document_title: Invention of radio

question_text: who invented the first radio and what year

question_tokens: who invented the first radio and what year

text: Invention of radio

text: Wireless communication theories and methods previous to radio[edit]

text: Before the discovery of electromagnetic waves and the development of radio communication there were many wireless telegraph systems proposed or tried out.[4] Early researchers may not have understood or disclosed which physical effects were responsible for transmitting signals. These experimenters used the existing theories of the day or novel theories of their own on how wireless signals could be transmitted.

text: In April 1872 William Henry Ward received U.S. Patent 126,356 for a wireless telegraphy system where he theorized that convection currents in the atmosphere could carry signals like a telegraph wire.[5] A few months after Ward received his patent, Mahlon Loomis of West Virginia received U.S. Patent 129,971 for a "wireless telegraph" in July 1872. The patented system claimed to utilize atmospheric electricity to eliminate the overhead wire used by the existing telegraph systems. It did not contain diagrams or specific methods and it did not refer to or incorporate any known scientific theory. It was similar to William Henry Ward's patent.[6][7]

text: In the United States, Thomas Edison, in the mid-1880s, patented an electromagnetic induction system he called "grasshopper telegraphy", which allowed telegraphic signals to jump the short distance between a running train and telegraph wires running parallel to the tracks.[8] In the United Kingdom, William Preece was able to develop an electromagnetic induction telegraph system that, with antenna wires many kilometers long, could transmit across gaps of about 5 kilometres (3.1 miles). Inventor Nathan Stubblefield, between 1885 and 1892,[9] also worked on an induction transmission system.

text: Forms of Wireless telephony is recorded in four patents for the photophone, invented jointly by Alexander Graham Bell and Charles Sumner Tainter in 1880. The photophone allowed for the transmission of sound on a beam of light, and on June 3, 1880 Bell and Tainter transmitted the world's first wireless telephone message on their newly invented form of light telecommunication.[10][11]

text: In the early 1890s Nikola Tesla began his research into high frequency electricity. Tesla was aware of Hertz's experiments with electromagnetic waves from 1889 on[12][13] but, (like many scientists of that time) thought, even if radio waves existed, they would probably only travel in straight lines making them useless for long range transmission.[14]

text: Tesla's primary interest in wireless phenomenon was as a power distribution system[14] although he proposed by 1893 that it could also incorporate communication. His laboratory work and later large scale experiments at Colorado Springs led him to the conclusion that he could build a conduction based worldwide wireless system that would use the Earth itself (via injecting very large amounts of electric current into the ground) as the means to conduct the signal to overcome this limitation.[15] He went on to try to implement his ideas of power transmission and wireless telecommunication in his very large but unsuccessful Wardenclyffe Tower project.[16]

text: Development of electromagnetism[edit]

text:
Experiments and Theory

Joseph Henry

Michael Faraday

Hans Christian Ørsted

text: Various scientists proposed that electricity and magnetism were linked. Around 1800 Alessandro Volta developed the first means of producing an electric current. In 1802 Gian Domenico Romagnosi may have suggested a relationship between electricity and magnetism but his reports went unnoticed.[17][18] In 1820 Hans Christian Ørsted performed a simple and today widely known experiment on electric current and magnetism. He demonstrated that a wire carrying a current could deflect a magnetized compass needle.[19] Ørsted's work influenced André-Marie Ampère to produce a theory of electromagnetism. Several scientists speculated that light might be connected with electricity or magnetism.

text: In 1831, Michael Faraday began a series of experiments in which he discovered electromagnetic induction. The relation was mathematically modelled by Faraday's law, which subsequently became one of the four Maxwell equations. Faraday proposed that electromagnetic forces extended into the empty space around the conductor, but did not complete his work involving that proposal. In 1846 Michael Faraday speculated that light was a wave disturbance in a force field".[20]

text: Expanding upon a series of experiments by Felix Savary,[21][22][23][24] between 1842 and 1850 Joseph Henry performed experiments detecting inductive magnetic effects over a distance of 200 feet (61 m).[25][26][27] He was the first (1838–42) to produce high frequency AC electrical oscillations, and to point out and experimentally demonstrate that the discharge of a capacitor under certain conditions is oscillatory, or, as he puts it, consists "of a principal discharge in one direction and then several reflex actions backward and forward, each more feeble than the preceding until equilibrium is attained"[citation needed]. This view was also later adopted by Helmholtz,[28] the mathematical demonstration of this fact was first given by Lord Kelvin in his paper on "Transient Electric Currents".[29][30]

text: Maxwell and the theoretical prediction of electromagnetic waves[edit]

text:
Maxwell and electromagnetic waves

James Clerk Maxwell

Oliver Heaviside

text: Between 1861 and 1865, based on the earlier experimental work of Faraday and other scientists and on his own modification to Ampere's law, James Clerk Maxwell developed his theory of electromagnetism, which predicted the existence of electromagnetic waves. In 1873 Maxwell described the theoretical basis of the propagation of electromagnetic waves in his paper to the Royal Society, "A Dynamical Theory of the Electromagnetic Field." This theory united all previously unrelated observations, experiments and equations of electricity, magnetism, and optics into a consistent theory.[31] His set of equations—Maxwell's equations—demonstrated that electricity, magnetism, and light are all manifestations of the same phenomenon, the electromagnetic field. Subsequently, all other classic laws or equations of these disciplines were special cases of Maxwell's equations. Maxwell's work in electromagnetism has been called the "second great unification in physics".[32]

text: Oliver Heaviside, later reformulated Maxwell's original equations into the set of four vector equations that are generally known today as Maxwell's equations.[33] Neither Maxwell nor Heaviside transmitted or received radio waves; however, their equations for electromagnetic fields established principles for radio design, and remain the standard expression of classical electromagnetism.

text: Of Maxwell's work, Albert Einstein wrote:[34]

text: "Imagine [Maxwell's] feelings when the differential equations he had formulated proved to him that electromagnetic fields spread in the form of polarised waves, and at the speed of light! To few men in the world has such an experience been vouchsafed... it took physicists some decades to grasp the full significance of Maxwell's discovery, so bold was the leap that his genius forced upon the conceptions of his fellow-workers."

text: Other physicists were equally impressed with Maxwell's work, such as Richard Feynman who commented:[35]

text: "From a long view of the history of the world—seen from, say, ten thousand years from now—there can be little doubt that the most significant event of the 19th century will be judged as Maxwell's discovery of the laws of electromagnetism. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade."

text: Experiments and proposals[edit]

text: Berend Wilhelm Feddersen,[36] a German physicist, in 1859, as a private scholar in Leipzig, succeeded in experiments with the Leyden jar to prove that electric sparks were composed of damped oscillations.

text: In 1870 the German physicist Wilhelm von Bezold discovered and demonstrated the fact that the advancing and reflected oscillations produced in conductors by a capacitor discharge gave rise to interference phenomena.[37][38] Professors Elihu Thomson and E. J. Houston in 1876 made a number of experiments and observations on high frequency oscillatory discharges.[39] In 1883 George FitzGerald suggested[40] at a British Association meeting that electromagnetic waves could be generated by the discharge of a capacitor, but the suggestion was not followed up, possibly because no means was known for detecting the waves.[30]

text: Hertz experimentally verifies Maxwell's theory[edit]

text: It had been proposed in 1879 to German physicist Heinrich Rudolf Hertz that he try to prove Maxwell's theory of electromagnetism but he could not see any way to build an apparatus to experimentally test the theory.[41] His observation, in the autumn of 1886, of discharging a Leyden jar into a large coil producing a spark in an adjacent coil gave him the idea of how to build a test apparatus.[42][43] Between 1886 and 1888 Hertz conducted scientific experiments that validated Maxwell's theory.[44] He designed a method of detecting spark-gap radio waves by observing that another unpowered spark-gap, acting as an antenna, would absorb the radio energy and convert it back into an electric spark. Hertz published his results in a series of papers between 1887 and 1890,[45] and again in complete book form in 1893.[46]

text: The first of the papers published, "On Very Rapid Electric Oscillations", gives an account of the chronological course of his investigation, as far as it was carried out up to the end of the year 1886 and the beginning of 1887.[47]

text: For the first time, electromagnetic radio waves ("Hertzian waves")[48] were intentionally and unequivocally proven to have been transmitted through free space by a spark-gap device, and detected over a short distance.[49]

text: Hertz was able to have some control over the frequencies of his radiated waves by altering the inductance and capacitance of his transmitting and receiving antennas. He focused the electromagnetic waves using a corner reflector and a parabolic reflector, to demonstrate that radio behaved the same as light, as Maxwell's electromagnetic theory had predicted more than 20 years earlier.[30]

text: Hertz did not devise a system for practical utilization of electromagnetic waves, nor did he describe any potential applications of the technology. Hertz was asked by his students at the University of Bonn what use there might be for these waves. He replied, “It's of no use whatsoever. This is just an experiment that proves Maestro Maxwell was right, we just have these mysterious electromagnetic waves that we cannot see with the naked eye. But they are there."[50]

text: Hertz died in 1894, and the art of radio wave communication was left to others to implement into a practical form. After Hertz's experiments, Sir William Crookes published an article in February 1892 in The Fortnightly Review on 'Some possibilities of electricity' with his thoughts on possibility of wireless communication based on the research of Lodge and Hertz,[51] and the American physicist Amos Emerson Dolbear brought similar attention to the idea.[52]

text: Pre-Hertz radio wave detection[edit]

text: Towards the end of 1875, while experimenting with the telegraph, Thomas Edison noted a phenomenon that he termed "etheric force", announcing it to the press on November 28. He abandoned this research when Elihu Thomson, among others, ridiculed the idea, claiming it was electromagnetic induction.

text: In 1879 the experimenter and inventor David Edward Hughes working in London discovered that a bad contact in a Bell telephone he was using in his experiments seemed to be sparking when he worked on a nearby induction balance (an early form of metal detector).[53][54] He developed an improved detector to pick up this unknown "extra current" based on his new microphone design (similar to later detectors known as coherers or crystal detectors)[53][55] and developed a way to interrupt his induction balance to produce a series of sparks. By trial and error experiments he eventually found he could pick up these "aerial waves" as he carried his telephone device down the street out to a range of 500 yards (460 m).

text: On February 20, 1880, he demonstrated his experiment to representatives of the Royal Society including Thomas Henry Huxley, Sir George Gabriel Stokes, and William Spottiswoode, then president of the Society. Stokes was convinced the phenomenon Hughes was demonstrating was merely electromagnetic induction, not a type of conduction through the air.[56][57][58] Hughes was not a physicist and seems to have accepted Stokes observations and did not pursue the experiments any further.[57] His work may have been mentioned in William Crookes' 1892 Fortnightly Review review of 'Some possibilities of electricity' article as an unnamed individual whose experiment Crookes participate in.[51]

text: Development of radio waves[edit]

text:
Early experimenters

Édouard Branly

Oliver Joseph Lodge

Jagadish Chandra Bose

text: The Branly detector[edit]

text: In 1890, Édouard Branly[59][60][61] demonstrated what he later called the "radio-conductor,"[62] which Lodge in 1893 named the coherer, the first sensitive device for detecting radio waves.[63] Shortly after the experiments of Hertz, Branly discovered that loose metal filings, which in a normal state have a high electrical resistance, lose this resistance in the presence of electric oscillations and become practically conductors of electricity. This Branly showed by placing metal filings in a glass box or tube, and making them part of an ordinary electric circuit. According to the common explanation, when electric waves are set up in the neighborhood of this circuit, electromotive forces are generated in it which appear to bring the filings more closely together, that is, to cohere, and thus their electrical resistance decreases, from which cause this piece of apparatus was termed by Sir Oliver Lodge a coherer.[64] Hence the receiving instrument, which may be a telegraph relay, that normally would not indicate any sign of current from the small battery, can be operated when electric oscillations are set up.[65] Branly further found that when the filings had once cohered they retained their low resistance until shaken apart, for instance, by tapping on the tube.[66] The coherer, however, was not sensitive enough to be used reliably as radio developed.[67]

text: Lodge's demonstrations[edit]

text: One of the first investigators to notice and measure stationary waves on wires produced by direct coupling (resonance) with the coatings of a Leyden jar was Sir Oliver Lodge, entitled "Experiments On The Discharge Of Leyden Jars" (1891).[68] On June 1, 1894, Oliver Lodge at the Royal Institution lectures, delivered "The Work of Hertz and Some of His Successors".[69] Lodge performed a transmission on August 14, 1894[70] at a meeting of the British Association for the Advancement of Science at Oxford University.[71][72] Also in 1894, Lodge would state that Alexander Muirhead clearly foresaw the telegraphic importance of the transmission of transverse Hertzian waves.[25] A convenient method of establishing stationary electric waves on wires is one which generally attribute to Ernst Lecher,[73] and call the Lecher arrangement.[25] In fact, it originated with Lodge and Hertz, whilst Edouard Sarasin and Lucien de la Rive gave it an improved form.[25][74]

text: On that day in August 1894, Lodge demonstrated the reception of Morse code signalling via radio waves using a "coherer". He later improved Branly's coherer by adding a "trembler" which dislodged clumped filings,[75] thus restoring the device's sensitivity.[76] In August 1898 he got U.S. Patent 609,154, "Electric Telegraphy", that made wireless signals using a Ruhmkorff coil for the transmitter and a Branly coherer for the detector. This patent utilized the concept of "syntonic" tuning. In 1912 Lodge sold the patent to Marconi.

text: In 1894 Lodge showed that the Branly coherer could be employed to transmit telegraphic signals, and in order that the filings should not remain "cohered" after the cessation of the electric oscillations, he devised an electro-mechanical "tapper" on the principle of the ordinary "buzzer," or electric door-bell, the hammer of which was caused to tap the glass tube as long as the electric oscillations continued. The filings thus virtually take the place of a key in the ordinary telegraph circuit. In the normal state the key is open; in the presence of electrical oscillations the key is closed. Thus, by opening and closing the key for a longer or shorter period, signals corresponding to dots and dashes may be produced. In other words, by setting up electric oscillations for periods of time corresponding to dots and dashes, messages may be transmitted from the sending station, and if, at the receiving station, a recording instrument (controlled by the coherer), such as the ordinary Morse register, be provided, a record of the message in dots and dashes may be obtained. Dr. Lodge in fact used a tapper operated continuously by clockwork.[65]

text: In 1894, with the help of the Branly filings tube, Lodge gave a couple of demonstrations, one in June at the Royal Institution at Oxford and one in August at Oxford, to the British Association, using Hertz oscillators for transmitting signals, using a Morse key in connection with the sending coil, and a Thomson marine galvanometer for receiving them—sending the signals from one room to another through walls, and so on. Lodge sent them also across the quadrangle of Liverpool College, but he applied very small power and did not try for big distances. At that time Dr. Alexander Muirhead was struck with its applicability to practical telegraphy, and when in 1896 Sir William Preece told the British Association meeting (as it happened in his laboratory) at Liverpool that an Italian gentleman (at that time unknown) was interesting the Post Office in a secret box, Lodge knew practically what the box must contain, and immediately afterwards (the same day) he showed to a few friends a Morse tape instrument, very roughly working on that plan. Mr. Marconi and Sir William Preece together interested the whole world in the subject; great power was applied to the sender, and the matter became of financial importance. However, the American Patent Office gave Lodge a telegraphic patent based on his work, as published in 1894, after proof that this book had reached America in or before 1895.[77]

text: J. C. Bose[edit]

text: In November 1894, the Indian physicist, Jagadish Chandra Bose, demonstrated publicly the use of radio waves in Calcutta, but he was not interested in patenting his work.[78] Bose ignited gunpowder and rang a bell at a distance using electromagnetic waves,[79] confirming that communication signals can be sent without using wires. He sent and received radio waves over distance but did not commercially exploit this achievement.

text: Bose demonstrated the ability of the signal to travel from the lecture room, and through an intervening room and passage, to a third room 75 feet (23 m) distant from the radiator, thus passing through three solid walls on the way, as well as the body of the chairman (who happened to be the Lieutenant-Governor). The receiver at this distance still had energy enough to make a contact which set a bell ringing, discharged a pistol, and exploded a miniature mine. To get this result from his small radiator, Bose set up an apparatus which curiously anticipated the lofty 'antennae' of modern wireless telegraphy— a circular metal plate at the top of a pole, 20 feet (6.1 m) high, being put in connection with the radiator and a similar one with the receiving apparatus.[80]

text: The form of 'Coherer' devised by Professor Bose, and described by him at the end of his paper 'On a new Electro Polariscope' allowed for the sensibility and range to appear to leave little to be desired at the time.[80] In 1896, the British, Daily Chronicle reported on his UHF experiments: "The inventor (J. C. Bose) has transmitted signals to a distance of nearly a mile and herein lies the first and obvious and exceedingly valuable application of this new theoretical marvel."

text: After Bose's Friday Evening Discourses at the Royal Institution, The Electric Engineer expressed 'surprise that no secret was at any time made as to its construction, so that it has been open to all the world to adopt it for practical and possibly money-making purposes.' Bose was sometimes criticised as unpractical for making no profit from his inventions.[80]

text: In 1899, Bose announced the development of an "iron-mercury-iron coherer with telephone detector" in a paper presented at the Royal Society, London.[81] Later he received U.S. Patent 755,840, "Detector for electrical disturbances" (1904), for a specific electromagnetic receiver. Bose would continue with his research and made other contributions to the development of radio.[82]

text: Adaptations of radio waves[edit]

text: Popov's lightning detector[edit]

text: In 1894-95 the Russian physicist Alexander Stepanovich Popov conducted experiments developing a radio receiver, an improved version of coherer-based design by Oliver Lodge. His design with coherer auto-tapping mechanism was designed as a lightning detector to help the forest service track lightning strikes that could start fires. His receiver proved to be able to sense lightning strikes at distances of up to 30 km. Popov built a version of the receiver that was capable of automatically recording lightning strikes on paper rolls. Popov presented his radio receiver to the Russian Physical and Chemical Society on May 7, 1895 — the day has been celebrated in the Russian Federation as "Radio Day" promoted in eastern European countries as the inventor of radio.[83][84][85] The paper on his findings was published the same year (December 15, 1895). Popov had recorded, at the end of 1895, that he was hoping for distant signaling with radio waves.[86] He did not apply for a patent for this invention.

text: Tesla's boat[edit]

text: In 1898 Nicola Tesla developed a radio/coherer based remote-controlled boat, with a form of secure communication[87][88] between transmitter and receiver,[89] which he demonstrated in 1898. Tesla called his invention a "teleautomaton" and he hoped to sell it as a guided naval torpedo.[90]

text: Radio based wireless telegraphy[edit]

text: Marconi[edit]

text: Guglielmo Marconi studied at the Leghorn Technical School, and acquainted himself with the published writings of Professor Augusto Righi of the University of Bologna.[91] In 1894, Sir William Preece delivered a paper to the Royal Institution in London on electric signalling without wires.[92][93] In 1894 at the Royal Institution lectures, Lodge delivered "The Work of Hertz and Some of His Successors".[69] Marconi is said to have read, while on vacation in 1894, about the experiments that Hertz did in the 1880s. Marconi also read about Tesla's work.[94] It was at this time that Marconi began to understand that radio waves could be used for wireless communications. Marconi's early apparatus was a development of Hertz’s laboratory apparatus into a system designed for communications purposes. At first Marconi used a transmitter to ring a bell in a receiver in his attic laboratory. He then moved his experiments out-of-doors on the family estate near Bologna, Italy, to communicate further. He replaced Hertz’s vertical dipole with a vertical wire topped by a metal sheet, with an opposing terminal connected to the ground. On the receiver side, Marconi replaced the spark gap with a metal powder coherer, a detector developed by Edouard Branly and other experimenters. Marconi transmitted radio signals for about 1.5 miles (2.4 km) at the end of 1895.[95]

text: Marconi was awarded a patent for radio with British patent No. 12,039, Improvements in Transmitting Electrical Impulses and Signals and in Apparatus There-for. The complete specification was filed March 2, 1897. This was Marconi's initial patent for the radio, though it used various earlier techniques of various other experimenters and resembled the instrument demonstrated by others (including Popov). During this time spark-gap wireless telegraphy was widely researched. In July, 1896, Marconi got his invention and new method of telegraphy to the attention of Preece, then engineer-in-chief to the British Government Telegraph Service, who had for the previous twelve years interested himself in the development of wireless telegraphy by the inductive-conductive method. On June 4, 1897, he delivered "Signalling through Space without Wires".[96] Preece devoted considerable time to exhibiting and explaining the Marconi apparatus at the Royal Institution in London, stating that Marconi invented a new relay which had high sensitiveness and delicacy.[97]

text: The Marconi Company Ltd. was founded by Marconi in 1897, known as the Wireless Telegraph Trading Signal Company. Also in 1897, Marconi established the radio station at Niton, Isle of Wight, England. Marconi's wireless telegraphy was inspected by the Post Office Telegraph authorities; they made a series of experiments with Marconi's system of telegraphy without connecting wires, in the Bristol Channel. The October wireless signals of 1897 were sent from Salisbury Plain to Bath, a distance of 34 miles (55 km).[100] Around 1900 Marconi developed an empirical law that, for simple vertical sending and receiving antennas of equal height, the maximum working telegraphic distance varied as the square of the height of the antenna.[101] This became known as Marconi's law.

text: Other experimental stations were established at Lavernock Point, near Penarth; on the Flat Holmes, an island in mid-channel, and at Brean Down, a promontory on the Somerset side. Signals were obtained between the first and last-named points, a distance of, approximately, 8 miles (13 km). The receiving instrument used was a Morse inkwriter[102] of the Post Office pattern.[103][104] In 1898, Marconi opened a radio factory in Hall Street, Chelmsford, England, employing around 50 people. In 1899, Marconi announced his invention of the "iron-mercury-iron coherer with telephone detector" in a paper presented at Royal Society, London.

text: In May, 1898, communication was established for the Corporation of Lloyds between Ballycastle and the Lighthouse on Rathlin Island in the North of Ireland. In July, 1898, the Marconi telegraphy was employed to report the results of yacht races at the Kingstown Regatta for the Dublin Express newspaper. A set of instruments were fitted up in a room at Kingstown, and another on board a steamer, the Flying Huntress. The aerial conductor on shore was a strip of wire netting attached to a mast 40 feet (12 m) high, and several hundred messages were sent and correctly received during the progress of the races.

text: At this time His Majesty King Edward VII, then Prince of Wales, had the misfortune to injure his knee, and was confined on board the royal yacht Osltorm in Cowes Bay.[105] Marconi fitted up his apparatus on board the royal yacht by request, and also at Osborne House, Isle of Wight, and kept up wireless communication for three weeks between these stations. The distances covered were small; but as the yacht moved about, on some occasions high hills were interposed, so that the aerial wires were overtopped by hundreds of feet, yet this was no obstacle to communication. These demonstrations led the Corporation of Trinity House to afford an opportunity for testing the system in practice between the South Foreland Lighthouse, near Dover, and the East Goodwin Lightship, on the Goodwin Sands. This installation was set in operation on December 24, 1898, and proved to be of value. It was shown that when once the apparatus was set up it could be worked by ordinary seamen with very little training.

text: At the end of 1898 electric wave telegraphy established by Marconi had demonstrated its utility, especially for communication between ship and ship and ship and shore.[106]

text: The Haven Hotel station and Wireless Telegraph Mast was where much of Marconi's research work on wireless telegraphy was carried out after 1898.[107] In 1899, he transmitted messages across the English Channel. Also in 1899, Marconi delivered "Wireless Telegraphy" to the Institution of Electrical Engineers.[106] In addition, in 1899, W. H. Preece delivered "Aetheric Telegraphy", stating that the experimental stage in wireless telegraphy had been passed in 1894 and inventors were then entering the commercial stage.[108] Preece, continuing in the lecture, details the work of Marconi and other British inventors. In October, 1899, the progress of the yachts in the international race between the Columbia and Shamrock was successfully reported by aerial telegraphy, as many as 4,000 words having been (as is said) despatched from the two ship stations to the shore stations. Immediately afterward the apparatus was placed by request at the service of the United States Navy Board, and some highly interesting experiments followed under Marconi's personal supervision.[109] The Marconi Company was renamed Marconi's Wireless Telegraph Company in 1900.

text: In 1901, Marconi claimed to have received daytime transatlantic radio frequency signals at a wavelength of 366 metres (820 kHz).[111][112][113] Marconi established a wireless transmitting station at Marconi House, Rosslare Strand, Co. Wexford in 1901 to act as a link between Poldhu in Cornwall and Clifden in Co. Galway. His announcement on 12 December 1901, using a 152.4-metre (500 ft) kite-supported antenna for reception, stated that the message was received at Signal Hill in St John's, Newfoundland (now part of Canada) via signals transmitted by the company's new high-power station at Poldhu, Cornwall. The message received had been prearranged and was known to Marconi, consisting of the Morse letter 'S' - three dots. Bradford has recently contested the reported success, however, based on theoretical work as well as a reenactment of the experiment. It is now well known that long-distance transmission at a wavelength of 366 meters is not possible during the daytime, because the skywave is heavily absorbed by the ionosphere.[citation needed] It is possible that what was heard was only random atmospheric noise, which was mistaken for a signal, or that Marconi may have heard a shortwave harmonic of the signal.[112][113] The distance between the two points was about 3,500 kilometres (2,200 mi).

text: The Poldhu to Newfoundland transmission claim has been criticized.[114] There are various science historians, such as Belrose and Bradford, who have cast doubt that the Atlantic was bridged in 1901, but other science historians have taken the position that this was the first transatlantic radio transmission. Critics have claimed that it is more likely that Marconi received stray atmospheric noise from atmospheric electricity in this experiment.[115] The transmitting station in Poldhu, Cornwall used a spark-gap transmitter that could produce a signal in the medium frequency range and with high power levels.

text: Marconi transmitted from England to Canada and the United States.[116] In this period, a particular electromagnetic receiver, called the Marconi magnetic detector[117] or hysteresis magnetic detector,[118] was developed further by Marconi and was successfully used in his early transatlantic work (1902) and in many of the smaller stations for a number of years.[119][120] In 1902, a Marconi station was established in the village of Crookhaven, County Cork, Ireland to provide marine radio communications to ships arriving from the Americas. A ship's master could contact shipping line agents ashore to enquire which port was to receive their cargo without the need to come ashore at what was the first port of landfall.[121] Ireland was also, due to its western location, to play a key role in early efforts to send trans-Atlantic messages. Marconi transmitted from his station in Glace Bay, Nova Scotia, Canada across the Atlantic, and on 18 January 1903 a Marconi station sent a message of greetings from Theodore Roosevelt, the President of the United States, to the King of the United Kingdom, marking the first transatlantic radio transmission originating in the United States

text: In 1904, Marconi inaugurated an ocean daily newspaper, the Cunard Daily Bulletin, on the R.M.S. "Campania." At the start, passing events were printed in a little pamphlet of four pages called the Cunard Bulletin. The title would read Cunard Daily Bulletin, with subheads for "Marconigrams Direct to the Ship."[122] All the passenger ships of the Cunard Company were fitted with Marconi's system of wireless telegraphy, by means of which constant communication was kept up, either with other ships or with land stations on the eastern or western hemisphere. The RMS Lucania, in October 1903, with Marconi on board, was the first vessel to hold communications with both sides of the Atlantic. The Cunard Daily Bulletin, a thirty-two page illustrated paper published on board these boats recorded news received by wireless telegraphy, and was the first ocean newspaper. In August 1903, in agreement was made with the British Government by which the Cunard Co. were to build two steamers, to be, with all other Cunard ships, at the disposal of the British Admiralty for hire or purchase whenever they might be required, the Government lending the company £2,600,000 to build the ships and granting them a subsidy of £150,000 a year. One was the RMS Lusitania and another was the RMS Mauritania.[123]

text: In June and July 1923, Marconi's shortwave transmissions took place at night on 97 meters from Poldhu Wireless Station, Cornwall, to his yacht Elettra in the Cape Verde Islands. In September 1924, Marconi transmitted during daytime and nighttime on 32 meters from Poldhu to his yacht in Beirut. In July 1924, Marconi entered into contracts with the British General Post Office (GPO) to install telegraphy circuits from London to Australia, India, South Africa and Canada as the main element of the Imperial Wireless Chain. The UK-to-Canada shortwave "Beam Wireless Service" went into commercial operation on 25 October 1926. Beam Wireless Services from the UK to Australia, South Africa and India went into service in 1927. Electronic components for the system were built at Marconi's New Street wireless factory in Chelmsford.[124]

text: Marconi was awarded the 1909 Nobel Prize in Physics with Karl Ferdinand Braun for contributions to radio sciences. Marconi's demonstrations of the use of radio for wireless communications, equipping ships with life saving wireless communications,[125] establishing the first transatlantic radio service,[116] and building the first stations for the British shortwave service, have marked his place in history. Shortly after the turn of the 20th century, the US Patent Office re-awarded Marconi a patent for radio. The U.S. Patent RE11,913 was granted on June 4, 1901. Marconi's U.S. Patent 676,332 was awarded on June 11, 1901, also. This system was more advanced than his previous works. The United States Supreme Court, decision of MARCONI WIRELESS T. CO. OF AMERICA v. U.S., 320 U.S. 1 (1943) stated that “Marconi's reputation as the man who first achieved successful radio transmission … is not here in question” this statement is followed by “Marconi's patent involved no invention over Lodge, Tesla, and Stone”. The 1943 decision didn't overturn Marconi's original patents, or his reputation as the first person to develop practical radiotelegraphic communication. It just said that the adoption of adjustable transformers in the transmitting and receiving circuits, which was an improvement of the initial invention, was anticipated by patents issued to Oliver Lodge and John Stone. (This decision wasn't unanimous).[126]

text: Braun[edit]

text: Ferdinand Braun's major contributions were the introduction of a closed tuned circuit in the generating part of the transmitter, and its separation from the radiating part (the antenna) by means of inductive coupling, and later on the usage of crystals for receiving purposes. Braun experimented at first at the University of Strasbourg. Braun had written extensively on wireless subjects and was well known through his many contributions to the Electrician and other scientific journals.[127] In 1899, he would apply for the patents, Electro telegraphy by means of condensers and induction coils and Wireless electro transmission of signals over surfaces.[128]

text: Pioneers working on wireless devices eventually came to a limit of distance they could cover. Connecting the antenna directly to the spark gap produced only a heavily damped pulse train. There were only a few cycles before oscillations ceased. Braun's circuit afforded a much longer sustained oscillation because the energy encountered less loss swinging between coil and Leyden Jars. Also, by means of inductive antenna coupling[129] the radiator was matched to the generator.

text: In spring 1899 Braun, accompanied by his colleagues Cantor and Zenneck, went to Cuxhaven to continue their experiments at the North Sea. On February 6, 1899, he would apply for the United States Patent, Wireless Electric Transmission of Signals Over Surfaces. Not before long he bridged a distance of 42 km to the city of Mutzing. On 24 September 1900 radio telegraphy signals were exchanged regularly with the island of Heligoland over a distance of 62 km. Lightvessels in the river Elbe and a coast station at Cuxhaven commenced a regular radio telegraph service. On August 6, 1901, he would apply for Means for Tuning and Adjusting Electric Circuits.

text: By 1904, the closed circuit system of wireless telegraphy, connected with the name of Braun, was well known and generally adopted in principle. The results of Braun's experiments, published in the Electrician, possess interest, apart from the method employed. Braun showed how the problem could be satisfactorily and economically solved.[130] The closed circuit oscillator has the advantage, as was known, of being able to draw upon the kinetic energy in the oscillator circuit, and thus, because such a circuit can be given a much greater capacity than can be obtained with a radiating aerial alone, much more energy can be stored up and radiated by its employment.[130] The emission is also prolonged, both results tending towards the attainment of the much desired train of undamped waves. The energy available, though greater than with the open system, was still inconsiderable unless very high potentials, with the attendant drawbacks, were used.[130][131] Braun avoided the use of extremely high potentials for charging the gap and also makes use of a less wasteful gap by sub-dividing it.[130][132] The chief point in his new arrangement, however, is not the sub-division of the gap merely but their arrangement, by which they are charged in parallel, at low voltages, and discharge in series. The Nobel Prize awarded to Braun in 1909 depicts this design.[133]

text: Stone Stone[edit]

text: John Stone Stone labored as an early telephone engineer and was influential in developing wireless communication technology, and holds dozens of key patents in the field of "space telegraphy". Patents of Stone for radio, together with their equivalents in other countries, form a very voluminous contribution to the patent literature of the subject. More than seventy United States patents have been granted to this patentee alone. In many cases these specifications are learned contributions to the literature of the subject, filled with valuable references to other sources of information.[134]

text: Stone has had issued to him a large number of patents embracing a method for impressing oscillations on a radiator system and emitting the energy in the form of waves of predetermined length whatever may be the electrical dimensions of the oscillator.[135] On February 8, 1900, he filed for a selective system in U.S. Patent 714,756. In this system, two simple circuits are associated inductively, each having an independent degree of freedom, and in which the restoration of electric oscillations to zero potential the currents are superimposed, giving rise to compound harmonic currents which permit the resonator system to be syntonized with precision to the oscillator.[135] Stone's system, as stated in U.S. Patent 714,831, developed free or unguided simple harmonic electromagnetic signal waves of a definite frequency to the exclusion of the energy of signal waves of other frequencies, and an elevated conductor and means for developing therein forced simple electric vibrations of corresponding frequency.[136] In these patents Stone devised a multiple inductive oscillation circuit with the object of forcing on the antenna circuit a single oscillation of definite frequency. In the system for receiving the energy of free or unguided simple harmonic electromagnetic signal waves of a definite frequency to the exclusion of the energy of signal waves of other frequencies, he claimed an elevated conductor and a resonant circuit associated with said conductor and attuned to the frequency of the waves, the energy of which is to be received.[136] A coherer made on what is called the Stone system[137] was employed in some of the portable wireless outfits of the United States Army. The Stone Coherer has two small steel plugs between which are placed loosely packed carbon granules. This is a self-decohering device; though not as sensitive as other forms of detectors it is well suited to the rough usage of portable outfits.[137]

text: Naval wireless[edit]

text: US Navy[edit]

text: In 1899 the United States Navy Board issued a report on the results of investigations of the Marconi system of wireless telegraphy.[138] The report noted that the system was well adapted for use in squadron signalling, under conditions of rain, fog, darkness and motion of speed although dampness affected the performance.[139] They also noted that when two stations were transmitting simultaneously both would be received and that the system had the potential to affect the compass. They reported ranges of from 85 miles for large ships with tall masts (43m) to 7 miles for smaller boats. The board recommended that the system was given a trial by the US navy.

text: Royal Navy[edit]

text: The HMS Hector became the first British warship to have wireless telegraphy installed when she conducted the first trials of the new equipment for the Royal Navy.[140][141] Starting in December 1899, the HMS Hector and HMS Jaseur were outfitted with wireless equipment. In 1901, HMS Jaseur received signals from the Marconi transmitter on the Isle of Wight and from the HMS Hector (25 January).[142]

text: Wireless telephony[edit]

text: Fessenden[edit]

text: In late 1886, Reginald A. Fessenden began working directly for Thomas Edison at the inventor's new laboratory in West Orange, New Jersey. Fessenden quickly made major advances, especially in receiver design, as he worked to develop audio reception of signals. The United States Weather Bureau began, early in 1900, a systematic course of experimentation in wireless telegraphy, employing him as a specialist.[143] Fessenden evolved the heterodyne principle here where two signals combined to produce a third signal.

text: In 1900, construction began on a large radio transmitting alternator. Fessenden, experimenting with a high-frequency spark transmitter, successfully transmitted speech on December 23, 1900, over a distance of about 1.6 kilometres (0.99 mi), the first audio radio transmission. Early in 1901 the Weather Bureau officially installed Fessenden at Wier's Point, Roanoke Island, North Carolina, and he made experimental transmissions across water to a station located about 5 miles (8.0 km) west of Cape Hatteras, the distance between the two stations being almost exactly 50 miles (80 km).[143] An alternator of 1 kW output at 10 kilohertz was built in 1902. The credit for the development of this machine is due to Charles Proteus Steinmetz, Caryl D. Haskins, Ernst Alexanderson, John T. H. Dempster, Henry Geisenhoner, Adam Stein, Jr., and F. P. Mansbendel.[30]

text: In a paper written by Fessenden in 1902, it was asserted that important advances had been made, one of which was overcoming largely the loss of energy experienced in other systems. In an interview with a New York Journal correspondent, Fessenden stated that in his early apparatus he did not use an air transformer at the sending end, nor a concentric cylinder for emitters and antennae,[143][144] and had used capacity, but arranged in a manner entirely different from that in other systems, and that he did not employ a coherer or any form of imperfect contact. Fessenden asserted that he had paid particular attention to selective and multiplex systems, and was well satisfied with the results in that direction.[143] On August 12, 1902, 13 patents were issued to Fessenden, covering various methods, devices, and systems for signaling without wires.[143] These patents involved many new principles, the chef-d'oeuvre of which was a method for distributing capacity and inductance instead of localizing these coefficients of the oscillator as in previous systems.[135]

text: By the summer of 1906, a machine producing 50 kilohertz was installed at the Brant Rock station, and in the fall of 1906, what was called an electric alternating dynamo was working regularly at 75 kilohertz, with an output of 0.5 kW.[30] Fessenden[145] used this for wireless telephoning to Plymouth, Massachusetts, a distance of approximately 11 miles (18 km).[30] In the following year machines were constructed having a frequency of 96 kilohertz[146] and outputs of 1 kW and 2 kW. Fessenden believed that the damped wave-coherer system was essentially and fundamentally incapable of development into a practical system.[30] He would employ a two-phase high frequency alternator method[147] and the continuous production of waves[148] with changing constants of sending circuit.[30][149] Fessenden would also use duplex and multiplex commutator methods.[150] On December 11, 1906, operation of the wireless transmission in conjunction with the wire lines took place.[151][30] In July 1907 the range was considerably extended and speech was successfully transmitted between Brant Rock and Jamaica, on Long Island, a distance of nearly 200 miles (320 km), in daylight and mostly over land,[152] the mast at Jamaica being approximately 180 feet (55 m) high.[30]

text: Fleming[edit]

text: In November 1904, the English physicist John Ambrose Fleming invented the two-electrode vacuum-tube rectifier, which he called the Fleming oscillation valve.[153] for which he obtained GB patent 24850 and U.S. Patent 803,684.[154] This "Fleming Valve" was sensitive and reliable, and so it replaced the crystal diode used in receivers used for long-distance wireless communication. It had an advantage, that it could not be permanently injured or set out of adjustment by any exceptionally strong stray signal, such as those due to atmospheric electricity.[155] Fleming earned a Hughes Medal in 1910 for his electronic achievements. Marconi used this device as a radio detector.[when?]

text: The Supreme Court of the United States would eventually invalidate the US patent because of an improper disclaimer and, additionally, maintained the technology in the patent was known art when filed.[156] This invention was the first vacuum tube. Fleming's diode was used in radio receivers for many decades afterward, until it was superseded by improved solid state electronic technology more than 50 years later.

text: De Forest[edit]

text: Lee De Forest[157][158][159] had an interest in wireless telegraphy and he invented the Audion in 1906. He was president and secretary of the De Forest Radio Telephone and Telegraph Company (1913).[160][161] The De Forest System was adopted by the United States Government, and had been demonstrated to other Governments including those of Great Britain, Denmark, Germany, Russia, and British Indies, all of which purchased De Forest apparatus previous to the Great War. De Forest is one of the fathers of the "electronic age", as the Audion helped to usher in the widespread use of electronics.[162]

text: De Forest made the Audion tube from a vacuum tube. He also made the "Oscillion", an undamped wave transmitter. He developed the De Forest method of wireless telegraphy and founded the American De Forest Wireless Telegraph Company. De Forest was a distinguished electrical engineer and the foremost American contributor to the development of wireless telegraphy and telephony. The elements of his device takes relatively weak electrical signals and amplifies them. The Audion Detector, Audion Amplifier, and the "Oscillion" transmitter had furthered the radio art and the transmission of written or audible speech. In World War I, the De Forest system was a factor in the efficiency of the United States Signal Service, and was also installed by the United States Government in Alaska.[162]

text: Radio invention timeline[edit]

text: Below is a brief selection of important events and individuals related to the development of radio, from 1860 to 1910.[163]


document_title: Daddy's Home 2

question_text: who is roger in daddy's home 2

question_tokens: who is roger in daddy 's home 2

root

text: Daddy's Home 2

text: Synopsis[edit]

text: This article needs an improved plot summary. (November 2017)

tag: table

text: Following the events of the first film, Brad and Dusty (Will Ferrell and Mark Wahlberg) must deal with their intrusive fathers, Don and Kurt (John Lithgow and Mel Gibson) during the holidays, along with Dusty's conflict with his stepchild's biological father, Roger (John Cena).

tag: p

text: Cast[edit]

text:
Will Ferrell as Brad Whitaker, Don's son, Sara's husband, Dylan and Megan's stepfather, and Griffy's father
Mark Wahlberg as Dusty Mayron, Kurt's son, Sara's ex-husband, Karen's husband, Dylan and Megan's father and Adrianna's stepfather
Linda Cardellini as Sara Whitaker, Brad's wife, Dusty's ex-wife, mother of Dylan, Megan, and Griffy
John Cena as Roger, Karen's ex-husband, and Adrianna's father
John Lithgow as Don Whitaker, Brad's father, Sara's father in law, Dylan and Megan's step grandfather, and Griffy's grandfather
Mel Gibson as Kurt Mayron, Dusty's father, Sara's ex-father in law, Karen's father in law, Dylan and Megan's grandfather, and Adrianna's stepgrandfather
Scarlett Estevez as Megan Mayron, Dusty and Sara's daughter, Dylan and Griffy's sister, Brad's stepdaughter, and Adrianna's stepsister
Owen Vaccaro as Dylan Mayron, Dusty and Sara's son, Megan and Griffy's brother, Brad's stepson, and Adrianna's stepbrother
Alessandra Ambrosio as Karen Mayron, Dusty's wife, Adrianna's mother, and Dylan and Megan's stepmother
Didi Costine as Adrianna, Dusty's stepdaughter, Karen and Roger's daughter, and Dylan and Megan's stepsister
Chesley Sullenberger as Brad's stepfather

tag: ul

text: Production[edit]

text: In April 2016, the sequel was announced, with Will Ferrell and Mark Wahlberg reprising their roles, Sean Anders and John Morris writing the script, and Anders directing.[3] In January 2017, it was reported that Mel Gibson and John Lithgow were being sought to play the main characters' fathers in the film.[4] The two were later confirmed to star, along with Linda Cardellini, John Cena, Owen Vaccaro and Scarlett Estevez, reprising their roles.[5][6]

text: Filming[edit]

text: Principal photography began on March 20, 2017.[7] Scenes were filmed in Concord, Massachusetts, Framingham, Massachusetts, and Great Barrington, Massachusetts.

text: Release[edit]

text: The film was released on November 10, 2017.[8]

text: Box office[edit]

text: As of November 26, 2017[update], Daddy's Home 2 has grossed $72.7 million in the United States and Canada, and $15 million in other territories, for a worldwide total of $87.7 million, against a production budget of $69 million.[2]

text: In the United States and Canada, Daddy's Home 2 opened alongside Murder on the Orient Express, and was projected to gross around $20 million from 3,575 theaters in its opening weekend.[9] The film made $10.7 million on its first day, including $1.5 million from Thursday night previews at 2,500 theaters, up from the $1.2 million made by the first film. It went on to debut to $30 million, finishing second at the box office, behind holdover Thor: Ragnarok ($57 million).[10] In its second weekend the film made $14.4 million (a drop of 51.3%), finishing 4th behind Justice League, Wonder and Thor: Ragnarok.[11]

text: Critical response[edit]

text: On review aggregator Rotten Tomatoes, the film has an approval rating of 18% based on 67 reviews, with an average rating of 3.9/10. The site's critical consensus reads, "A formulaic comedy that's unlikely to spread much yuletide merriment, Daddy's Home 2 can only muster a few stray yuks from its talented cast."[12] On Metacritic, which assigns a normalized rating to reviews, the film holds a weighted average score of 30 out of 100, based on 25 critics, indicating "generally unfavorable reviews".[13] Audiences polled by CinemaScore gave the film an average grade of "A–" on an A+ to F scale, up from the "B+" earned by the first film.[10]

text: Alonso Duralde of TheWrap was critical of what he described as the film's sloppiness and laziness, saying, "Director Sean Anders and his co-writer John Morris execute what are supposed to be the laughs with blunt force. The jokes announce themselves with heavy footsteps, and almost none of them land, stranding a talented cast with terrible material that they’re straining to sell."[14] Richard Roeper of the Chicago Sun-Times gave the film 1 out of 4 stars, saying: "After enduring last week’s lousy and lazy A Bad Moms Christmas, I would have bet it would be many a year before we’d see another holiday comedy more sour and cynical and profoundly unfunny. I sit corrected."[15]

text: Glenn Kenny of RogerEbert.com gave the film 2.5 out of 4 stars, writing, "I found the sequel better than the original—the writing sharper, the jokes fresher and smarter, the comic interaction between the lead characters consistently engaging. I mentioned this to my incredulous wife, who said, 'So you’re saying it's the Godfather, Part 2 of the Daddy’s Home series'."[16]


document_title: Limit of a function

question_text: when do we say a limit does not exist

question_tokens: when do we say a limit does not exist

text: Limit of a function

text: History[edit]

text:
Although implicit in the development of calculus of the 17th and 18th centuries, the modern idea of the limit of a function goes back to Bolzano who, in 1817, introduced the basics of the epsilon-delta technique to define continuous functions. However, his work was not known during his lifetime (Felscher 2000). Cauchy discussed variable quantities, infinitesimals, and limits and defined continuity of

y
=
f
(
x
)

{\displaystyle y=f(x)}

 by saying that an infinitesimal change in x necessarily produces an infinitesimal change in y in his 1821 book Cours d'analyse, while (Grabiner 1983) claims that he only gave a verbal definition. Weierstrass first introduced the epsilon-delta definition of limit in the form it is usually written today. He also introduced the notations lim and limx→x0 (Burton 1997).

text: The modern notation of placing the arrow below the limit symbol is due to Hardy in his book A Course of Pure Mathematics in 1908 (Miller 2004).

text: Motivation[edit]

text: Imagine a person walking over a landscape represented by the graph of y = f(x). Her horizontal position is measured by the value of x, much like the position given by a map of the land or by a global positioning system. Her altitude is given by the coordinate y. She is walking towards the horizontal position given by x = p. As she gets closer and closer to it, she notices that her altitude approaches L. If asked about the altitude of x = p, she would then answer L.

text: What, then, does it mean to say that her altitude approaches L? It means that her altitude gets nearer and nearer to L except for a possible small error in accuracy. For example, suppose we set a particular accuracy goal for our traveler: she must get within ten meters of L. She reports back that indeed she can get within ten meters of L, since she notes that when she is within fifty horizontal meters of p, her altitude is always ten meters or less from L.

text: The accuracy goal is then changed: can she get within one vertical meter? Yes. If she is anywhere within seven horizontal meters of p, then her altitude always remains within one meter from the target L. In summary, to say that the traveler's altitude approaches L as her horizontal position approaches p means that for every target accuracy goal, however small it may be, there is some neighborhood of p whose altitude fulfills that accuracy goal.

text: The initial informal statement can now be explicated:

text: This explicit statement is quite close to the formal definition of the limit of a function with values in a topological space.

text: To say that

text: means that ƒ(x) can be made as close as desired to L by making x close enough, but not equal, to p.

text: The following definitions (known as (ε, δ)-definitions) are the generally accepted ones for the limit of a function in various contexts.

text: Functions of a single variable[edit]

text: Suppose f : R → R is defined on the real line and p,L ∈ R. It is said the limit of f, as x approaches p, is L and written

text: if the following property holds:

text: For every real ε > 0, there exists a real δ > 0 such that for all real x, 0 < | x − p | < δ implies | f(x) − L | < ε.

text: The value of the limit does not depend on the value of f(p), nor even that p be in the domain of f.

text: A more general definition applies for functions defined on subsets of the real line. Let (a, b) be an open interval in R, and p a point of (a, b). Let f be a real-valued function defined on all of (a, b) except possibly at p itself. It is then said that the limit of f, as x approaches p, is L if, for every real ε > 0, there exists a real δ > 0 such that 0 < | x − p | < δ and x ∈ (a, b) implies | f(x) − L | < ε.

text: Here again the limit does not depend on f(p) being well-defined.

text:
The letters ε and δ can be understood as "error" and "distance", and in fact Cauchy used ε as an abbreviation for "error" in some of his work (Grabiner 1983), though in his definition of continuity he used an infinitesimal

α

{\displaystyle \alpha }

 rather than either ε or δ (see Cours d'Analyse). In these terms, the error (ε) in the measurement of the value at the limit can be made as small as desired by reducing the distance (δ) to the limit point. As discussed below this definition also works for functions in a more general context. The idea that δ and ε represent distances helps suggest these generalizations.

text: Existence and one-sided limits[edit]

text: Alternatively x may approach p from above (right) or below (left), in which case the limits may be written as

text: or

text: respectively. If these limits exist at p and are equal there, then this can be referred to as the limit of f(x) at p. If the one-sided limits exist at p, but are unequal, there is no limit at p (the limit at p does not exist). If either one-sided limit does not exist at p, the limit at p does not exist.

text: A formal definition is as follows. The limit of f(x) as x approaches p from above is L if, for every ε > 0, there exists a δ > 0 such that |f(x) − L| < ε whenever 0 < x − p < δ. The limit of f(x) as x approaches p from below is L if, for every ε > 0, there exists a δ > 0 such that |f(x) − L| < ε whenever 0 < p − x < δ.

text: If the limit does not exist then the oscillation of f at p is non-zero.

text: More general subsets[edit]

text: Apart from open intervals, limits can be defined for functions on arbitrary subsets of R, as follows. Let f be a real-valued function defined on a subset S of the real line. Let p be a limit point of S—that is, p is the limit of some sequence of distinct elements of S. The limit of f, as x approaches p from values in S, is L if, for every ε > 0, there exists a δ > 0 such that 0 < |x − p| < δ and x ∈ S implies |f(x) − L| < ε.

text: This limit is often written

text:
The condition that f be defined on S is that S be a subset of the domain of f. This generalization includes as special cases limits on an interval, as well as left-handed limits of real-valued functions (e.g., by taking S to be an open interval of the form

(
−
∞
,
a
)

{\displaystyle (-\infty ,a)}

), and right-handed limits (e.g., by taking S to be an open interval of the form

(
a
,
∞
)

{\displaystyle (a,\infty )}

).

text: Deleted versus non-deleted limits[edit]

text: The definition of limit given here does not depend on how (or whether) f is defined at p. Bartle (1967), refers to this as a deleted limit, because it excludes the value of f at p. The corresponding non-deleted limit does depend on the value of f at p, if p is in the domain of f:

text: A number L is the non-deleted limit of f as x approaches p if, for every ε > 0, there exists a δ > 0 such that | x − p | < δ and x ∈ Dm(f) implies | f(x) − L | < ε.

text: The definition is the same, except that the neighborhood | x − p | < δ now includes the point p, in contrast to the (deleted) neighborhood 0 < | x − p | < δ. This makes the definition of a non-deleted limit less general. One of the advantages of working with non-deleted limits is that they allow to state the theorem about limits of compositions without any constraints on the functions (other than the existence of their non-deleted limits) (Hubbard (2015)).

text: Bartle (1967) notes that although by "limit" some authors do mean this non-deleted limit, deleted limits are the most popular. For example, Apostol (1974), Courant (1924), Hardy (1921), Rudin (1964), Whittaker & Watson (1902) all by "limit" mean the deleted version.

text: Examples[edit]

text: Non-existence of one-sided limit(s)[edit]

text: The function

text:
has no limit at

x

0

=
1

{\displaystyle x_{0}=1}

 (the left-hand limit does not exist due to the oscillatory nature of the sine function, and the right-hand limit does not exist due to the asymptotic behaviour of the reciprocal function), but has a limit at every other x-coordinate.

text: The function

text: (the Dirichlet function) has no limit at any x-coordinate.

text: Non-equality of one-sided limits[edit]

text: The function

text: has a limit at every non-zero x-coordinate (the limit equals 1 for negative x and equals 2 for positive x). The limit at x = 0 does not exist (the left-hand limit equals 1, whereas the right-hand limit equals 2).

text: Limits at only one point[edit]

text: The functions

text: and

text: both have a limit at x = 0 and it equals 0.

text: Limits at countably many points[edit]

text: The function

text:
has a limit at any x-coordinate of the form

π
2

+
2
n
π

{\displaystyle {\frac {\pi }{2}}+2n\pi }

, where n is any integer.

text: Functions on metric spaces[edit]

text: Suppose M and N are subsets of metric spaces A and B, respectively, and f : M → N is defined between M and N, with x ∈ M, p a limit point of M and L ∈ N. It is said that the limit of f as x approaches p is L and write

text: if the following property holds:

text: For every ε > 0, there exists a δ > 0 such that dB(f(x), L) < ε whenever 0 < dA(x, p) < δ.

text: Again, note that p need not be in the domain of f, nor does L need to be in the range of f, and even if f(p) is defined it need not be equal to L.

text: An alternative definition using the concept of neighbourhood is as follows:

text: if, for every neighbourhood V of L in B, there exists a neighbourhood U of p in A such that f(U ∩ M − {p}) ⊆ V.

text: Functions on topological spaces[edit]

text: Suppose X,Y are topological spaces with Y a Hausdorff space. Let p be a limit point of Ω ⊆ X, and L ∈Y. For a function f : Ω → Y, it is said that the limit of f as x approaches p is L (i.e., f(x) → L as x → p) and written

text: if the following property holds:

text: For every open neighborhood V of L, there exists an open neighborhood U of p such that f(U ∩ Ω − {p}) ⊆ V.

text: This last part of the definition can also be phrased "there exists an open punctured neighbourhood U of p such that f(U∩Ω) ⊆ V ".

text: Note that the domain of f does not need to contain p. If it does, then the value of f at p is irrelevant to the definition of the limit. In particular, if the domain of f is X − {p} (or all of X), then the limit of f as x → p exists and is equal to L if, for all subsets Ω of X with limit point p, the limit of the restriction of f to Ω exists and is equal to L. Sometimes this criterion is used to establish the non-existence of the two-sided limit of a function on R by showing that the one-sided limits either fail to exist or do not agree. Such a view is fundamental in the field of general topology, where limits and continuity at a point are defined in terms of special families of subsets, called filters, or generalized sequences known as nets.

text: Alternatively, the requirement that Y be a Hausdorff space can be relaxed to the assumption that Y be a general topological space, but then the limit of a function may not be unique. In particular, one can no longer talk about the limit of a function at a point, but rather a limit or the set of limits at a point.

text: A function is continuous in a limit point p of and in its domain if and only if f(p) is the (or, in the general case, a) limit of f(x) as x tends to p.

text: Limits involving infinity[edit]

text: Limits at infinity[edit]

text: For f(x) a real function, the limit of f as x approaches infinity is L, denoted

text:
means that for all

ε
>
0

{\displaystyle \varepsilon >0}

, there exists c such that

|

f
(
x
)
−
L

|

<
ε

{\displaystyle |f(x)-L|<\varepsilon }

 whenever x > c. Or, symbolically:

text: Similarly, the limit of f as x approaches negative infinity is L, denoted

text:
means that for all

ε
>
0

{\displaystyle \varepsilon >0}

 there exists c such that

|

f
(
x
)
−
L

|

<
ε

{\displaystyle |f(x)-L|<\varepsilon }

 whenever x < c. Or, symbolically:

text: For example

text: Infinite limits[edit]

text: Limits can also have infinite values. When infinities are not considered legitimate values, which is standard (but see below), a formalist will insist upon various circumlocutions. For example, rather than say that a limit is infinity, the proper thing is to say that the function "diverges" or "grows without bound". In particular, the following informal example of how to pronounce the notation is arguably inappropriate in the classroom (or any other formal setting). In any case, for example the limit of f as x approaches a is infinity, denoted

text:
means that for all

ε
>
0

{\displaystyle \varepsilon >0}

 there exists

δ
>
0

{\displaystyle \delta >0}

 such that

f
(
x
)
>
ε

{\displaystyle f(x)>\varepsilon }

 whenever

|

x
−
a

|

<
δ

{\displaystyle |x-a|<\delta }

.

text: These ideas can be combined in a natural way to produce definitions for different combinations, such as

text: For example

text: Limits involving infinity are connected with the concept of asymptotes.

text: These notions of a limit attempt to provide a metric space interpretation to limits at infinity. However, note that these notions of a limit are consistent with the topological space definition of limit if

text:
a neighborhood of −∞ is defined to contain an interval [−∞, c) for some c ∈ R
a neighborhood of ∞ is defined to contain an interval (c, ∞] where c ∈ R
a neighborhood of a∈R is defined in the normal way metric space R

text: In this case, R is a topological space and any function of the form f: X → Y with X, Y⊆ R is subject to the topological definition of a limit. Note that with this topological definition, it is easy to define infinite limits at finite points, which have not been defined above in the metric sense.

text: Alternative notation[edit]

text:
Many authors[1] allow for the projectively extended real line to be used as a way to include infinite values as well as extended real line. With this notation, the extended real line is given as R ∪ {−∞, +∞} and the projectively extended real line is R ∪ {∞} where a neighborhood of ∞ is a set of the form {x: |x| > c}. The advantage is that one only needs three definitions for limits (left, right, and central) to cover all the cases. As presented above, for a completely rigorous account, we would need to consider 15 separate cases for each combination of infinities (five directions: −∞, left, central, right, and +∞; three bounds: −∞, finite, or +∞). There are also noteworthy pitfalls. For example, when working with the extended real line,

x

−
1

{\displaystyle x^{-1}}

 does not possess a central limit (which is normal):

text: In contrast, when working with the projective real line, infinities (much like 0) are unsigned, so, the central limit does exist in that context:

text:
In fact there are a plethora of conflicting formal systems in use. In certain applications of numerical differentiation and integration, it is, for example, convenient to have signed zeroes. A simple reason has to do with the converse of

lim

x
→

0

−

x

−
1

=
−
∞

{\displaystyle \lim _{x\to 0^{-}}{x^{-1}}=-\infty }

, namely, it is convenient for

lim

x
→
−
∞

x

−
1

=
−
0

{\displaystyle \lim _{x\to -\infty }{x^{-1}}=-0}

 to be considered true. Such zeroes can be seen as an approximation to infinitesimals.

text: Limits at infinity for rational functions[edit]

text: There are three basic rules for evaluating limits at infinity for a rational function f(x) = p(x)/q(x): (where p and q are polynomials):

text:
If the degree of p is greater than the degree of q, then the limit is positive or negative infinity depending on the signs of the leading coefficients;
If the degree of p and q are equal, the limit is the leading coefficient of p divided by the leading coefficient of q;
If the degree of p is less than the degree of q, the limit is 0.

text: If the limit at infinity exists, it represents a horizontal asymptote at y = L. Polynomials do not have horizontal asymptotes; such asymptotes may however occur with rational functions.

text: Functions of more than one variable[edit]

text: By noting that |x − p| represents a distance, the definition of a limit can be extended to functions of more than one variable. In the case of a function f : R2 → R,

text: if

text: where ||(x,y) − (p,q)|| represents the Euclidean distance. This can be extended to any number of variables.

text: Sequential limits[edit]

text: Let f : X → Y be a mapping from a topological space X into a Hausdorff space Y, p ∈ X and L ∈ Y.

text: If L is the limit (in the sense above) of f as x approaches p, then it is a sequential limit as well, however the converse need not hold in general. If in addition X is metrizable, then L is the sequential limit of f as x approaches p if and only if it is the limit (in the sense above) of f as x approaches p.

text: Other characterizations[edit]

text: In terms of sequences[edit]

text: For functions on the real line, one way to define the limit of a function is in terms of the limit of sequences. (This definition is usually attributed to Eduard Heine.) In this setting:

text:
if and only if for all sequences

x

n

{\displaystyle x_{n}}

 (with

x

n

{\displaystyle x_{n}}

 not equal to a for all n) converging to

a

{\displaystyle a}

 the sequence

f
(

x

n

)

{\displaystyle f(x_{n})}

 converges to

L

{\displaystyle L}

. It was shown by Sierpiński in 1916 that proving the equivalence of this definition and the definition above, requires and is equivalent to a weak form of the axiom of choice. Note that defining what it means for a sequence

x

n

{\displaystyle x_{n}}

 to converge to

a

{\displaystyle a}

 requires the epsilon, delta method.

text:
Similarly as it was the case of Weierstrass's definition, a more general Heine definition applies to functions defined on subsets of the real line. Let f be a real-valued function with the domain Dm(f). Let a be the limit of a sequence of elements of Dm(f). Then the limit (in this sense) of f is L as x approaches p if for every sequence

x

n

{\displaystyle x_{n}}

 ∈ Dm(f) \ {a} (so that for all n,

x

n

{\displaystyle x_{n}}

 is not equal to a) that converges to a, the sequence

f
(

x

n

)

{\displaystyle f(x_{n})}

 converges to

L

{\displaystyle L}

. This is the same as the definition of a sequential limit in the preceding section obtained by regarding the subset Dm(f) of R as a metric space with the induced metric.

text: In non-standard calculus[edit]

text: In non-standard calculus the limit of a function is defined by:

text:
if and only if for all

x
∈

R

∗

{\displaystyle x\in \mathbb {R} ^{*}}

,

f

∗

(
x
)
−
L

{\displaystyle f^{*}(x)-L}

 is infinitesimal whenever

x
−
a

{\displaystyle x-a}

 is infinitesimal. Here

R

∗

{\displaystyle \mathbb {R} ^{*}}

 are the hyperreal numbers and

f

∗

{\displaystyle f^{*}}

 is the natural extension of f to the non-standard real numbers. Keisler proved that such a hyperreal definition of limit reduces the quantifier complexity by two quantifiers.[2] On the other hand, Hrbacek writes that for the definitions to be valid for all hyperreal numbers they must implicitly be grounded in the ε-δ method, and claims that, from the pedagogical point of view, the hope that non-standard calculus could be done without ε-δ methods cannot be realized in full.[3] Bŀaszczyk et al. detail the usefulness of microcontinuity in developing a transparent definition of uniform continuity, and characterize Hrbacek's criticism as a "dubious lament".[4]

text: In terms of nearness[edit]

text:
At the 1908 international congress of mathematics F. Riesz introduced an alternate way defining limits and continuity in concept called "nearness". A point

x

{\displaystyle x}

 is defined to be near a set

A
⊆

R

{\displaystyle A\subseteq \mathbb {R} }

 if for every

r
>
0

{\displaystyle r>0}

 there is a point

a
∈
A

{\displaystyle a\in A}

 so that

|

x
−
a

|

<
r

{\displaystyle |x-a|<r}

. In this setting the

text:
if and only if for all

A
⊆

R

{\displaystyle A\subseteq \mathbb {R} }

,

L

{\displaystyle L}

 is near

f
(
A
)

{\displaystyle f(A)}

 whenever

a

{\displaystyle a}

 is near

A

{\displaystyle A}

. Here

f
(
A
)

{\displaystyle f(A)}

 is the set

{
f
(
x
)

|

x
∈
A
}

{\displaystyle \{f(x)|x\in A\}}

. This definition can also be extended to metric and topological spaces.

text: Relationship to continuity[edit]

text: The notion of the limit of a function is very closely related to the concept of continuity. A function ƒ is said to be continuous at c if it is both defined at c and its value at c equals the limit of f as x approaches c:

text: (We have here assumed that c is a limit point of the domain of f.)

text: Properties[edit]

text: If a function f is real-valued, then the limit of f at p is L if and only if both the right-handed limit and left-handed limit of f at p exist and are equal to L.

text: The function f is continuous at p if and only if the limit of f(x) as x approaches p exists and is equal to f(p). If f : M → N is a function between metric spaces M and N, then it is equivalent that f transforms every sequence in M which converges towards p into a sequence in N which converges towards f(p).

text: If N is a normed vector space, then the limit operation is linear in the following sense: if the limit of f(x) as x approaches p is L and the limit of g(x) as x approaches p is P, then the limit of f(x) + g(x) as x approaches p is L + P. If a is a scalar from the base field, then the limit of af(x) as x approaches p is aL.

text: If f is a real-valued (or complex-valued) function, then taking the limit is compatible with the algebraic operations, provided the limits on the right sides of the equations below exist (the last identity only holds if the denominator is non-zero). This fact is often called the algebraic limit theorem.

text: In each case above, when the limits on the right do not exist, or, in the last case, when the limits in both the numerator and the denominator are zero, nonetheless the limit on the left, called an indeterminate form, may still exist—this depends on the functions f and g. These rules are also valid for one-sided limits, for the case p = ±∞, and also for infinite limits using the rules

text:
q + ∞ = ∞ for q ≠ −∞
q × ∞ = ∞ if q > 0
q × ∞ = −∞ if q < 0
q / ∞ = 0 if q ≠ ± ∞

text: (see extended real number line).

text: Note that there is no general rule for the case q / 0; it all depends on the way 0 is approached. Indeterminate forms—for instance, 0/0, 0×∞, ∞−∞, and ∞/∞—are also not covered by these rules, but the corresponding limits can often be determined with L'Hôpital's rule or the Squeeze theorem.

text: Limits of compositions of functions[edit]

text: In general, from knowing that

text:
it does not follow that

lim

x
→
a

f
(
g
(
x
)
)
=
c

{\displaystyle \lim _{x\to a}f(g(x))=c}

. However, this "chain rule" does hold if one of the following additional conditions holds:

text:
f(b) = c (that is, f is continuous at b), or
g does not take the value b near a (that is, there exists a

δ
>
0

{\displaystyle \delta >0}

 such that if

0
<

|

x
−
a

|

<
δ

{\displaystyle 0<|x-a|<\delta }

 then

|

g
(
x
)
−
b

|

>
0

{\displaystyle |g(x)-b|>0}

).

text: As an example of this phenomenon, consider the following functions that violates both additional restrictions:

text: Since the value at f(0) is a removable discontinuity,

text: Thus, the naïve chain rule would suggest that the limit of f(f(x)) is 0. However, it is the case that

text: and so

text: Limits of special interest[edit]

text: Rational functions[edit]

text:
For

n

{\displaystyle n}

 a nonnegative integer and constants

a

1

,

a

2

,

a

3

,
…
,

a

n

{\displaystyle a_{1},a_{2},a_{3},\ldots ,a_{n}}

 and

b

1

,

b

2

,

b

3

,
…
,

b

n

{\displaystyle b_{1},b_{2},b_{3},\ldots ,b_{n}}

,

text:
lim

x
→
∞

a

1

x

n

+

a

2

x

n
−
1

+

a

3

x

n
−
2

+
.
.
.
+

a

n

b

1

x

n

+

b

2

x

n
−
1

+

b

3

x

n
−
2

+
.
.
.
+

b

n

=

a

1

b

1

{\displaystyle \lim _{x\to \infty }{\frac {a_{1}{x}^{n}+a_{2}{x}^{n-1}+a_{3}{x}^{n-2}+...+a_{n}}{b_{1}{x}^{n}+b_{2}{x}^{n-1}+b_{3}{x}^{n-2}+...+b_{n}}}={\frac {a_{1}}{b_{1}}}}

text:
This can be proven by dividing both the numerator and denominator by

x

n

{\displaystyle x^{n}}

. If the numerator is a polynomial of higher degree, the limit does not exist. If the denominator is of higher degree, the limit is 0.

text: Trigonometric functions[edit]

text:
lim

x
→
0

sin

x

x

=
1

{\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1}

lim

x
→
0

1
−
cos

x

x

=
0

{\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x}}=0}

lim

x
→
∞

x
sin

(

1
x

)

=
1

{\displaystyle \lim _{x\to \infty }x\sin \left({\frac {1}{x}}\right)=1}

text: Exponential functions[edit]

text:
lim

x
→
0

(
1
+
x

)

1
x

=

lim

r
→
∞

(
1
+

1
r

)

r

=
e

{\displaystyle \lim _{x\to 0}(1+x)^{\frac {1}{x}}=\lim _{r\to \infty }\left(1+{\frac {1}{r}}\right)^{r}=e}

lim

x
→
0

e

x

−
1

x

=
1

{\displaystyle \lim _{x\to 0}{\frac {e^{x}-1}{x}}=1}

lim

x
→
0

e

a
x

−
1

b
x

=

a
b

{\displaystyle \lim _{x\to 0}{\frac {e^{ax}-1}{bx}}={\frac {a}{b}}}

lim

x
→
0

c

a
x

−
1

b
x

=

a
b

ln

c

{\displaystyle \lim _{x\to 0}{\frac {c^{ax}-1}{bx}}={\frac {a}{b}}\ln c}

lim

x
→

0

+

x

x

=
1

{\displaystyle \lim _{x\to 0^{+}}x^{x}=1}

text: Logarithmic functions[edit]

text:
lim

x
→
0

ln

(
1
+
x
)

x

=
1

{\displaystyle \lim _{x\to 0}{\frac {\ln(1+x)}{x}}=1}

lim

x
→
0

ln

(
1
+
a
x
)

b
x

=

a
b

{\displaystyle \lim _{x\to 0}{\frac {\ln(1+ax)}{bx}}={\frac {a}{b}}}

lim

x
→
0

log

c

(
1
+
a
x
)

b
x

=

a

b
ln

c

{\displaystyle \lim _{x\to 0}{\frac {\log _{c}(1+ax)}{bx}}={\frac {a}{b\ln c}}}

text: L'Hôpital's rule[edit]

text: This rule uses derivatives to find limits of indeterminate forms 0/0 or ±∞/∞, and only applies to such cases. Other indeterminate forms may be manipulated into this form. Given two functions f(x) and g(x), defined over an open interval I containing the desired limit point c, then if:

text:
lim

x
→
c

f
(
x
)
=

lim

x
→
c

g
(
x
)
=
0
,

{\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0,}

 or

lim

x
→
c

f
(
x
)
=
±

lim

x
→
c

g
(
x
)
=
±
∞

{\displaystyle \lim _{x\to c}f(x)=\pm \lim _{x\to c}g(x)=\pm \infty }

, and

f

{\displaystyle f}

 and

g

{\displaystyle g}

 are differentiable over

I
∖
{
c
}

{\displaystyle I\setminus \{c\}}

, and

g
′

(
x
)
≠
0

{\displaystyle g'(x)\neq 0}

 for all

x
∈
I
∖
{
c
}

{\displaystyle x\in I\setminus \{c\}}

, and

lim

x
→
c

f
′

(
x
)

g
′

(
x
)

{\displaystyle \lim _{x\to c}{\frac {f'(x)}{g'(x)}}}

 exists,

text: then:

text:
lim

x
→
c

f
(
x
)

g
(
x
)

=

lim

x
→
c

f
′

(
x
)

g
′

(
x
)

{\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}}

text: Normally, the first condition is the most important one.

text:
For example:

lim

x
→
0

sin

(
2
x
)

sin

(
3
x
)

=

lim

x
→
0

2
cos

(
2
x
)

3
cos

(
3
x
)

=

2
⋅
1

3
⋅
1

=

2
3

.

{\displaystyle \lim _{x\to 0}{\frac {\sin(2x)}{\sin(3x)}}=\lim _{x\to 0}{\frac {2\cos(2x)}{3\cos(3x)}}={\frac {2\cdot 1}{3\cdot 1}}={\frac {2}{3}}.}

text: Summations and integrals[edit]

text: Specifying an infinite bound on a summation or integral is a common shorthand for specifying a limit.

text:
A short way to write the limit

lim

n
→
∞

∑

i
=
s

n

f
(
i
)

{\displaystyle \lim _{n\to \infty }\sum _{i=s}^{n}f(i)}

 is

∑

i
=
s

∞

f
(
i
)

{\displaystyle \sum _{i=s}^{\infty }f(i)}

. An important example of limits of sums such as these are series.

text:
A short way to write the limit

lim

x
→
∞

∫

a

x

f
(
t
)

d
t

{\displaystyle \lim _{x\to \infty }\int _{a}^{x}f(t)\;dt}

 is

∫

a

∞

f
(
t
)

d
t

{\displaystyle \int _{a}^{\infty }f(t)\;dt}

.

text:
A short way to write the limit

lim

x
→
−
∞

∫

x

b

f
(
t
)

d
t

{\displaystyle \lim _{x\to -\infty }\int _{x}^{b}f(t)\;dt}

 is

∫

−
∞

b

f
(
t
)

d
t

{\displaystyle \int _{-\infty }^{b}f(t)\;dt}

.


document_title: National Honor Society

question_text: percentage of high school students in national honor society

text: National Honor Society

text: History[edit]

text: The National Association of Secondary School Principals (NASSP) officially established NHS in 1921. Though many local and regional honor societies existed prior to 1921, no nationwide organization had been founded. Under the leadership of Dr. Edward Rynearson, principal of the Fifth Avenue High School in Pittsburgh, the organization grew from the original Alpha Chapter at the Fifth Avenue School to more than 1,000 chapters by 1930. Equipped with a constitution, an emblem and motto, and a group of dedicated principals as coordinators, the new NHS organization quickly developed into one of the country's leading educational groups.

text:
Four main purposes have guided chapters of NHS from the beginning:
1. to create enthusiasm for scholarship
2. to stimulate a desire to render service
3. to promote leadership
4. to develop character in the students of secondary schools

text: Membership[edit]

text: Over one million students are estimated to participate in the National Honor Society. NHS and NJHS chapters are found in all 50 states, the District of Columbia, Puerto Rico, many U.S. territories, and Canada. They can also be found in areas of Asia; Pakistan has three schools maintaining an active chapter. They further can be found in international and American schools throughout the globe.

text: Motto[edit]

text: The NHS motto is noblesse oblige.[2] This is French for "nobility obligates". The Dictionnaire de l'Académie française defines it thus:

text:
Whoever claims to be noble must conduct himself nobly.
(Figuratively) One must act in a fashion that conforms to one's position, and with the reputation that one has earned.

tag: ol

text: Scholarship[edit]

text: Since 1946, the National Honor Society has given out more than $12 million in scholarship grants. Among all the applicants, 300 will receive $1,500 each while 25 will get $3,000 each (with an additional $1,500 added to the national semifinalist award), and one will receive the award of $15,500 plus $12,500 to the national finalist award.[3]

text: Parent and sister organizations[edit]

text:
National Association of Secondary School Principals
National Junior Honor Society
National Elementary Honor Society
National Association of Student Councils[4]