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20,700 | On March 10, 1876, Bell used "the instrument" in Boston to call Thomas Watson who was in another room but out of earshot. He said, "Mr. Watson, come here – I want to see you" and Watson soon appeared at his side. | https://en.wikipedia.org/wiki?curid=852 |
20,701 | Continuing his experiments in Brantford, Bell brought home a working model of his telephone. On August 3, 1876, from the telegraph office in Brantford, Ontario, Bell sent a tentative telegram to the village of Mount Pleasant distant, indicating that he was ready. He made a telephone call via telegraph wires and faint v... | https://en.wikipedia.org/wiki?curid=852 |
20,702 | The first two-way (reciprocal) conversation over a line occurred between Cambridge and Boston (roughly 2.5 miles) on October 9, 1876. During that conversation, Bell was on Kilby Street in Boston and Watson was at the offices of the Walworth Manufacturing Company. | https://en.wikipedia.org/wiki?curid=852 |
20,703 | Bell and his partners, Hubbard and Sanders, offered to sell the patent outright to Western Union for $100,000, equal to $ today. The president of Western Union balked, countering that the telephone was nothing but a toy. Two years later, he told colleagues that if he could get the patent for $25 million (equal to $ tod... | https://en.wikipedia.org/wiki?curid=852 |
20,704 | Bell began a series of public demonstrations and lectures to introduce the new invention to the scientific community as well as the general public. A short time later, his demonstration of an early telephone prototype at the 1876 Centennial Exposition in Philadelphia brought the telephone to international attention. In... | https://en.wikipedia.org/wiki?curid=852 |
20,705 | On January 14, 1878, at Osborne House, on the Isle of Wight, Bell demonstrated the device to Queen Victoria, placing calls to Cowes, Southampton and London. These were the first publicly witnessed long-distance telephone calls in the UK. The queen considered the process to be "quite extraordinary" although the sound wa... | https://en.wikipedia.org/wiki?curid=852 |
20,706 | The Bell Telephone Company was created in 1877, and by 1886, more than 150,000 people in the U.S. owned telephones. Bell Company engineers made numerous other improvements to the telephone, which emerged as one of the most successful products ever. In 1879, the Bell company acquired Edison's patents for the carbon micr... | https://en.wikipedia.org/wiki?curid=852 |
20,707 | Emperor Pedro II of Brazil was the first person to buy stock in Bell's company, the Bell Telephone Company. One of the first telephones in a private residence was installed in his palace in Petrópolis, his summer retreat from Rio de Janeiro. | https://en.wikipedia.org/wiki?curid=852 |
20,708 | In January 1915, Bell made the first ceremonial transcontinental telephone call. Calling from the AT&T head office at 15 Dey Street in New York City, Bell was heard by Thomas Watson at 333 Grant Avenue in San Francisco. "The New York Times" reported: | https://en.wikipedia.org/wiki?curid=852 |
20,709 | As is sometimes common in scientific discoveries, simultaneous developments can occur, as evidenced by a number of inventors who were at work on the telephone. Over a period of 18 years, the Bell Telephone Company faced 587 court challenges to its patents, including five that went to the U.S. Supreme Court, but none wa... | https://en.wikipedia.org/wiki?curid=852 |
20,710 | On January 13, 1887, the U.S. Government moved to annul the patent issued to Bell on the grounds of fraud and misrepresentation. After a series of decisions and reversals, the Bell company won a decision in the Supreme Court, though a couple of the original claims from the lower court cases were left undecided. By the ... | https://en.wikipedia.org/wiki?curid=852 |
20,711 | During a deposition filed for the 1887 trial, Italian inventor Antonio Meucci also claimed to have created the first working model of a telephone in Italy in 1834. In 1886, in the first of three cases in which he was involved, Meucci took the stand as a witness in the hope of establishing his invention's priority. Meuc... | https://en.wikipedia.org/wiki?curid=852 |
20,712 | The value of the Bell patent was acknowledged throughout the world, and patent applications were made in most major countries, but when Bell delayed the German patent application, the electrical firm of Siemens & Halske set up a rival manufacturer of Bell telephones under their own patent. The Siemens company produced ... | https://en.wikipedia.org/wiki?curid=852 |
20,713 | On July 11, 1877, a few days after the Bell Telephone Company was established, Bell married Mabel Hubbard (1857–1923) at the Hubbard estate in Cambridge, Massachusetts. His wedding present to his bride was to turn over 1,487 of his 1,497 shares in the newly formed Bell Telephone Company. Shortly thereafter, the newlywe... | https://en.wikipedia.org/wiki?curid=852 |
20,714 | The Bell family home was in Cambridge, Massachusetts, until 1880 when Bell's father-in-law bought a house in Washington, D.C.; in 1882 he bought a home in the same city for Bell's family, so they could be with him while he attended to the numerous court cases involving patent disputes. | https://en.wikipedia.org/wiki?curid=852 |
20,715 | Bell was a British subject throughout his early life in Scotland and later in Canada until 1882 when he became a naturalized citizen of the United States. In 1915, he characterized his status as: "I am not one of those hyphenated Americans who claim allegiance to two countries." Despite this declaration, Bell has been ... | https://en.wikipedia.org/wiki?curid=852 |
20,716 | By 1885, a new summer retreat was contemplated. That summer, the Bells had a vacation on Cape Breton Island in Nova Scotia, spending time at the small village of Baddeck. Returning in 1886, Bell started building an estate on a point across from Baddeck, overlooking Bras d'Or Lake. By 1889, a large house, christened "Th... | https://en.wikipedia.org/wiki?curid=852 |
20,717 | Until the end of his life, Bell and his family would alternate between the two homes, but "Beinn Bhreagh" would, over the next 30 years, become more than a summer home as Bell became so absorbed in his experiments that his annual stays lengthened. Both Mabel and Bell became immersed in the Baddeck community and were ac... | https://en.wikipedia.org/wiki?curid=852 |
20,718 | Although Alexander Graham Bell is most often associated with the invention of the telephone, his interests were extremely varied. According to one of his biographers, Charlotte Gray, Bell's work ranged "unfettered across the scientific landscape" and he often went to bed voraciously reading the "Encyclopædia Britannica... | https://en.wikipedia.org/wiki?curid=852 |
20,719 | Bell worked extensively in medical research and invented techniques for teaching speech to the deaf. During his Volta Laboratory period, Bell and his associates considered impressing a magnetic field on a record as a means of reproducing sound. Although the trio briefly experimented with the concept, they could not dev... | https://en.wikipedia.org/wiki?curid=852 |
20,720 | Bell's own home used a primitive form of air conditioning, in which fans blew currents of air across great blocks of ice. He also anticipated modern concerns with fuel shortages and industrial pollution. Methane gas, he reasoned, could be produced from the waste of farms and factories. At his Canadian estate in Nova Sc... | https://en.wikipedia.org/wiki?curid=852 |
20,721 | Bell and his assistant Charles Sumner Tainter jointly invented a wireless telephone, named a photophone, which allowed for the transmission of both sounds and normal human conversations on a beam of light. Both men later became full associates in the Volta Laboratory Association. | https://en.wikipedia.org/wiki?curid=852 |
20,722 | On June 21, 1880, Bell's assistant transmitted a wireless voice telephone message a considerable distance, from the roof of the Franklin School in Washington, D.C., to Bell at the window of his laboratory, some away, 19 years before the first voice radio transmissions. | https://en.wikipedia.org/wiki?curid=852 |
20,723 | Bell believed the photophone's principles were his life's "greatest achievement", telling a reporter shortly before his death that the photophone was "the greatest invention [I have] ever made, greater than the telephone". The photophone was a precursor to the fiber-optic communication systems which achieved popular wo... | https://en.wikipedia.org/wiki?curid=852 |
20,724 | Bell is also credited with developing one of the early versions of a metal detector through the use of an induction balance, after the shooting of U.S. President James A. Garfield in 1881. According to some accounts, the metal detector worked flawlessly in tests but did not find Guiteau's bullet, partly because the met... | https://en.wikipedia.org/wiki?curid=852 |
20,725 | Bell's own detailed account, presented to the American Association for the Advancement of Science in 1882, differs in several particulars from most of the many and varied versions now in circulation, by concluding that extraneous metal was not to blame for failure to locate the bullet. Perplexed by the peculiar results... | https://en.wikipedia.org/wiki?curid=852 |
20,726 | The March 1906 "Scientific American" article by American pioneer William E. Meacham explained the basic principle of hydrofoils and hydroplanes. Bell considered the invention of the hydroplane as a very significant achievement. Based on information gained from that article, he began to sketch concepts of what is now ca... | https://en.wikipedia.org/wiki?curid=852 |
20,727 | During his world tour of 1910–11, Bell and Baldwin met with Forlanini in France. They had rides in the Forlanini hydrofoil boat over Lake Maggiore. Baldwin described it as being as smooth as flying. On returning to Baddeck, a number of initial concepts were built as experimental models, including the "Dhonnas Beag" (Sc... | https://en.wikipedia.org/wiki?curid=852 |
20,728 | In 1891, Bell had begun experiments to develop motor-powered heavier-than-air aircraft. The AEA was first formed as Bell shared the vision to fly with his wife, who advised him to seek "young" help as Bell was at the age of 60. | https://en.wikipedia.org/wiki?curid=852 |
20,729 | In 1898, Bell experimented with tetrahedral box kites and wings constructed of multiple compound tetrahedral kites covered in maroon silk. The tetrahedral wings were named "Cygnet" I, II, and III, and were flown both unmanned and manned ("Cygnet I" crashed during a flight carrying Selfridge) in the period from 1907 to ... | https://en.wikipedia.org/wiki?curid=852 |
20,730 | Bell was a supporter of aerospace engineering research through the Aerial Experiment Association (AEA), officially formed at Baddeck, Nova Scotia, in October 1907 at the suggestion of his wife Mabel and with her financial support after the sale of some of her real estate. The AEA was headed by Bell and the founding mem... | https://en.wikipedia.org/wiki?curid=852 |
20,731 | The AEA's work progressed to heavier-than-air machines, applying their knowledge of kites to gliders. Moving to Hammondsport, the group then designed and built the "Red Wing", framed in bamboo and covered in red silk and powered by a small air-cooled engine. On March 12, 1908, over Keuka Lake, the biplane lifted off on... | https://en.wikipedia.org/wiki?curid=852 |
20,732 | Their final aircraft design, the "Silver Dart", embodied all of the advancements found in the earlier machines. On February 23, 1909, Bell was present as the "Silver Dart" flown by J. A. D. McCurdy from the frozen ice of Bras d'Or made the first aircraft flight in Canada. Bell had worried that the flight was too danger... | https://en.wikipedia.org/wiki?curid=852 |
20,733 | Bell, along with many members of the scientific community at the time, took an interest in the popular science of heredity which grew out of the publication of Charles Darwin's book "On the Origin of Species" in 1859. On his estate in Nova Scotia, Bell conducted meticulously recorded breeding experiments with rams and ... | https://en.wikipedia.org/wiki?curid=852 |
20,734 | In November 1883, Bell presented a paper at a meeting of the National Academy of Sciences titled "Upon the Formation of a Deaf Variety of the Human Race". The paper is a compilation of data on the hereditary aspects of deafness. Bell's research indicated that a hereditary tendency toward deafness, as indicated by the p... | https://en.wikipedia.org/wiki?curid=852 |
20,735 | A review of Bell's "Memoir upon the Formation of a Deaf Variety of the Human Race" appearing in an 1885 issue of the "American Annals of the Deaf and Dumb" states that "Dr. Bell does not advocate legislative interference with the marriages of the deaf for several reasons one of which is that the results of such marriag... | https://en.wikipedia.org/wiki?curid=852 |
20,736 | Historians have noted that Bell explicitly opposed laws regulating marriage, and never mentioned sterilization in any of his writings. Even after Bell agreed to engage with scientists conducting eugenic research, he consistently refused to support public policy that limited the rights or privileges of the deaf. | https://en.wikipedia.org/wiki?curid=852 |
20,737 | Bell's interest and research on heredity attracted the interest of Charles Davenport, a Harvard professor and head of the Cold Spring Harbor Laboratory. In 1906, Davenport, who was also the founder of the American Breeder's Association, approached Bell about joining a new committee on eugenics chaired by David Starr Jo... | https://en.wikipedia.org/wiki?curid=852 |
20,738 | In 1921, a Second International Congress of Eugenics was held in New York at the Museum of Natural History and chaired by Davenport. Although Bell did not present any research or speak as part of the proceedings, he was named as honorary president as a means to attract other scientists to attend the event. A summary of... | https://en.wikipedia.org/wiki?curid=852 |
20,739 | Bell died of complications arising from diabetes on August 2, 1922, at his private estate in Cape Breton, Nova Scotia, at age 75. Bell had also been affected by pernicious anemia. His last view of the land he had inhabited was by moonlight on his mountain estate at 2:00 a.m. While tending to him after his long illness,... | https://en.wikipedia.org/wiki?curid=852 |
20,740 | Bell's coffin was constructed of Beinn Bhreagh pine by his laboratory staff, lined with the same red silk fabric used in his tetrahedral kite experiments. To help celebrate his life, his wife asked guests not to wear black (the traditional funeral color) while attending his service, during which soloist Jean MacDonald ... | https://en.wikipedia.org/wiki?curid=852 |
20,741 | Upon the conclusion of Bell's funeral, for one minute at 6:25 p.m. Eastern Time, "every phone on the continent of North America was silenced in honor of the man who had given to mankind the means for direct communication at a distance". | https://en.wikipedia.org/wiki?curid=852 |
20,742 | Alexander Graham Bell was buried atop Beinn Bhreagh mountain, on his estate where he had resided increasingly for the last 35 years of his life, overlooking Bras d'Or Lake. He was survived by his wife Mabel, his two daughters, Elsie May and Marian, and nine of his grandchildren. | https://en.wikipedia.org/wiki?curid=852 |
20,743 | Honors and tributes flowed to Bell in increasing numbers as his invention became ubiquitous and his personal fame grew. Bell received numerous honorary degrees from colleges and universities to the point that the requests almost became burdensome. During his life, he also received dozens of major awards, medals, and ot... | https://en.wikipedia.org/wiki?curid=852 |
20,744 | A large number of Bell's writings, personal correspondence, notebooks, papers, and other documents reside in both the United States Library of Congress Manuscript Division (as the "Alexander Graham Bell Family Papers"), and at the Alexander Graham Bell Institute, Cape Breton University, Nova Scotia; major portions of w... | https://en.wikipedia.org/wiki?curid=852 |
20,745 | A number of historic sites and other marks commemorate Bell in North America and Europe, including the first telephone companies in the United States and Canada. Among the major sites are: | https://en.wikipedia.org/wiki?curid=852 |
20,746 | In 1880, Bell received the Volta Prize with a purse of 50,000 French francs (approximately US$ in today's dollars) for the invention of the telephone from the French government. Among the luminaries who judged were Victor Hugo and Alexandre Dumas, "fils". The Volta Prize was conceived by Napoleon III in 1852, and named... | https://en.wikipedia.org/wiki?curid=852 |
20,747 | In partnership with Gardiner Greene Hubbard, Bell helped establish the publication "Science" during the early 1880s. In 1898, Bell was elected as the second president of the National Geographic Society, serving until 1903, and was primarily responsible for the extensive use of illustrations, including photography, in t... | https://en.wikipedia.org/wiki?curid=852 |
20,748 | The "bel" (B) and the smaller "decibel" (dB) are units of measurement of sound pressure level (SPL) invented by Bell Labs and named after him. Since 1976, the IEEE's Alexander Graham Bell Medal has been awarded to honor outstanding contributions in the field of telecommunications. | https://en.wikipedia.org/wiki?curid=852 |
20,749 | In 1936, the US Patent Office declared Bell first on its list of the country's greatest inventors, leading to the US Post Office issuing a commemorative stamp honoring Bell in 1940 as part of its 'Famous Americans Series'. The First Day of Issue ceremony was held on October 28 in Boston, Massachusetts, the city where B... | https://en.wikipedia.org/wiki?curid=852 |
20,750 | The 150th anniversary of Bell's birth in 1997 was marked by a special issue of commemorative £1 banknotes from the Royal Bank of Scotland. The illustrations on the reverse of the note include Bell's face in profile, his signature, and objects from Bell's life and career: users of the telephone over the ages; an audio w... | https://en.wikipedia.org/wiki?curid=852 |
20,751 | Alexander Graham Bell was ranked 57th among the 100 Greatest Britons (2002) in an official BBC nationwide poll, and among the Top Ten Greatest Canadians (2004), and the 100 Greatest Americans (2005). In 2006, Bell was also named as one of the 10 greatest Scottish scientists in history after having been listed in the Na... | https://en.wikipedia.org/wiki?curid=852 |
20,752 | Alexander Graham Bell, who could not complete the university program of his youth, received at least a dozen honorary degrees from academic institutions, including eight honorary LL.D.s (Doctorate of Laws), two Ph.D.s, a D.Sc., and an M.D.: | https://en.wikipedia.org/wiki?curid=852 |
20,753 | The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of as approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the ... | https://en.wikipedia.org/wiki?curid=9633 |
20,754 | The (natural) exponential function is the unique function that equals its own derivative and satisfies the equation ; hence one can also define as . The natural logarithm, or logarithm to base , is the inverse function to the natural exponential function. The natural logarithm of a number can be defined directly as the... | https://en.wikipedia.org/wiki?curid=9633 |
20,755 | The number is sometimes called Euler's number (not to be confused with Euler's constant formula_2)after the Swiss mathematician Leonhard Euleror Napier's constantafter John Napier. The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest. | https://en.wikipedia.org/wiki?curid=9633 |
20,756 | The number is of great importance in mathematics, alongside 0, 1, , and . All five appear in one formulation of Euler's identity formula_3 and play important and recurring roles across mathematics. Like the constant , is irrational (it cannot be represented as a ratio of integers) and transcendental (it is not a root o... | https://en.wikipedia.org/wiki?curid=9633 |
20,757 | The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of logarithms to the base formula_4. It is assumed that the table was written by William Oughtred. | https://en.wikipedia.org/wiki?curid=9633 |
20,758 | The discovery of the constant itself is credited to Jacob Bernoulli in 1683, the following expression (which is equal to ): | https://en.wikipedia.org/wiki?curid=9633 |
20,759 | The first known use of the constant, represented by the letter , was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler introduced the letter as the base for natural logarithms, writing in a letter to Christian Goldbach on 25 November 1731. Euler started to use the letter fo... | https://en.wikipedia.org/wiki?curid=9633 |
20,760 | In mathematics, the most common typographical convention is to typeset the constant as "", in italics, although sometimes "e" in roman is used. However, the ISO 80000-2:2019 standard recommends typesetting constants in an upright style. | https://en.wikipedia.org/wiki?curid=9633 |
20,761 | If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial $1 is multiplied by 1.5 twice, yielding at the end of the year. Compounding quarterly yields , and compounding monthly yields . If there are compounding intervals, the interest for each interval will be and th... | https://en.wikipedia.org/wiki?curid=9633 |
20,762 | Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger and, thus, smaller compounding intervals. Compounding weekly () yields $2.692596..., while compounding daily () yields $2.714567... (approximately two cents more). The limit as grows large is the number that came to be known as ... | https://en.wikipedia.org/wiki?curid=9633 |
20,763 | More generally, an account that starts at $1 and offers an annual interest rate of will, after years, yield dollars with continuous compounding. | https://en.wikipedia.org/wiki?curid=9633 |
20,764 | The number itself also has applications in probability theory, in a way that is not obviously related to exponential growth. Suppose that a gambler plays a slot machine that pays out with a probability of one in and plays it times. As increases, the probability that gambler will lose all bets approaches . For , this is... | https://en.wikipedia.org/wiki?curid=9633 |
20,765 | This is an example of a Bernoulli trial process. Each time the gambler plays the slots, there is a one in "n" chance of winning. Playing "n" times is modeled by the binomial distribution, which is closely related to the binomial theorem and Pascal's triangle. The probability of winning times out of "n" trials is: | https://en.wikipedia.org/wiki?curid=9633 |
20,766 | The normal distribution with zero mean and unit standard deviation is known as the "standard normal distribution", given by the probability density function | https://en.wikipedia.org/wiki?curid=9633 |
20,767 | The constraint of unit variance (and thus also unit standard deviation) results in the in the exponent, and the constraint of unit total area under the curve formula_9 results in the factor formula_10. This function is symmetric around , where it attains its maximum value formula_10, and has inflection points at . | https://en.wikipedia.org/wiki?curid=9633 |
20,768 | Another application of , also discovered in part by Jacob Bernoulli along with Pierre Remond de Montmort, is in the problem of derangements, also known as the "hat check problem": guests are invited to a party, and at the door, the guests all check their hats with the butler, who in turn places the hats into boxes, eac... | https://en.wikipedia.org/wiki?curid=9633 |
20,769 | As the number of guests tends to infinity, approaches . Furthermore, the number of ways the hats can be placed into the boxes so that none of the hats are in the right box is rounded to the nearest integer, for every positive . | https://en.wikipedia.org/wiki?curid=9633 |
20,770 | The maximum value of formula_14 occurs at formula_15. Equivalently, for any value of the base , it is the case that the maximum value of formula_16 occurs at formula_15 (Steiner's problem, discussed below). | https://en.wikipedia.org/wiki?curid=9633 |
20,771 | This is useful in the problem of a stick of length that is broken into equal parts. The value of that maximizes the product of the lengths is then either | https://en.wikipedia.org/wiki?curid=9633 |
20,772 | The quantity formula_16 is also a measure of information gleaned from an event occurring with probability formula_21, so that essentially the same optimal division appears in optimal planning problems like the secretary problem. | https://en.wikipedia.org/wiki?curid=9633 |
20,773 | The number occurs naturally in connection with many problems involving asymptotics. An example is Stirling's formula for the asymptotics of the factorial function, in which both the numbers and appear: | https://en.wikipedia.org/wiki?curid=9633 |
20,774 | The principal motivation for introducing the number , particularly in calculus, is to perform differential and integral calculus with exponential functions and logarithms. A general exponential has a derivative, given by a limit: | https://en.wikipedia.org/wiki?curid=9633 |
20,775 | The parenthesized limit on the right is independent of the Its value turns out to be the logarithm of to base . Thus, when the value of is set this limit is equal and so one arrives at the following simple identity: | https://en.wikipedia.org/wiki?curid=9633 |
20,776 | Consequently, the exponential function with base is particularly suited to doing calculus. (as opposed to some other number as the base of the exponential function) makes calculations involving the derivatives much simpler. | https://en.wikipedia.org/wiki?curid=9633 |
20,777 | The logarithm with this special base is called the natural logarithm, and is denoted as ; it behaves well under differentiation since there is no undetermined limit to carry through the calculations. | https://en.wikipedia.org/wiki?curid=9633 |
20,778 | Thus, there are two ways of selecting such special numbers . One way is to set the derivative of the exponential function equal to , and solve for . The other way is to set the derivative of the base logarithm to and solve for . In each case, one arrives at a convenient choice of base for doing calculus. It turns out t... | https://en.wikipedia.org/wiki?curid=9633 |
20,779 | Other characterizations of are also possible: one is as the limit of a sequence, another is as the sum of an infinite series, and still others rely on integral calculus. So far, the following two (equivalent) properties have been introduced: | https://en.wikipedia.org/wiki?curid=9633 |
20,780 | As in the motivation, the exponential function is important in part because it is the unique nontrivial function that is its own derivative (up to multiplication by a constant): | https://en.wikipedia.org/wiki?curid=9633 |
20,781 | for all real , with equality if and only if . Furthermore, is the unique base of the exponential for which the inequality holds for all . This is a limiting case of Bernoulli's inequality. | https://en.wikipedia.org/wiki?curid=9633 |
20,782 | For proof, the inequality formula_39, from above, evaluated at formula_40 and simplifying gives formula_41. So formula_42 for all positive "x". | https://en.wikipedia.org/wiki?curid=9633 |
20,783 | converges if and only if (or approximately between 0.0660 and 1.4447), due to a theorem of Leonhard Euler. | https://en.wikipedia.org/wiki?curid=9633 |
20,784 | The real number is irrational. Euler proved this by showing that its simple continued fraction expansion is infinite. (See also Fourier's proof that is irrational.) | https://en.wikipedia.org/wiki?curid=9633 |
20,785 | Furthermore, by the Lindemann–Weierstrass theorem, is transcendental, meaning that it is not a solution of any non-constant polynomial equation with rational coefficients. It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare with Liouville number); t... | https://en.wikipedia.org/wiki?curid=9633 |
20,786 | It is conjectured that is normal, meaning that when is expressed in any base the possible digits in that base are uniformly distributed (occur with equal probability in any sequence of given length). | https://en.wikipedia.org/wiki?curid=9633 |
20,787 | Because this series is convergent for every complex value of , it is commonly used to extend the definition of to the complex numbers. This, with the Taylor series for and, allows one to derive Euler's formula: | https://en.wikipedia.org/wiki?curid=9633 |
20,788 | The number can be represented in a variety of ways: as an infinite series, an infinite product, a continued fraction, or a limit of a sequence. Two of these representations, often used in introductory calculus courses, are the limit | https://en.wikipedia.org/wiki?curid=9633 |
20,789 | Many other series, sequence, continued fraction, and infinite product representations of have been proved. | https://en.wikipedia.org/wiki?curid=9633 |
20,790 | In addition to exact analytical expressions for representation of , there are stochastic techniques for estimating . One such approach begins with an infinite sequence of independent random variables , ..., drawn from the uniform distribution on [0, 1]. Let be the least number such that the sum of the first observation... | https://en.wikipedia.org/wiki?curid=9633 |
20,791 | The number of known digits of has increased substantially during the last decades. This is due both to the increased performance of computers and to algorithmic improvements. | https://en.wikipedia.org/wiki?curid=9633 |
20,792 | Since around 2010, the proliferation of modern high-speed desktop computers has made it feasible for most amateurs to compute trillions of digits of within acceptable amounts of time. On Dec 5, 2020, a record-setting calculation was made, giving to 31,415,926,535,897 (approximately ) digits. | https://en.wikipedia.org/wiki?curid=9633 |
20,793 | A faster method involves two recursive function formula_63 and formula_64. The functions are defined as formula_65. | https://en.wikipedia.org/wiki?curid=9633 |
20,794 | The expression formula_66 produces the digits of . This method uses binary splitting to compute with fewer single-digit arithmetic operations and reduced bit complexity. Combining this with Fast Fourier Transform-based methods of multiplying integers makes computing the digits very fast. | https://en.wikipedia.org/wiki?curid=9633 |
20,795 | During the emergence of internet culture, individuals and organizations sometimes paid homage to the number . | https://en.wikipedia.org/wiki?curid=9633 |
20,796 | In an early example, the computer scientist Donald Knuth let the version numbers of his program Metafont approach . The versions are 2, 2.7, 2.71, 2.718, and so forth. | https://en.wikipedia.org/wiki?curid=9633 |
20,797 | In another instance, the IPO filing for Google in 2004, rather than a typical round-number amount of money, the company announced its intention to raise 2,718,281,828 USD, which is billion dollars rounded to the nearest dollar. | https://en.wikipedia.org/wiki?curid=9633 |
20,798 | that appeared in the heart of Silicon Valley, and later in Cambridge, Massachusetts; Seattle, Washington; and Austin, Texas. It read "{first 10-digit prime found in consecutive digits of }.com". The first 10-digit prime in is 7427466391, which starts at the 99th digit. Solving this problem and visiting the advertised (... | https://en.wikipedia.org/wiki?curid=9633 |
20,799 | Solving this second problem finally led to a Google Labs webpage where the visitor was invited to submit a résumé. | https://en.wikipedia.org/wiki?curid=9633 |
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