philosophical journal, Volume 2 Printed for Archibald Constable, 1820"The Gazetteer for Scotland: Overview of Thomas Drummond". Retrieved 2010-06-14.The life of Thomas Telford, Civil Engineer: With an introductory history of roads and travelling in Great Britain J. Murray, 1867John Rennie 1761–1821 Manchester University Press NDThe industrial archaeology of Scotland, Volume 2 Macmillan of Canada, 1977 - Social Science"Ayrshire brothers' invention to transform America's railways". BBC. 6 June 2016. Retrieved 7 June 2016."Laying lines". Railway Strategies (103). 6 January 2014. Retrieved 7 June 2016."US Patent Application No: 2008/0072,783 - Railway Rail Handling Apparatus and Method". PatentBuddy. Retrieved 7 June 2016.University of Glasgow :: World Changing:: Establishing fundamental principles in aircraft design"William John Macquorn Rankine". Retrieved 2014-01-13."William Murdoch - The Scot Who Lit The World". Retrieved 2010-06-14."Electric Scotland: Significant Scots - Robert Stirling". Retrieved 2010-06-14."The Gazetteer for Scotland: Overview of Prof. George Forbes". Retrieved 2010-06-14."Encyclopædia Britannica: Sir Dugald Clerk". Retrieved 2010-06-14."How Stuff Works: Could Salter's Duck have solved the oil crisis?". Retrieved 2010-06-14."Pelamis founder honoured for key role in marine energy". The Scottish Government. 28 March 2012. Retrieved 2012-03-29."Significant Scots: Henry Bell". Retrieved 2010-06-15."The Gazetteer for Scotland: Overview of Sir William Fairbairn". Retrieved 2010-06-16.The Dynamics of Victorian Business: Problems And Perspectives to the 1870s By Roy ChurchUniversity of Glasgow :: World Changing:: Establishing the Royal Air ForceThe Edinburgh History of Scottish Literature: From Columba to the Union (Until 10707). By Ian Brown"Electric Scotland: Significant Scots - David Mushet". Retrieved 2010-06-17.Houses of glass: a nineteenth-century building type By Georg Kohlmaier, Barna von Sartory, John C. HarveyDictionary of energy By Cutler J. Cleveland, Chris MorrisMaterials processing defects By Swadhin Kumar Ghosh, M. PredeleanuIron: An illustrated weekly journal for iron and steel .., Volume 63 by Sholto PercyRepertory of patent inventions and other discoveries and improvements in arts, manufactures and agriculture MacIntosh 1846American narrow gauge railroads By George Woodman HiltonNature: international journal of science 1917 MacMillanAnnual report of the Indiana State Board of Agriculture, Volume 2 By Indiana. State Board of Agriculture, Indiana. Geological SurveyGreat Scots By Betty KirkpatrickThe English cyclopædia: a new dictionary of universal knowledge, Volume 1 edited by Charles KnightThe new American cyclopaedia: a popular dictionary of general knowledgeJournal of the Society of Arts, Volume 6 By Society of Arts (Great Britain)"The Fresno Scraper - American Society of Mechanical Engineers" (PDF). Retrieved 2010-11-12.The complete guide to trees of Britain and Northern Europe Alan F. Mitchell, David More"William Ged (Scottish goldsmith)". Retrieved 2010-06-13."roller printing (textile industry)". Retrieved 2010-06-13."Arbroath & District Stamp & Postcard Club". Retrieved 2010-06-19.http://www.gracesguide.co.uk/MacNiven_and_CameronCommunication and empire: media, markets, and globalization, 1860–1930 by Dwayne Roy Winseck, Robert M. PikeMilitary communications: from ancient times to the 21st century By Christopher H. SterlingRadiolocation in Ubiquitous Wireless Communication by Danko Antolovic"it was his Scottish protégé, William Dickson, who... ", The Scotsman, 23 March 2002The worldwide history of telecommunications by Anton A. Huurdemanhttp://www.bbc.co.uk/historyofthebbc/resources/in-depth/reith_1.shtml"Radar Personalities: Sir Robert Watson-Watt". Retrieved 2008-12-31."Who Invented the ATM? The James Goodfellow Story". Retrieved 2011-08-26.Encyclopaedic visions: scientific dictionaries and enlightenment culture By Natasha J. YeoThe Early history of surgery William John Bishop - 1995Twenty Medical Classics of the Jefferson Era http://www.hsl.virginia.edu/historical/rare_books/classics/#CullenPicture Postcards By C W Hillhttp://www.scotland.gov.uk/News/Releases/2012/03/ebookref08032012. Retrieved 2012-04-04.Lyle, T. K.; Miller, S.; Ashton, N. H. (1980). "William Stewart Duke-Elder. 22 April 1898-27 March 1978". Biographical Memoirs of Fellows of the Royal Society 26: 85. doi:10.1098/rsbm.1980.0003http://www.nationalgalleries.org/visit/about-the-portrait-gallery/Taylor, Miles (2004). "'Bull, John (supp. fl. 1712–)'". Oxford Dictionary of National Biography. Oxford University Press. doi:10.1093/ref:odnb/68195.http://global.britannica.com/EBchecked/topic/304946/John-BullErnest William Hobson. John Napier and the invention of logarithms, 1614. The University Press, 1914.Davis, William L, Bob Figgins, David Hedengren, and Daniel B. Klein. "Economic Professors' Favorite Economic Thinkers, Journals, and Blogs", Econ Journal Watch 8(2): 126–146, May 2011.[1]M Skousen (2007). The Big Three in Economics: Adam Smith, Karl Marx, And John Maynard Keynes p3,5,6.E. K. Hunt (2002). History of Economic Thought: A Critical Perspective, p.3. ISBN 0-7656-0606-2Willcox, William Bradford; Arnstein, Walter L. (1966). The Age of Aristocracy, 1688 to 1830. Volume III of A History of England, edited by Lacey Baldwin Smith (Sixth Edition, 1992 ed.). Lexington, Massachusetts. p. 133. ISBN 0-669-24459-7.The Discovery of Hypnosis- The Complete Writings of James Braid, the Father of Hypnotherapy James Braid, Donald Robertson (ed.) 2009Manson-Bahr, Patrick (1962). Patrick Manson. The Father of Tropical Medicine. Thomas NelsonJames Hutt

Nein….. CloseTo all our readers in the U.S., We will get straight to the point: This Wednesday we ask you to protect Wikipedia's independence. Time is running out in 2017 to help us. We depend on donations averaging about $15. Only a tiny portion of our readers give. If everyone reading this gave $3, we could keep Wikipedia thriving for years to come. The price of your Wednesday coffee is all we need. When I made Wikipedia a non-profit, people warned me I’d regret it. Over a decade later, it’s the only top ten site run by a non-profit and a community of volunteers. Has it crossed my mind how much we could have made if it had ads? Sure. But it wouldn’t be the same. We wouldn’t trust it. Most people ignore my messages. But I hope you’ll think about how useful it is to have unlimited access to reliable, neutral information. Please help keep Wikipedia online and growing. Thank you. — Jimmy Wales, Wikipedia FounderHow often would you like to donate?Just OnceGive MonthlySelect an amount (USD)$3$5$10$20$30$50$100Select a payment methodSecure transactionMaybe laterNatural numberFrom Wikipedia, the free encyclopediaThis article is about “positive integers” and “non-negative integers”. For all the numbers …, −2, −1, 0, 1, 2, …, see Integer.Natural numbers can be used for counting (one apple, two apples, three apples, …)Real numbers (ℝ) include the rational (ℚ), which include the integers (ℤ), which include the natural numbers (ℕ)In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). In common language, words used for counting are "cardinal numbers" and words used for ordering are "ordinal numbers".Some definitions, including the standard ISO 80000-2,[1]begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, ….[2][3][4][5]Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers (including negative integers).[6]The natural numbers are the basis from which many other number sets may be built by extension: the integers, by including (if not yet in) the neutral element 0 and an additive inverse (−n) for each nonzero natural number n; the rational numbers, by including a multiplicative inverse (1/n) for each nonzero integer n (and also the product of these inverses by integers); the real numbers by including with the rationals the limits of (converging) Cauchy sequences of rationals; the complex numbers, by including with the real numbers the unresolved square root of minus one (and also the sums and products of thereof); and so on.[7][8]These chains of extensions make the natural numbers canonically embedded (identified) in the other number systems.Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics.In common language, for example in primary school, natural numbers may be called counting numbers[9]both to intuitively exclude the negative integers and zero, and also to contrast the discreteness of counting to the continuity of measurement, established by the real numbers.The natural numbers can, at times, appear as a convenient set of names (labels), that is, as what linguists call nominal numbers, foregoing many or all of the properties of being a number in a mathematical sense.Contents1 History 1.1 Ancient roots 1.2 Modern definitions2 Notation3 Properties 3.1 Addition 3.2 Multiplication 3.3 Relationship between addition and multiplication 3.4 Order 3.5 Division 3.6 Algebraic properties satisfied by the natural numbers4 Generalizations5 Formal definitions 5.1 Peano axioms 5.2 Constructions based on set theory 5.2.1 Von Neumann construction 5.2.2 Other constructions6 See also7 Notes8 References9 External linksHistoryAncient rootsThe Ishango bone (on exhibition at the Royal Belgian Institute of Natural Sciences)[10][11][12]is believed to have been used 20,000 years ago for natural number arithmetic.The most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set.The first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers. The ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The Babylonians had a place-value system based essentially on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context.[13]A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation (within other numbers) dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number.[14]The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica.[15][16]The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus (the calculation of the date of Easter), beginning with Dionysius Exiguus in 525, without being denoted by a numeral (standard Roman numerals do not have a symbol for 0); instead nulla (or the genitive form nullae) from nullus, the Latin word for "none", was employed to denote a 0 value.[17]The first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all.[18]Independent studies also occurred at around the same time in India, China, and Mesoamerica.[19]Modern definitionsThis section needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.(October 2014)(Learn how and when to remove this template message)In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school[which?]of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man".[20]In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics.[21]In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural but a consequence of definitions. Later, two classes of such formal definitions were constructed; later, they were shown to be equivalent in most practical applications.Set-theoretical definitions of natural numbers were initiated by Frege and he initially defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox. Therefore, this formalism was modified so that a natural number is defined as a particular set, and any set that can be put into one-to-one correspondence with that set is said to have that number of elements.[22]The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic. It is based on an axiomatization of the properties of ordinal numbers: each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several weak systems of set theory. One such system is ZFC with the axiom of infinity replaced by its negation. Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein's theorem.[23]With all these definitions it is convenient to include 0 (corresponding to the empty set) as a natural number. Including 0 is now the common convention among set theorists[24]and logicians.[25]Other mathematicians also include 0[5]although many have kept the older tradition and take 1 to be the first natural number.[26]Computer scientists often start from zero when enumerating items like loop counters and string- or array- elements.[27][28]NotationThe double-struck capital N symbol, often used to denote the set of all natural numbers (see List of mathematical symbols).Mathematicians use N or ℕ (an N in blackboard bold) to refer to the set of all natural numbers. Older texts have also occasionally employed J as the symbol for this set.[29]This set is countably infinite: it is infinite but countable by definition. This is also expressed by saying that the cardinal number of the set is aleph-naught (ℵ0).[30]To be unambiguous about whether 0 is included or not, sometimes an index (or superscript) "0" is added in the former case, and a superscript "*" or subscript ">0" is added in the latter case:[1]ℕ0 = ℕ0 = {0, 1, 2, …}ℕ* = ℕ+ = ℕ1 = ℕ>0 = {1, 2, …}.Alternatively, natural numbers may be distinguished from positive integers with the index notation, but it must be understood by context that since both symbols are used, the natural numbers contain zero.[31]ℕ = {0, 1, 2, …}.ℤ+= {1, 2, …}.PropertiesAdditionOne can recursively define an addition operator on the natural numbers by setting a + 0 = a and a + S(b) = S(a + b) for all a, b. Here S should be read as "successor". This turns the natural numbers (ℕ, +) into a commutative monoid with identity element 0, the so-called free object with one generator. This monoid satisfies the cancellation property and can be embedded in a group (in the mathematical sense of the word group). The smallest group containing the natural numbers is the integers.If 1 is defined as S(0), then b + 1 = b + S(0) = S(b + 0) = S(b). That is, b + 1 is simply the successor of b.MultiplicationAnalogously, given that addition has been defined, a multiplication operator × can be defined via a × 0 = 0 and a × S(b) = (a × b) + a. This turns (ℕ*, ×) into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers.Relationship between addition and multiplicationAddition and multiplication are compatible, which is expressed in the distribution law: a × (b + c) = (a × b) + (a × c). These properties of addition and multiplication make the natural numbers an instance of a commutative semiring. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that ℕ is not closed under subtraction (i.e., subtracting one natural from another does not always result in another natural), means that ℕ is not a ring; instead it is a semiring (also known as a rig).If the natural numbers are taken as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that they begin with a + 1 = S(a) and a × 1 = a.OrderIn this section, juxtaposed variables such as ab indicate the product a × b, and the standard order of operations is assumed.A total order on the natural numbers is defined by letting a ≤ b if and only if there exists another natural number c where a + c = b. This order is compatible with the arithmetical operations in the following sense: if a, b and c are natural numbers and a ≤ b, then a + c ≤ b + c and ac ≤ bc.An important property of the natural numbers is that they are well-ordered: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an ordinal number; for the natural numbers, this is denoted as ω (omega).DivisionIn this section, juxtaposed variables such as ab indicate the product a × b, and the standard order of operations is assumed.While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of division with remainder is available as a substitute: for any two natural numbers a and b with b ≠ 0 there are natural numbers q and r such thata = bq + r and r < b.The number q is called the quotient and r is called the remainder of the division of a by b. The numbers q and r are uniquely determined by a and b. This Euclidean division is key to several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number theory.Algebraic properties satisfied by the natural numbersThe addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties:Closure under addition and multiplication: for all natural numbers a and b, both a + b and a × b are natural numbers.Associativity: for all natural numbers a, b, and c, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.Commutativity: for all natural numbers a and b, a + b = b + a and a × b = b × a.Existence of identity elements: for every natural number a, a + 0 = a and a × 1 = a.Distributivity of multiplication over addition for all natural numbers a, b, and c, a × (b + c) = (a × b) + (a × c).No nonzero zero divisors: if a and b are natural numbers such that a × b = 0, then a = 0 or b = 0 (or both).GeneralizationsTwo important generalizations of natural numbers arise from the two uses of counting and ordering: cardinal numbers and ordinal numbers.A natural number can be used to express the size of a finite set; more precisely, a cardinal number is a measure for the size of a set, which is even suitable for infinite sets. This concept of "size" relies on maps between sets, such that two sets have the same size, exactly if there exists a bijection between them. The set of natural numbers itself, and any bijective image of it, is said to be countably infinite and to have cardinality aleph-null (ℵ0).Natural numbers are also used as linguistic ordinal numbers: "first", "second", "third", and so forth. This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of any well-ordered countably infinite set. This assignment can be generalized to general well-orderings with a cardinality beyond countability, to yield the ordinal numbers. An ordinal number may also be used to describe the notion of "size" for a well-ordered set, in a sense different from cardinality: if there is an order isomorphism (more than a bijection!) between two well-ordered sets, they have the same ordinal number. The first ordinal number that is not a natural number is expressed as ω; this is also the ordinal number of the set of natural numbers itself.Many well-ordered sets with cardinal number ℵ0 have an ordinal number greater than ω (the latter is the lowest possible). The least ordinal of cardinality ℵ0 (i.e., the initial ordinal) is ω.For finite well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite, sequence.A countable non-standard model of arithmetic satisfying the Peano Arithmetic (i.e., the first-order Peano axioms) was developed by Skolem in 1933. The hypernatural numbers are an uncountable model that can be constructed from the ordinary natural numbers via the ultrapower construction.Georges Reeb used to claim provocatively that The naïve integers don't fill up ℕ. Other generalizations are discussed in the article on numbers.Formal definitionsPeano axiomsMain article: Peano axiomsMany properties of the natural numbers can be derived from the Peano axioms.[32][33]Axiom One: 0 is a natural number.Axiom Two: Every natural number has a successor.Axiom Three: 0 is not the successor of any natural number.Axiom Four: If the successor of [math] x {\displaystyle x} [/math] equals the successor of [math] y {\displaystyle y} [/math], then [math] x {\displaystyle x} [/math] equals [math] y {\displaystyle y} [/math].Axiom Five (the axiom of induction): If a statement is true of 0, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number.These are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axioms have 1 in place of 0. In ordinary arithmetic, the successor of [math] x {\displaystyle x} [/math] is [math] x + 1 {\displaystyle x+1} [/math]. Replacing Axiom Five by an axiom schema one obtains a (weaker) first-order theory called Peano Arithmetic.Constructions based on set theoryMain article: Set-theoretic definition of natural numbersVon Neumann constructionIn the area of mathematics called set theory, a special case of the von Neumann ordinal construction[34]defines the natural numbers as follows:Set 0 = { }, the empty set,Define S(a) = a ∪ {a} for every set a. S(a) is the successor of a, and S is called the successor function.By the axiom of infinity, there exists a set which contains 0 and is closed under the successor function. Such sets are said to be 'inductive'. The intersection of all such inductive sets is defined to be the set of natural numbers. It can be checked that the set of natural numbers satisfies the Peano axioms.It follows that each natural number is equal to the set of all natural numbers less than it:0 = { },1 = 0 ∪ {0} = {0} = {{ }},2 = 1 ∪ {1} = {0, 1} = {{ }, {{ }}},3 = 2 ∪ {2} = {0, 1, 2} = {{ }, {{ }}, {{ }, {{ }}}},n = n−1 ∪ {n−1} = {0, 1, …, n−1} = {{ }, {{ }}, …, {{ }, {{ }}, …}}, etc.With this definition, a natural number n is a particular set with n elements, and n ≤ m if and only if n is a subset of m.Also, with this definition, different possible interpretations of notations like ℝn (n-tuples versus mappings of n into ℝ) coincide.Even if one does not accept the axiom of infinity and therefore cannot accept that the set of all natural numbers exists, it is still possible to define any one of these sets.Other constructionsAlthough the standard construction is useful, it is not the only possible construction. Zermelo's construction goes as follows:Set 0 = { }Define S(a) = {a},It then follows that0 = { },1 = {0} = {{ }},2 = {1} = {{{ }}},n = {n−1} = {{{…}}}, etc.Each natural number is then equal to the set containing just the natural number preceding it.See alsoMathematics portalIntegerSet-theoretic definition of natural numbersPeano axiomsCanonical representation of a positive integerCountable setNumber#Classification for other number systems (rational, real, complex etc.)Notes"Standard number sets and intervals". ISO 80000-2:2009. International Organization for Standardization. p. 6.Weisstein, Eric W. "Natural Number". MathWorld."natural number", Dictionary by Merriam-Webster: America's most-trusted online dictionary, Merriam-Webster, retrieved 4 October 2014Carothers (2000) says: "ℕ is the set of natural numbers (positive integers)" (p. 3)Mac Lane & Birkhoff (1999) include zero in the natural numbers: 'Intuitively, the set ℕ = {0, 1, 2, ...} of all natural numbers may be described as follows: ℕ contains an "initial" number 0; ...'. They follow that with their version of the Peano Postulates. (p. 15)Jack G. Ganssle & Michael Barr (2003). Embedded Systems Dictionary. p. 138 (integer), 247 (signed integer), & 276 (unsigned integer). ISBN 1578201209. integer 1. n. Any whole number.Mendelson (2008) says: "The whole fantastic hierarchy of number systems is built up by purely set-theoretic means from a few simple assumptions about natural numbers." (Preface, p. x)Bluman (2010): "Numbers make up the foundation of mathematics." (p. 1)Weisstein, Eric W. "Counting Number". MathWorld.Introduction, Royal Belgian Institute of Natural Sciences, Brussels, Belgium.Flash presentation, Royal Belgian Institute of Natural Sciences, Brussels, Belgium.The Ishango Bone, Democratic Republic of the Congo, on permanent display at the Royal Belgian Institute of Natural Sciences, Brussels, Belgium. UNESCO's Portal to the Heritage of AstronomyGeorges Ifrah, The Universal History of Numbers, Wiley, 2000, ISBN 0-471-37568-3"A history of Zero". MacTutor History of Mathematics. Retrieved 2013-01-23. … a tablet found at Kish … thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty placeMann, Charles C. (2005), 1491: New Revelations Of The Americas Before Columbus, Knopf, p. 19, ISBN 9781400040063.Evans, Brian (2014), "Chapter 10. Pre-Columbian Mathematics: The Olmec, Maya, and Inca Civilizations", The Development of Mathematics Throughout the Centuries: A Brief History in a Cultural Context, John Wiley & Sons, ISBN 9781118853979.Michael L. Gorodetsky (2003-08-25). "Cyclus Decemnovennalis Dionysii – Nineteen year cycle of Dionysius". Группа Квантовых и Прецизионных Измерений. Retrieved 2012-02-13.This convention is used, for example, in Euclid's Elements, see Book VII, definitions 1 and 2.Morris Kline, Mathematical Thought From Ancient to Modern Times, Oxford University Press, 1990 [1972], ISBN 0-19-506135-7The English translation is from Gray. In a footnote, Gray attributes the German quote to: "Weber 1891/92, 19, quoting from a lecture of Kronecker's of 1886."Gray, Jeremy (2008), Plato's Ghost: The Modernist Transformation of Mathematics, Princeton University Press, p. 153Weber, Heinrich L. 1891-2. Kronecker. Jahresbericht der Deutschen Mathematiker-Vereinigung 2:5-23. (The quote is on p. 19.)"Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." (Eves 1990, p. 606)Eves 1990, Chapter 15L. Kirby; J. Paris, Accessible Independence Results for Peano Arithmetic, Bulletin of the London Mathematical Society 14 (4): 285. doi:10.1112/blms/14.4.285, 1982.Bagaria, Joan. "Set Theory". The Stanford Encyclopedia of Philosophy (Winter 2014 Edition).Goldrei, Derek (1998). "3". Classic set theory : a guided independent study (1. ed., 1. print ed.). Boca Raton, Fla. [u.a.]: Chapman & Hall/CRC. p. 33. ISBN 0-412-60610-0.This is common in texts about Real analysis. See, for example, Carothers (2000, p. 3) or Thomson, Bruckner & Bruckner (2000, p. 2).Brown, Jim (1978). "In Defense of Index Origin 0". ACM SIGAPL APL Quote Quad. 9 (2): 7. doi:10.1145/586050.586053. Retrieved 19 January 2015.Hui, Roger. "Is Index Origin 0 a Hindrance?". J Home. Retrieved 19 January 2015.Rudin, W. (1976). Principles of Mathematical Analysis (PDF). New York: McGraw-Hill. p. 25. ISBN 978-0-07-054235-8.Weisstein, Eric W. "Cardinal Number". MathWorld.Grimaldi, Ralph P. (2003). A review of discrete and combinatorial mathematics (5th ed.). Boston, MA: Addison-Wesley. p. 133. ISBN 978-0201726343.G.E. Mints (originator), "Peano axioms", Encyclopedia of Mathematics, Springer, in cooperation with the European Mathematical Society, retrieved 8 October 2014Hamilton (1988) calls them "Peano's Postulates" and begins with "1. 0 is a natural number." (p. 117f)Halmos (1960) uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I) 0 ∈ ω (where, of course, 0 = ∅" (ω is the set of all natural numbers). (p. 46)Morash (1991) gives "a two-part axiom" in which the natural numbers begin with 1. (Section 10.1: An Axiomatization for the System of Positive Integers)Von Neumann 1923ReferencesBluman, Allan (2010), Pre-Algebra DeMYSTiFieD (Second ed.), McGraw-Hill ProfessionalCarothers, N.L. (2000), Real analysis, Cambridge University Press, ISBN 0-521-49756-6Clapham, Christopher; Nicholson, James (2014), The Concise Oxford Dictionary of Mathematics (Fifth ed.), Oxford University PressDedekind, Richard (1963), Essays on the Theory of Numbers, Dover, ISBN 0-486-21010-3 Dedekind, Richard (2007), Essays on the Theory of Numbers, Kessinger Publishing, LLC, ISBN 0-548-08985-XEves, Howard (1990), An Introduction to the History of Mathematics (6th ed.), Thomson, ISBN 978-0-03-029558-4Halmos, Paul (1960), Naive Set Theory, Springer Science & Business MediaHamilton, A. G. (1988), Logic for Mathematicians (Revised ed.), Cambridge University PressJames, Robert C.; James, Glenn (1992), Mathematics Dictionary (Fifth ed.), Chapman & HallLandau, Edmund (1966), Foundations of Analysis (Third ed.), Chelsea Pub Co, ISBN 0-8218-2693-XMac Lane, Saunders; Birkhoff, Garrett (1999), Algebra (3rd ed.), American Mathematical SocietyMendelson, Elliott (2008) [1973], Number Systems and the Foundations of Analysis, Dover PublicationsMorash, Ronald P. (1991), Bridge to Abstract Mathematics: Mathematical Proof and Structures (Second ed.), Mcgraw-Hill CollegeMusser, Gary L.; Peterson, Blake E.; Burger, William F. (2013), Mathematics for Elementary Teachers: A Contemporary Approach (10th ed.), Wiley Global Education, ISBN 978-1118457443Szczepanski, Amy F.; Kositsky, Andrew P. (2008), The Complete Idiot's Guide to Pre-algebra, Penguin GroupThomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008), Elementary Real Analysis (Second ed.), ClassicalRealAnalysis.info, ISBN 9781434843678Von Neumann, Johann (1923), "Zur Einführung der transfiniten Zahlen", Acta litterarum ac scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae, Sectio scientiarum mathematicarum, 1: 199–208 Von Neumann, John (January 2002) [1923], "On the introduction of transfinite numbers", in Jean van Heijenoort, From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (3rd ed.), Harvard University Press, pp. 346–354, ISBN 0-674-32449-8 - English translation of von Neumann 1923.External linksHazewinkel, Michiel, ed. (2001) [1994], "Natural number", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4Axioms and Construction of Natural NumbersEssays on the Theory of Numbers by Richard Dedekind at Project Gutenberg[show]vteNumber systems[show]vteClasses of natural numbersCategories:Cardinal numbersElementary mathematicsIntegersNumber theoryNumbersNavigation menuNot logged inTalkContributionsCreate accountLog inArticleTalkReadEditView historySearchMain pageContentsFeatured contentCurrent eventsRandom articleDonate to WikipediaWikipedia storeInteractionHelpAbout WikipediaCommunity portalRecent changesContact pageToolsWhat links hereRelated changesUpload fileSpecial pagesPermanent linkPage informationWikidata itemCite this pagePrint/exportCreate a bookDownload as PDFPrintable versionIn other projectsWikimedia CommonsLanguagesAfrikaansAlemannischአማርኛالعربيةAragonésঅসমীয়াAzərbaycancaتۆرکجهবাংলাBân-lâm-gúБашҡортсаБеларускаяБеларуская (тарашкевіца)‎Българскиབོད་ཡིགBosanskiBrezhonegCatalàЧӑвашлаČeštinaCymraegDanskDeutschEestiΕλληνικάEmiliàn e rumagnòlEspañolEsperantoEuskaraفارسیFøroysktFrançaisGaeilgeGalego贛語Хальмг한국어Հայերենहिन्दीHornjoserbsceHrvatskiBahasa IndonesiaInterlinguaИронIsiXhosaÍslenskaItalianoעבריתBasa Jawaಕನ್ನಡქართულიҚазақшаKiswahiliKurdîລາວLatinaLatviešuLëtzebuergeschLietuviųLa .lojban.LumbaartMagyarМакедонскиMalagasyമലയാളംमराठीمصرىBahasa MelayuMirandésМонголမြန်မာဘာသာNederlandsनेपालीनेपाल भाषा日本語NordfriiskNorskNorsk nynorskOʻzbekcha/ўзбекчаਪੰਜਾਬੀپنجابیPatoisភាសាខ្មែរPiemontèisPlattdüütschPolskiPortuguêsRomânăРусскийसंस्कृतम्SarduScotsShqipSicilianuසිංහලSimple EnglishSlovenčinaSlovenščinaŚlůnskiSoomaaligaکوردیСрпски / srpskiSrpskohrvatski / српскохрватскиSuomiSvenskaTagalogதமிழ்Татарча/tatarçaతెలుగుไทยТоҷикӣTürkçeTürkmençeУкраїнськаاردوTiếng ViệtVõro文言West-VlamsWinarayייִדישYorùbá粵語Žemaitėška中文Edit linksThis page was last edited on 28 November 2017, at 18:20.Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. 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