Introductio in analysin infinitorum
Introductio in analysin infinitorum is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis. Written in Latin and published in 1748, the Introductio contains 18 chapters in the first part and 22 chapters in the second, it has Eneström numbers E101 and E102. Carl Boyer's lectures at the 1950 International Congress of Mathematicians compared the influence of Euler's Introductio to that of Euclid's Elements, calling the Elements the foremost textbook of ancient times, the Introductio "the foremost textbook of modern times". Boyer wrote: The analysis of Euler comes close to the modern orthodox discipline, the study of functions by means of infinite processes through infinite series, it is doubtful that any other didactic work includes as large a portion of original material that survives in the college courses today... Can be read with comparative ease by the modern student... The prototype of modern textbooks; the first translation into English was that by John D. Blanton, published in 1988.
The second, by Ian Bruce, is available online. A list of the editions of Introductio has been assembled by V. Frederick Rickey. Chapter 1 is on the concepts of functions. Chapter 4 introduces infinite series through rational functions. According to Henk Bos, The Introduction is meant as a survey of concepts and methods in analysis and analytic geometry preliminary to the study of the differential and integral calculus. Made of this survey a masterly exercise in introducing as much as possible of analysis without using differentiation or integration. In particular, he introduced the elementary transcendental functions, the logarithm, the exponential function, the trigonometric functions and their inverses without recourse to integral calculus —, no mean feat, as the logarithm was traditionally linked to the quadrature of the hyperbola and the trigonometric functions to the arc-length of the circle. Euler accomplished this feat by introducing exponentiation ax for arbitrary constant a in the positive real numbers.
He noted. For a > 1 these functions are monotonic increasing and form bijections of the real line with positive real numbers. Each base a corresponds to an inverse function called the logarithm to base a, in chapter 6. In chapter 7, Euler introduces e as the number whose hyperbolic logarithm is 1; the reference here is to Gregoire de Saint-Vincent who performed a quadrature of the hyperbola y = 1/x through description of the hyperbolic logarithm. Section 122 labels the logarithm to base e the "natural or hyperbolic logarithm...since the quadrature of the hyperbola can be expressed through these logarithms". Here he gives the exponential series: exp = ∑ k = 0 ∞ z k k! = 1 + z + z 2 2 + z 3 6 + z 4 24 + ⋯ Then in chapter 8 Euler is prepared to address the classical trigonometric functions as "transcendental quantities that arise from the circle." He presents Euler's formula. Chapter 9 considers trinomial factors in polynomials. Chapter 16 is concerned with a topic in number theory. Continued fractions are the topic of chapter 18.
J. C. Scriba review of 1983 reprint of 1885 German edition MR715928 Doru Stefanescu MR1025504 Marco Panza MR2384380 Ricardo Quintero Zazueta MR1823258 Ernst Hairer & Gerhard Wanner Analysis by its History, chapter 1, pp 1 to 79, Undergraduate Texts in Mathematics #70, ISBN 978-0-387-77036-9 MR1410751
In computing, floating-point arithmetic is arithmetic using formulaic representation of real numbers as an approximation so as to support a trade-off between range and precision. For this reason, floating-point computation is found in systems which include small and large real numbers, which require fast processing times. A number is, in general, represented to a fixed number of significant digits and scaled using an exponent in some fixed base. A number that can be represented is of the following form: significand × base exponent, where significand is an integer, base is an integer greater than or equal to two, exponent is an integer. For example: 1.2345 = 12345 ⏟ significand × 10 ⏟ base − 4 ⏞ exponent. The term floating point refers to the fact that a number's radix point can "float"; this position is indicated as the exponent component, thus the floating-point representation can be thought of as a kind of scientific notation. A floating-point system can be used to represent, with a fixed number of digits, numbers of different orders of magnitude: e.g. the distance between galaxies or the diameter of an atomic nucleus can be expressed with the same unit of length.
The result of this dynamic range is that the numbers that can be represented are not uniformly spaced. Over the years, a variety of floating-point representations have been used in computers. In 1985, the IEEE 754 Standard for Floating-Point Arithmetic was established, since the 1990s, the most encountered representations are those defined by the IEEE; the speed of floating-point operations measured in terms of FLOPS, is an important characteristic of a computer system for applications that involve intensive mathematical calculations. A floating-point unit is a part of a computer system specially designed to carry out operations on floating-point numbers. A number representation specifies some way of encoding a number as a string of digits. There are several mechanisms. In common mathematical notation, the digit string can be of any length, the location of the radix point is indicated by placing an explicit "point" character there. If the radix point is not specified the string implicitly represents an integer and the unstated radix point would be off the right-hand end of the string, next to the least significant digit.
In fixed-point systems, a position in the string is specified for the radix point. So a fixed-point scheme might be to use a string of 8 decimal digits with the decimal point in the middle, whereby "00012345" would represent 0001.2345. In scientific notation, the given number is scaled by a power of 10, so that it lies within a certain range—typically between 1 and 10, with the radix point appearing after the first digit; the scaling factor, as a power of ten, is indicated separately at the end of the number. For example, the orbital period of Jupiter's moon Io is 152,853.5047 seconds, a value that would be represented in standard-form scientific notation as 1.528535047×105 seconds. Floating-point representation is similar in concept to scientific notation. Logically, a floating-point number consists of: A signed digit string of a given length in a given base; this digit string is referred to mantissa, or coefficient. The length of the significand determines the precision; the radix point position is assumed always to be somewhere within the significand—often just after or just before the most significant digit, or to the right of the rightmost digit.
This article follows the convention that the radix point is set just after the most significant digit. A signed integer exponent. To derive the value of the floating-point number, the significand is multiplied by the base raised to the power of the exponent, equivalent to shifting the radix point from its implied position by a number of places equal to the value of the exponent—to the right if the exponent is positive or to the left if the exponent is negative. Using base-10 as an example, the number 152,853.5047, which has ten decimal digits of precision, is represented as the significand 1,528,535,047 together with 5 as the exponent. To determine the actual value, a decimal point is placed after the first digit of the significand and the result is multiplied by 105 to give 1.528535047×105, or 152,853.5047. In storing such a number, the base need not be stored, since it will be the same for the entire range of supported numbers, can thus be inferred. Symbolically, this final value is: s b p − 1 × b e, where s is the
The Bibliotheca Teubneriana, or Teubner editions of Greek and Latin texts, comprise the most thorough modern collection published of ancient Greco-Roman literature. The series, whose full name is the Bibliotheca Scriptorum Graecorum et Romanorum Teubneriana, consists of critical editions by leading scholars (now always with a full critical apparatus on each page, although during the nineteenth century there were editiones minores, published either without critical apparatuses or with abbreviated textual appendices, editiones maiores, published with a full apparatus. Teubneriana is an abbreviation used to denote a single volume of the series the whole collection. Today, the only comparable publishing ventures, producing authoritative scholarly reference editions of numerous ancient authors, are the Oxford Classical Texts and the Collection Budé. In 1811, Benedictus Gotthelf Teubner refounded in his own name a printing operation he had directed since 1806, the Weinedelsche Buchdruckerei, giving rise to the Leipzig publishing house of B.
G. Teubner; the volumes of the Bibliotheca Teubneriana began to appear in 1849. Although today Teubner editions are expensive, they were introduced to fill the need unmet, for low-priced but high-quality editions. Prior to the introduction of the Teubner series, accurate editions of antique authors could only be purchased by libraries and rich private scholars because of their expense. Students and other individuals of modest means had to rely on editions which were affordable but filled with errors. To satisfy the need for accurate and affordable editions Teubner introduced the Bibliotheca Teubneriana. In the 19th century, Teubner offered both affordable editiones maiores for scholars, low-priced editiones minores for students. Editiones minores were dropped from the series and Teubner began to offer only scholarly reference editions of ancient authors. During the period between the end of World War II and German reunification, the publishing house of B. G. Teubner split into two firms, Teubner KG BSB B. G. Teubner Verlagsgesellschaft, in Leipzig in East Germany, Verlag B. G. Teubner / BG Teubner GmbH in Stuttgart in West Germany.
Both offered volumes in the Bibliotheca Teubneriana. After the fall of the Berlin wall and the reunification of Germany, B. G. Teubner was reunited and subsequently consolidated its headquarters at Wiesbaden. In late 1999, B. G. Teubner Verlag announced their intention to concentrate on technical publishing. All their Classical Studies titles, including the Biblotheca Teubneriana, were sold to K. G. Saur, a publisher based in Munich. Although new volumes began to appear with the imprint in aedibus K. G. Saur, the name of the series remained unchanged. In 2006, the publishing firm of Walter de Gruyter acquired K. G. Saur and their entire publishing range, including the Bibliotheca Teubneriana. Since January 2007, the Bibliotheca Teubneriana is being published by Walter de Gruyter GmbH & Co. KG; as of 1 May 2007, the new North American distributor of titles from the Bibliotheca Teubneriana is Walter de Gruyter, Inc. While the typography of the Greek Teubners has been subject to innovations over the years, an overview of the whole series shows a great deal of consistency.
The old-fashioned, cursive font used in most of the existing volumes is recognized by classicists and associated with Teubner. This type was in regular use for verse and prose texts. In older Teubners, several old-fashioned features of the type are still found which would be smoothed away, for example, omega with bent-in ends, medial sigma, not closed, phi with a bent stem. Teubner used an upright type, designed to match the original cursive type, in some editions. In the example shown, the cursive type is still used in the critical apparatus. In other editions, this upright font is used throughout. Beginning in the 1990s, the digital production of books has been marked by new digital fonts, sometimes based on Teubner's older traditions. In the 1990s, individual editions of Euripides' tragedies were digitally typeset in a font based on the original Teubner cursive. There have been recent innovations in upright type. One of these, which may be seen in Bernabé's edition of the Orphica, seems to be the current standard for new Teubners from K.
G. Saur; some Teubner Greek editions made a bold typographic departure from the tradition outlined above. E. J. Kenney considered this twentieth-century experiment to be a refreshing break from the Porsonian norm, emblematic of the best kind of modernist simplicity and directness: More there has been a welcome and long overdue return to the older and purer models; the pleasing modification of M. E. Pinder's "Griechische Antiqua" used by Teubner in some of their editions represents a lost opportunit
History of logarithms
The history of logarithms is the story of a correspondence between multiplication on the positive real numbers and addition on the real number line, formalized in seventeenth century Europe and was used to simplify calculation until the advent of the digital computer. The Napierian logarithms were published first in 1614. Henry Briggs introduced common logarithms. Tables of logarithms were published in many forms over four centuries; the idea of logarithms was used to construct the slide rule, which became ubiquitous in science and engineering until the 1970s. A breakthrough generating the natural logarithm was the result of a search for an expression of area against a rectangular hyperbola, required the assimilation of a new function into standard mathematics; as the common log of ten is one, of a hundred is two, a thousand is three, the concept of common logarithms is close to the decimal-positional number system. The common log is said to have base 10. In his book The Sand Reckoner, Archimedes used the myriad as the base of a number system designed to count the grains of sand in the universe.
As was noted in 2000: In antiquity Archimedes gave a recipe for reducing multiplication to addition by making use of geometric progression of numbers and relating them to an arithmetic progression. In 1616 Henry Briggs visited Napier at Edinburgh in order to discuss the suggested change to Napier's logarithms; the following year he again visited for a similar purpose. During these conferences the alteration proposed by Briggs was agreed upon, on his return from his second visit to Edinburgh, in 1617, he published the first chiliad of his logarithms. In 1624 Briggs published his Arithmetica Logarithmica, in folio, a work containing the logarithms of thirty thousand natural numbers to fourteen decimal places; this table was extended by Adriaan Vlacq, but to 10 places, by Alexander John Thompson to 20 places in 1952. Briggs was one of the first to use finite-difference methods to compute tables of functions, he completed a table of logarithmic sines and tangents for the hundredth part of every degree to fourteen decimal places, with a table of natural sines to fifteen places and the tangents and secants for the same to ten places, all of which were printed at Gouda in 1631 and published in 1633 under the title of Trigonometria Britannica.
In 1649, Alphonse Antonio de Sarasa, a former student of Grégoire de Saint-Vincent, related logarithms to the quadrature of the hyperbola, by pointing out that the area A under the hyperbola from x = 1 to x = t satisfies A = A + A. The natural logarithm was first described by Nicholas Mercator in his work Logarithmotechnia published in 1668, although the mathematics teacher John Speidell had in 1619 compiled a table of what were natural logarithms, based on Napier's work. Historian Tom Whiteside described the transition to the analytic function as follows: By the end of the 17th century we can say that much more than being a calculating device suitably well-tabulated, the logarithm function much on the model of the hyperbola-area, had been accepted into mathematics. When, in the 18 century, this geometric basis was discarded in favour of a analytical one, no extension or reformulation was necessary – the concept of "hyperbola-area" was transformed painlessly into "natural logarithm"; the Babylonians sometime in 2000–1600 BC may have invented the quarter square multiplication algorithm to multiply two numbers using only addition, subtraction and a table of quarter squares.
Thus, such a table served a similar purpose to tables of logarithms, which allow multiplication to be calculated using addition and table lookups. However, the quarter-square method could not be used for division without an additional table of reciprocals. Large tables of quarter squares were used to simplify the accurate multiplication of large numbers from 1817 onwards until this was superseded by the use of computers; the Indian mathematician Virasena worked with the concept of ardhaccheda: the number of times a number of the form 2n could be halved. For exact powers of 2, this equals the binary logarithm, but it differs from the logarithm for other numbers, he described a product formula for this concept and introduced analogous concepts for base 3 and base 4. Michael Stifel published Arithmetica integra in Nuremberg in 1544, which contains a table of integers and powers of 2, considered an early version of a table of binary logarithms. In the 16th and early 17th centuries an algorithm called prosthaphaeresis was used to approximate multiplication and division.
This used the trigonometric identity cos α cos β = 1 2 or similar to convert the multiplications to additions and table lookups. However, logarithms are more straightforward and require less work
Macmillan Publishers Ltd is an international publishing company owned by Holtzbrinck Publishing Group. It operates in more than thirty others. Macmillan was founded in 1843 by Daniel and Alexander Macmillan, two brothers from the Isle of Arran, Scotland. Daniel was the business brain, while Alexander laid the literary foundations, publishing such notable authors as Charles Kingsley, Thomas Hughes, Francis Turner Palgrave, Christina Rossetti, Matthew Arnold and Lewis Carroll. Alfred Tennyson joined the list in 1884, Thomas Hardy in 1886 and Rudyard Kipling in 1890. Other major writers published by Macmillan included W. B. Yeats, Rabindranath Tagore, Nirad C. Chaudhuri, Seán O'Casey, John Maynard Keynes, Charles Morgan, Hugh Walpole, Margaret Mitchell, C. P. Snow, Rumer Godden and Ram Sharan Sharma. Beyond literature, the company created such enduring titles as Nature, the Grove Dictionary of Music and Musicians and Sir Robert Harry Inglis Palgrave's Dictionary of Political Economy. George Edward Brett opened the first Macmillan office in the United States in 1869 and Macmillan sold its U.
S. operations to the Brett family, George Platt Brett, Sr. and George Platt Brett, Jr. in 1896, resulting in the creation of an American company, Macmillan Publishing called the Macmillan Company. With the split of the American company from its parent company in England, George Brett, Jr. and Harold Macmillan remained close personal friends. Macmillan Publishers re-entered the American market in 1954 under the name St. Martin's Press. Macmillan of Canada was founded in 1905. After retiring from politics in 1964, former Prime Minister of the United Kingdom Harold Macmillan became chairman of the company, serving until his death in December 1986, he had been with the family firm as a junior partner from 1920 to 1940, from 1945 to 1951 while he was in the opposition in Parliament. Holtzbrinck Publishing Group purchased the company in 1999. Pearson acquired the Macmillan name in America in 1998, following its purchase of the Simon & Schuster educational and professional group. Holtzbrinck purchased it from them in 2001.
McGraw-Hill continues to market its pre-kindergarten through elementary school titles under its Macmillan/McGraw-Hill brand. The US operations of Holtzbrinck Publishing changed its name to Macmillan in October 2017, its audio publishing imprint changed its name from Audio Renaissance to Macmillan Audio, while its distribution arm was renamed from Von Holtzbrinck Publishers Services to Macmillan Publishers Services. With Pan Macmillan's purchase of Kingfisher, a British children's publisher, Roaring Brook Press publisher Simon Boughton would take oversee Kingfisher's US business in October 2007. By some estimates, as of 2009 e-books account for three to five per cent of total book sales, are the fastest growing segment of the market. According to The New York Times and other major publishers "fear that massive discounting by retailers including Amazon, Barnes & Noble and Sony could devalue what consumers are willing to pay for books." In response, the publisher introduced a new boilerplate contract for its authors that established a royalty of 20 per cent of net proceeds on e-book sales, a rate five per cent lower than most other major publishers.
Following the announcement of the Apple iPad on 27 January 2010—a product that comes with access to the iBookstore—Macmillan gave Amazon.com two options: continue to sell e-books based on a price of the retailer's choice, with the e-book edition released several months after the hardcover edition is released, or switch to the agency model introduced to the industry by Apple, in which both are released and the price is set by the publisher. In the latter case, Amazon.com would receive a 30 per cent commission. Amazon responded by pulling all Macmillan books, both physical, from their website. On 31 January 2010, Amazon chose the agency model preferred by Macmillan. In April 2012, the United States Department of Justice filed United States v. Apple Inc. naming Apple and four other major publishers as defendants. The suit alleged that they conspired to fix prices for e-books, weaken Amazon.com's position in the market, in violation of antitrust law. In December 2013, a federal judge approved a settlement of the antitrust claims, in which Macmillan and the other publishers paid into a fund that provided credits to customers who had overpaid for books due to the price-fixing.
In 2010, Macmillan Education submitted to an investigation on grounds of fraudulent practices. The Macmillan division admitted to bribery in an attempt to secure a contract for an education project in southern Sudan; as a direct result of the investigation, sanctions were applied by the World Bank Group, namely a 6-year debarment declaring the company ineligible to be awarded Bank-financed contracts. In December 2011, Bedford and Worth Publishing Group, Macmillan's higher education group, changed its name to Macmillan Higher Education while retaining the Bedford and Worth name for its k–12 educational unit; that month, Brian Napack resigned as Macmillan president while staying on for transitional purposes. In May 2015, London-based Macmillan Science and Education merged with Berlin-based Springer Science+Business Media to form Springer Nature, jointly controlled by Holtzbrinck Publishing Group and BC Partners. US publishing divis
Edinburgh is the capital city of Scotland and one of its 32 council areas. Part of the county of Midlothian, it is located in Lothian on the Firth of Forth's southern shore. Recognised as the capital of Scotland since at least the 15th century, Edinburgh is the seat of the Scottish Government, the Scottish Parliament and the supreme courts of Scotland; the city's Palace of Holyroodhouse is the official residence of the monarch in Scotland. The city has long been a centre of education in the fields of medicine, Scots law, philosophy, the sciences and engineering, it is the second largest financial centre in the United Kingdom and the city's historical and cultural attractions have made it the United Kingdom's second most popular tourist destination, attracting over one million overseas visitors each year. Edinburgh is Scotland's second most populous city and the seventh most populous in the United Kingdom; the official population estimates are 488,050 for the Locality of Edinburgh, 513,210 for the City of Edinburgh, 1,339,380 for the city region.
Edinburgh lies at the heart of the Edinburgh and South East Scotland city region comprising East Lothian, Fife, Scottish Borders and West Lothian. The city is the annual venue of the General Assembly of the Church of Scotland, it is home to national institutions such as the National Museum of Scotland, the National Library of Scotland and the Scottish National Gallery. The University of Edinburgh, founded in 1582 and now one of four in the city, is placed 18th in the QS World University Rankings for 2019; the city is famous for the Edinburgh International Festival and the Fringe, the latter being the world's largest annual international arts festival. Historic sites in Edinburgh include Edinburgh Castle, the Palace of Holyroodhouse, the churches of St. Giles and the Canongate, the extensive Georgian New Town, built in the 18th/19th centuries. Edinburgh's Old Town and New Town together are listed as a UNESCO World Heritage site, managed by Edinburgh World Heritage since 1999. "Edin", the root of the city's name, derives from Eidyn, the name for this region in Cumbric, the Brittonic Celtic language spoken there.
The name's meaning is unknown. The district of Eidyn centred on the dun or hillfort of Eidyn; this stronghold is believed to have been located at Castle Rock, now the site of Edinburgh Castle. Eidyn was conquered by the Angles of Bernicia in the 7th century and by the Scots in the 10th century; as the language shifted to Old English, subsequently to modern English and Scots, The Brittonic din in Din Eidyn was replaced by burh, producing Edinburgh. Din became dùn in Scottish Gaelic, producing Dùn Èideann; the city is affectionately nicknamed Auld Reekie, Scots for Old Smoky, for the views from the country of the smoke-covered Old Town. Allan Ramsay said. A name the country people give Edinburgh from the cloud of smoke or reek, always impending over it."Thomas Carlyle said, "Smoke cloud hangs over old Edinburgh,—for since Aeneas Silvius's time and earlier, the people have the art strange to Aeneas, of burning a certain sort of black stones, Edinburgh with its chimneys is called'Auld Reekie' by the country people."A character in Walter Scott's The Abbot says "... yonder stands Auld Reekie--you may see the smoke hover over her at twenty miles' distance."Robert Chambers who said that the sobriquet could not be traced before the reign of Charles II attributed the name to a Fife laird, Durham of Largo, who regulated the bedtime of his children by the smoke rising above Edinburgh from the fires of the tenements.
"It's time now bairns, to tak' the beuks, gang to our beds, for yonder's Auld Reekie, I see, putting on her nicht -cap!"Some have called Edinburgh the Athens of the North for a variety of reasons. The earliest comparison between the two cities showed that they had a similar topography, with the Castle Rock of Edinburgh performing a similar role to the Athenian Acropolis. Both of them had fertile agricultural land sloping down to a port several miles away. Although this arrangement is common in Southern Europe, it is rare in Northern Europe; the 18th-century intellectual life, referred to as the Scottish Enlightenment, was a key influence in gaining the name. Such luminaries as David Hume and Adam Smith shone during this period. Having lost most of its political importance after the Union, some hoped that Edinburgh could gain a similar influence on London as Athens had on Rome. A contributing factor was the neoclassical architecture that of William Henry Playfair, the National Monument. Tom Stoppard's character Archie, of Jumpers, said playing on Reykjavík meaning "smoky bay", that the "Reykjavík of the South" would be more appropriate.
The city has been known by several Latin names, such as Aneda or Edina. The adjectival form of the latter, can be seen inscribed on educational buildings; the Scots poets Robert Fergusson and Robert Burns used Edina in their poems. Ben Jonson described it as "Britaine's other eye", Sir Walter Scott referred to it as "yon Empress of the North". Robert Louis Stevenson a son of the city, wrote, "Edinburgh is what Paris ought to be"; the colloquial pronunciation "Embra" or "Embro" has been used, as in Robert Garioch's Embro to the Ploy. The earliest known human habitation in the Edinburgh area was at Cramond, where evidence was found of a Mesolithi
In mathematics, the binary logarithm is the power to which the number 2 must be raised to obtain the value n. That is, for any real number x = log 2 n ⟺ 2 x = n. For example, the binary logarithm of 1 is 0, the binary logarithm of 2 is 1, the binary logarithm of 4 is 2, the binary logarithm of 32 is 5; the binary logarithm is the logarithm to the base 2. The binary logarithm function is the inverse function of the power of two function; as well as log2, alternative notations for the binary logarithm include lg, ld, lb, log. The first application of binary logarithms was in music theory, by Leonhard Euler: the binary logarithm of a frequency ratio of two musical tones gives the number of octaves by which the tones differ. Binary logarithms can be used to calculate the length of the representation of a number in the binary numeral system, or the number of bits needed to encode a message in information theory. In computer science, they count the number of steps needed for related algorithms. Other areas in which the binary logarithm is used include combinatorics, the design of sports tournaments, photography.
Binary logarithms are included in the standard C mathematical functions and other mathematical software packages. The integer part of a binary logarithm can be found using the find first set operation on an integer value, or by looking up the exponent of a floating point value; the fractional part of the logarithm can be calculated efficiently. The powers of two have been known since antiquity. IX.32 and IX.36. And the binary logarithm of a power of two is just its position in the ordered sequence of powers of two. On this basis, Michael Stifel has been credited with publishing the first known table of binary logarithms in 1544, his book Arithmetica Integra contains several tables that show the integers with their corresponding powers of two. Reversing the rows of these tables allow them to be interpreted as tables of binary logarithms. Earlier than Stifel, the 8th century Jain mathematician Virasena is credited with a precursor to the binary logarithm. Virasena's concept of ardhacheda has been defined as the number of times a given number can be divided evenly by two.
This definition gives rise to a function that coincides with the binary logarithm on the powers of two, but it is different for other integers, giving the 2-adic order rather than the logarithm. The modern form of a binary logarithm, applying to any number was considered explicitly by Leonhard Euler in 1739. Euler established the application of binary logarithms to music theory, long before their applications in information theory and computer science became known; as part of his work in this area, Euler published a table of binary logarithms of the integers from 1 to 8, to seven decimal digits of accuracy. The binary logarithm function may be defined as the inverse function to the power of two function, a increasing function over the positive real numbers and therefore has a unique inverse. Alternatively, it may be defined as ln n/ln 2, where ln is the natural logarithm, defined in any of its standard ways. Using the complex logarithm in this definition allows the binary logarithm to be extended to the complex numbers.
As with other logarithms, the binary logarithm obeys the following equations, which can be used to simplify formulas that combine binary logarithms with multiplication or exponentiation: log 2 x y = log 2 x + log 2 y log 2 x y = log 2 x − log 2 y log 2 x y = y log 2 x. For more, see list of logarithmic identities. In mathematics, the binary logarithm of a number n is written as log2 n. However, several other notations for this function have been used or proposed in application areas; some authors write the binary logarithm as the notation listed in The Chicago Manual of Style. Donald Knuth credits this notation to a suggestion of Edward Reingold, but its use in both information theory and computer science dates to before Reingold was active; the binary logarithm has been written as log n with a prior statement that the default base for the logarithm is 2. Another notation, used for the same function is ld n, from Latin logarithmus dualis aka logarithmus dyadis; the DIN 1302, ISO 31-11 and ISO 80000-2 standards recommend yet another lb n.
According to these standards, lg n should not be used for the binary logarithm, as it is instead reserved for the common logarithm log10 n. The number of digits in the binary representation of a positive integer n is the integral part of 1 + log2 n, i.e. ⌊ log 2 n ⌋ + 1. In information theory, t