1 (New York City Subway service), Service history, Skip-stop implementation
Beginning at 6:30 a.m. on Monday, August 21, 1989, the services were coordinated as the 1/9 and both ran between Van Cortlandt Park–242nd Street and South Ferry. The plan was to have skip-stop service begin north of 116th Street–Columbia University, but due to objections, most notably that riders did not want 125th Street to be a skip-stop station, skip-stop service was only implemented north of 137th Street–City College between the hours of 6:30 am and 7:00 pm weekdays. All 1 trains skipped Marble Hill–225th, 207th, 191st and 145th Streets, while all 9 trains skipped 238th, 215th, Dyckman and 157th Streets. On September 4, 1994, midday skip-stop service was discontinued, and 191st Street became a common station for skip-stop service.


1 (New York City Subway service), Service history, 9/11 and recovery
After the September 11 attacks in 2001, 1 trains had to be rerouted since the IRT Broadway–Seventh Avenue Line ran directly under the World Trade Center site and was heavily damaged in the collapse of the Twin Towers. It initially ran as a shuttle alongside the 2 and 3 trains to Times Square, due to debris that fell on the tracks south of Pennsylvania Station. When the debris was cleared by September 17, the 1 ran only between 242nd Street and 14th Street, making local stops north of 96th Street and express stops south of that point.


1 (New York City Subway service), Service history, 9/11 and recovery
Local service was replaced by the 2 and 3 trains, running express from Canal Street to Fulton Street due to debris covering the stops between them. The skip-stop service with the 9 train was suspended for the duration of the 9/11 emergency service plan. On September 19, after a few switching delays at 96th Street, service was changed.


1 (New York City Subway service), Service history, 9/11 and recovery
1 trains made all stops at all times from 242nd Street to New Lots Avenue via the Clark Street Tunnel and IRT Eastern Parkway Line, replacing the route of 3 trains within Brooklyn, as the 3 terminated at 14th Street during this period. All 1 trains continued running express with the 2 between the aforementioned streets until October 1, when it terminated at Chambers Street in Manhattan overnight.


1 (New York City Subway service), Service history, 9/11 and recovery
Over the next year, the section of the Broadway–Seventh Avenue Line's South Ferry spur south of Chambers Street was cleaned and the tunnel was rebuilt through the World Trade Center site. The tunnel was completed and opened to service on September 15, 2002, when 1 trains returned to the South Ferry Loop and 9 skip-stop service was reinstated. However, the Cortlandt Street station, which was directly underneath the World Trade Center, was demolished as part of the clean-up and was rebuilt as part of the World Trade Center Transportation Hub until September 8, 2018, when it reopened as WTC Cortlandt.


1 (New York City Subway service), Service history, 9/11 and recovery
On April 27, 2004, it was announced that New York City Transit was considering eliminating 9 service, and thus, the skip-stop pattern. By that time, riders at skip-stop stations were experiencing longer wait times, and fewer riders were benefiting from the service pattern overall. The MTA estimated that eliminating skip-stop service only added 2+1⁄2 to 3 minutes of travel time for passengers at the northernmost stations at 242nd Street and 238th Street, while many passengers would see trains frequencies double, resulting in decreased overall travel time because of less time waiting for trains. Though the MTA had planned to vote on the future of the skip-stop service in summer 2004, it approved the change on January 11, 2005. The 9 train was discontinued on May 27, 2005, and the 1 now makes all stops on the IRT Broadway–Seventh Avenue Line.


1 (New York City Subway service), Service history, 9/11 and recovery
On March 16, 2009, the new South Ferry station opened, replacing the original loop station. However, Hurricane Sandy flooded the station, requiring it to be shut down for repairs. Rector Street served as a temporary terminal for the 1 until April 4, 2013, when the 1 returned to the reopened loop station. The old loop station then served as a temporary terminal until the replacement South Ferry station reopened on June 27, 2017.


1 (New York City Subway service), Route, Service pattern
The 1 uses the following line with the same service pattern at all times.


1 (New York City Subway service), Route, Stations
The 1 runs on the IRT Broadway–Seventh Avenue Line in its entirety.


1 (Pole album)
1 is the debut studio album by German electronic music producer Pole. It was released by Kiff SM and Matador Records in 1998.


1 (Pole album), Reception
Sean Cooper of AllMusic wrote that Pole's "nine variations on damaged dub minimalism recall the best of Basic Channel techno while steering well clear of the monotony factor", praising 1 as "an excellent debut". In 2017, Pitchfork placed 1 at number 34 on its list of "The 50 Best IDM Albums of All Time".


1 (Royal Buckinghamshire Yeomanry) Signal Squadron
1 (Royal Buckinghamshire Yeomanry) Signal Squadron was a British Territorial Army Squadron of the Royal Corps of Signals.


1 (Royal Buckinghamshire Yeomanry) Signal Squadron, History
The squadron was formed at Bletchley on 1 April 1995 from a detachment of 5 Squadron of 39 Signal Regiment and 602 Signal Troop, perpetuating the traditions of the Royal Buckinghamshire Yeomanry from 20 October 1996, a title that had been dormant since the disbandment of B Company, 2nd Battalion Wessex Volunteers a decade earlier. The squadron left 39 Signal Regiment to become an independent unit on 1 July 1999.


1 (Royal Buckinghamshire Yeomanry) Signal Squadron, Structure
The Squadron consisted of a Headquarters and three Signal troops:


1 (Suburban Kids with Biblical Names EP)
#1 is an EP by Suburban Kids with Biblical Names, released in 2004 by Labrador Records. "Rent a Wreck" ended up being on their #3 album, and "Love Will" as well as "Trumpets and Violins" were bonus US/UK tracks. "Do It All or Don't Do It at All" is the only track from #1 that wasn't carried on to #3.


1 (The Black Heart Procession album)
1 is the debut studio album by the American indie band The Black Heart Procession. It was released on January 1, 1998, on Headhunter.


1 (The Black Heart Procession album), Reception
Ned Raggett from AllMusic rated the album 3.5 stars out of 5 and called the album "a lovely melancholia that avoids self-pity for deliberate reflection and consideration."


1 (The Black Heart Procession album), Track listing
Note: First pressing of this album in LP format was on clear vinyl.


1 (The Motors album)
1 (sometimes also known as The Motors 1) is the debut studio album by English rock band The Motors, originally released in October 1977. Three singles came from the album, "Dancing the Night Away", "Be What You Gotta Be" and "Cold Love".


1 (The Motors album)
American band Cheap Trick covered "Dancing The Night Away" for their album Next Position Please in 1983.


1 (The Motors album), Background and production
The Motors signed for Virgin Records on 13 May 1977. The Motors then consisted of Nick Garvey (lead vocals, guitars), Andrew McMaster (bass, keyboards, vocals), Ricky Slaughter (drums) and Rob Hendry (guitar, vocals). Soon afterwards Hendry was replaced by Bram Tchaikovsky.


1 (The Motors album), Background and production
The new line-up began by touring with the Heavy Metal Kids in June 1977. After the tour they then began recording the album with the record producer Robert John "Mutt" Lange and released "Dancing the Night Away" / "Whiskey And Wine", their debut single off the album in September 1977. The single climbed to number 42 in the UK Singles Chart and spent four weeks in the charts. The track was extended to go onto the album, which they released as a 12-inch single. Two other singles were also released from the album, "Be What You Gotta Be" and "Cold Love", but they failed to chart.


1 (The Motors album), Background and production
The album was released in October 1977. It entered the UK Albums Chart on 22 October 1977 and climbed to number 46 and spent four weeks in the chart. In the Melody Maker album chart it peaked at number 30. The album was re-released on CD on 22 March 1991 by Blue Plate / Virgin (Catalogue No 1821) and in 2006 by Captain Oi! Records (Catalogue No AHOYCD 276) with bonus tracks.


1 (The Motors album), Background and production
The album cover was designed by Cooke Key Associates for Virgin Records. The design of the album cover was altered for the US market, with a larger "The Motors" logo. The  image of the band and the name of the album "1" were both removed, therefore implying that the title of the album was "The Motors".


1 (Zara Larsson album)
1 is the debut studio album by Swedish singer Zara Larsson. The album was released on 1 October 2014 by TEN Music Group and Universal Music Group. The album is certified platinum in Sweden and gold in Denmark, for selling 40,000 and 10,000 units in the countries, respectively. The album is only available in Scandinavian countries.


1 (Zara Larsson album), Critical reception
Kyle Copier of A Music Blog, Yea? gave 1 a very positive review, stating "Where her colleagues, even older, more experienced ones, usually lack emotion, the change in tone from confident ex-girlfriend to shaken ex-lover in “She’s Not Me” from Part 1 to Part 2 is chilling." Copier praised her vocals for being beyond her age. Copier finished his review by writing: "For now, she’s an exciting, young, impressive artist with a ceiling that has yet to be seen and a killer debut- I’m sure that will do.


1 (Zara Larsson album), Critical reception
You can bet I’m going to do a much better job keeping tabs on Zara Larsson from here on out." Press Play OK gave 1 a more mixed review, writing: "Larsson’s songs unfortunately feel bereft of any substance based on life experience. Skipping A Beat takes a riff on that similar Olly Murs song, while Rooftop is simplistic."


1 (Zara Larsson album), Commercial performance
1 has charted in three countries, it reached number 33 in Denmark, number 28 in Norway, and topped the Swedish Albums Chart.


1 + 1 + 1 + 1 + ⋯
In mathematics, 1 + 1 + 1 + 1 + ⋯, also written ∑n=1∞n0{\displaystyle \sum _{n=1}^{\infty }n^{0}}, ∑n=1∞1n{\displaystyle \sum _{n=1}^{\infty }1^{n}}, or simply ∑n=1∞1{\displaystyle \sum _{n=1}^{\infty }1}, is a divergent series, meaning that its sequence of partial sums does not converge to a limit in the real numbers. The sequence 1n can be thought of as a geometric series with the common ratio 1. Unlike other geometric series with rational ratio (except −1), it converges in neither the real numbers nor in the p-adic numbers for some p. In the context of the extended real number line


1 + 1 + 1 + 1 + ⋯
since its sequence of partial sums increases monotonically without bound.


1 + 1 + 1 + 1 + ⋯
Where the sum of n0 occurs in physical applications, it may sometimes be interpreted by zeta function regularization, as the value at s = 0 of the Riemann zeta function:


1 + 1 + 1 + 1 + ⋯
The two formulas given above are not valid at zero however, but the analytic continuation is.


1 + 1 + 1 + 1 + ⋯
where the power series expansion for ζ(s) about s = 1 follows because ζ(s) has a simple pole of residue one there. In this sense 1 + 1 + 1 + 1 + ⋯ = ζ(0) = −1/2.


1 + 1 + 1 + 1 + ⋯
Emilio Elizalde presents a comment from others about the series:


1 + 1 + 1 + 1 + ⋯
In a short period of less than a year, two distinguished physicists, A. Slavnov and F. Yndurain, gave seminars in Barcelona, about different subjects. It was remarkable that, in both presentations, at some point the speaker addressed the audience with these words: 'As everybody knows, 1 + 1 + 1 + ⋯ = −1/2.' Implying maybe: If you do not know this, it is no use to continue listening.


1 + 2 (album)
1 + 2 is the debut album by Recoil, released in August 1986. The CD and cassette version were released two years later with Recoil's second release, Hydrology.


1 + 2 (album)
Wilder had always experimented with his own ideas, but when Daniel Miller heard some demos (recorded on a Portastudio, a 4-track cassette machine) and asked to reproduce them, Recoil became a musical entity. The early recordings show Wilder's position as a pioneer in sampling technology and demonstrated how he could completely change the "Depeche Mode" sound into something new. 1 + 2, his first collection of demos, though completed in the early 1980s, was inconspicuously released as a 12" EP the same year as his band Depeche Mode's fifth studio album, Black Celebration.


1 + 2 (album), Re-release
The CD of Hydrology Plus 1 + 2 was re-released in 2007, again on Mute Records. The track listing and artwork remain the same.


1 + 2 (album), Sample sources
The recording is largely built upon samples of other music, primarily by other Mute Records artists - to avoid legal issues during an era when such approach to music-making was not overtly common.


1 + 2 (album), Notes
This 1980s electronic music album–related article is a stub. You can help Wikipedia by .


1 + 2 + 3 + 4 + ⋯
The infinite series whose terms are the natural numbers 1 + 2 + 3 + 4 + ⋯ is a divergent series. The nth partial sum of the series is the triangular number


1 + 2 + 3 + 4 + ⋯
which increases without bound as n goes to infinity. Because the sequence of partial sums fails to converge to a finite limit, the series does not have a sum.


1 + 2 + 3 + 4 + ⋯
Although the series seems at first sight not to have any meaningful value at all, it can be manipulated to yield a number of mathematically interesting results. For example, many summation methods are used in mathematics to assign numerical values even to a divergent series. In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of −1/12, which is expressed by a famous formula


1 + 2 + 3 + 4 + ⋯
where the left-hand side has to be interpreted as being the value obtained by using one of the aforementioned summation methods and not as the sum of an infinite series in its usual meaning. These methods have applications in other fields such as complex analysis, quantum field theory, and string theory.


1 + 2 + 3 + 4 + ⋯
In a monograph on moonshine theory, Terry Gannon calls this equation "one of the most remarkable formulae in science".


1 + 2 + 3 + 4 + ⋯, Partial sums
The partial sums of the series 1 + 2 + 3 + 4 + 5 + 6 + ⋯ are 1, 3, 6, 10, 15, etc. The nth partial sum is given by a simple formula:


1 + 2 + 3 + 4 + ⋯, Partial sums
This equation was known to the Pythagoreans as early as the sixth century BCE. Numbers of this form are called triangular numbers, because they can be arranged as an equilateral triangle.


1 + 2 + 3 + 4 + ⋯, Partial sums
The infinite sequence of triangular numbers diverges to +∞, so by definition, the infinite series 1 + 2 + 3 + 4 + ⋯ also diverges to +∞. The divergence is a simple consequence of the form of the series: the terms do not approach zero, so the series diverges by the term test.


1 + 2 + 3 + 4 + ⋯, Summability
Among the classical divergent series, 1 + 2 + 3 + 4 + ⋯ is relatively difficult to manipulate into a finite value. Many summation methods are used to assign numerical values to divergent series, some more powerful than others. For example, Cesàro summation is a well-known method that sums Grandi's series, the mildly divergent series 1 − 1 + 1 − 1 + ⋯, to 1/2. Abel summation is a more powerful method that not only sums Grandi's series to 1/2, but also sums the trickier series 1 − 2 + 3 − 4 + ⋯ to 1/4.


1 + 2 + 3 + 4 + ⋯, Summability
Unlike the above series, 1 + 2 + 3 + 4 + ⋯ is not Cesàro summable nor Abel summable. Those methods work on oscillating divergent series, but they cannot produce a finite answer for a series that diverges to +∞. Most of the more elementary definitions of the sum of a divergent series are stable and linear, and any method that is both stable and linear cannot sum 1 + 2 + 3 + ⋯ to a finite value; see below. More advanced methods are required, such as zeta function regularization or Ramanujan summation. It is also possible to argue for the value of −+1/12 using some rough heuristics related to these methods.


1 + 2 + 3 + 4 + ⋯, Summability, Heuristics
Srinivasa Ramanujan presented two derivations of "1 + 2 + 3 + 4 + ⋯ = −+1/12" in chapter 8 of his first notebook. The simpler, less rigorous derivation proceeds in two steps, as follows.


1 + 2 + 3 + 4 + ⋯, Summability, Heuristics
The first key insight is that the series of positive numbers 1 + 2 + 3 + 4 + ⋯ closely resembles the alternating series 1 − 2 + 3 − 4 + ⋯. The latter series is also divergent, but it is much easier to work with; there are several classical methods that assign it a value, which have been explored since the 18th century.


1 + 2 + 3 + 4 + ⋯, Summability, Heuristics
In order to transform the series 1 + 2 + 3 + 4 + ⋯ into 1 − 2 + 3 − 4 + ⋯, one can subtract 4 from the second term, 8 from the fourth term, 12 from the sixth term, and so on. The total amount to be subtracted is 4 + 8 + 12 + 16 + ⋯, which is 4 times the original series. These relationships can be expressed using algebra. Whatever the "sum" of the series might be, call it c = 1 + 2 + 3 + 4 + ⋯. Then multiply this equation by 4 and subtract the second equation from the first:


1 + 2 + 3 + 4 + ⋯, Summability, Heuristics
The second key insight is that the alternating series 1 − 2 + 3 − 4 + ⋯ is the formal power series expansion of the function 1/(1 + x)2 but with x defined as 1. Accordingly, Ramanujan writes


1 + 2 + 3 + 4 + ⋯, Summability, Heuristics
Dividing both sides by −3, one gets c = −+1/12.


1 + 2 + 3 + 4 + ⋯, Summability, Heuristics
Generally speaking, it is incorrect to manipulate infinite series as if they were finite sums. For example, if zeroes are inserted into arbitrary positions of a divergent series, it is possible to arrive at results that are not self-consistent, let alone consistent with other methods. In particular, the step 4c = 0 + 4 + 0 + 8 + ⋯ is not justified by the additive identity law alone. For an extreme example, appending a single zero to the front of the series can lead to a different result.


1 + 2 + 3 + 4 + ⋯, Summability, Heuristics
One way to remedy this situation, and to constrain the places where zeroes may be inserted, is to keep track of each term in the series by attaching a dependence on some function. In the series 1 + 2 + 3 + 4 + ⋯, each term n is just a number. If the term n is promoted to a function n−s, where s is a complex variable, then one can ensure that only like terms are added. The resulting series may be manipulated in a more rigorous fashion, and the variable s can be set to −1 later. The implementation of this strategy is called zeta function regularization.


1 + 2 + 3 + 4 + ⋯, Summability, Zeta function regularization
In zeta function regularization, the series ∑n=1∞n{\textstyle \sum _{n=1}^{\infty }n} is replaced by the series ∑n=1∞n−s{\textstyle \sum _{n=1}^{\infty }n^{-s}}. The latter series is an example of a Dirichlet series. When the real part of s is greater than 1, the Dirichlet series converges, and its sum is the Riemann zeta function ζ(s). On the other hand, the Dirichlet series diverges when the real part of s is less than or equal to 1, so, in particular, the series 1 + 2 + 3 + 4 + ⋯ that results from setting s = –1 does not converge. The benefit of introducing the Riemann zeta function is that it can be defined for other values of s by analytic continuation. One can then define the zeta-regularized sum of 1 + 2 + 3 + 4 + ⋯ to be ζ(−1).


1 + 2 + 3 + 4 + ⋯, Summability, Zeta function regularization
From this point, there are a few ways to prove that ζ(−1) = −+1/12. One method, along the lines of Euler's reasoning, uses the relationship between the Riemann zeta function and the Dirichlet eta function η(s). The eta function is defined by an alternating Dirichlet series, so this method parallels the earlier heuristics. Where both Dirichlet series converge, one has the identities:


1 + 2 + 3 + 4 + ⋯, Summability, Zeta function regularization
The identity (1−21−s)ζ(s)=η(s){\displaystyle (1-2^{1-s})\zeta (s)=\eta (s)} continues to hold when both functions are extended by analytic continuation to include values of s for which the above series diverge. Substituting s = −1, one gets −3ζ(−1) = η(−1). Now, computing η(−1) is an easier task, as the eta function is equal to the Abel sum of its defining series, which is a one-sided limit:


1 + 2 + 3 + 4 + ⋯, Summability, Zeta function regularization
Dividing both sides by −3, one gets ζ(−1) = −+1/12.


1 + 2 + 3 + 4 + ⋯, Summability, Cutoff regularization
The method of regularization using a cutoff function can "smooth" the series to arrive at −+1/12. Smoothing is a conceptual bridge between zeta function regularization, with its reliance on complex analysis, and Ramanujan summation, with its shortcut to the Euler–Maclaurin formula. Instead, the method operates directly on conservative transformations of the series, using methods from real analysis.


1 + 2 + 3 + 4 + ⋯, Summability, Cutoff regularization
The idea is to replace the ill-behaved discrete series ∑n=0Nn{\displaystyle \textstyle \sum _{n=0}^{N}n} with a smoothed version


1 + 2 + 3 + 4 + ⋯, Summability, Cutoff regularization
where f is a cutoff function with appropriate properties. The cutoff function must be normalized to f(0) = 1; this is a different normalization from the one used in differential equations. The cutoff function should have enough bounded derivatives to smooth out the wrinkles in the series, and it should decay to 0 faster than the series grows. For convenience, one may require that f is smooth, bounded, and compactly supported. One can then prove that this smoothed sum is asymptotic to −+1/12 + CN2, where C is a constant that depends on f. The constant term of the asymptotic expansion does not depend on f: it is necessarily the same value given by analytic continuation, −+1/12.


1 + 2 + 3 + 4 + ⋯, Summability, Ramanujan summation
The Ramanujan sum of 1 + 2 + 3 + 4 + ⋯ is also −+1/12. Ramanujan wrote in his second letter to G. H. Hardy, dated 27 February 1913:


1 + 2 + 3 + 4 + ⋯, Summability, Ramanujan summation
Ramanujan summation is a method to isolate the constant term in the Euler–Maclaurin formula for the partial sums of a series. For a function f, the classical Ramanujan sum of the series ∑k=1∞f(k){\displaystyle \textstyle \sum _{k=1}^{\infty }f(k)} is defined as


1 + 2 + 3 + 4 + ⋯, Summability, Ramanujan summation
where f(2k−1) is the (2k − 1)-th derivative of f and B2k is the 2k-th Bernoulli number: B2 = 1/6, B4 = −+1/30, and so on. Setting f(x) = x, the first derivative of f is 1, and every other term vanishes, so


1 + 2 + 3 + 4 + ⋯, Summability, Ramanujan summation
To avoid inconsistencies, the modern theory of Ramanujan summation requires that f is "regular" in the sense that the higher-order derivatives of f decay quickly enough for the remainder terms in the Euler–Maclaurin formula to tend to 0. Ramanujan tacitly assumed this property. The regularity requirement prevents the use of Ramanujan summation upon spaced-out series like 0 + 2 + 0 + 4 + ⋯, because no regular function takes those values. Instead, such a series must be interpreted by zeta function regularization. For this reason, Hardy recommends "great caution" when applying the Ramanujan sums of known series to find the sums of related series.


1 + 2 + 3 + 4 + ⋯, Summability, Failure of stable linear summation methods
A summation method that is linear and stable cannot sum the series 1 + 2 + 3 + ⋯ to any finite value. (Stable means that adding a term to the beginning of the series increases the sum by the same amount.) This can be seen as follows. If


1 + 2 + 3 + 4 + ⋯, Summability, Failure of stable linear summation methods
by stability. By linearity, one may subtract the second equation from the first (subtracting each component of the second line from the first line in columns) to give


1 + 2 + 3 + 4 + ⋯, Summability, Failure of stable linear summation methods
Therefore, every method that gives a finite value to the sum 1 + 2 + 3 + ⋯ is not stable or not linear.


1 + 2 + 3 + 4 + ⋯, Physics
In bosonic string theory, the attempt is to compute the possible energy levels of a string, in particular, the lowest energy level. Speaking informally, each harmonic of the string can be viewed as a collection of D − 2 independent quantum harmonic oscillators, one for each transverse direction, where D is the dimension of spacetime. If the fundamental oscillation frequency is ω, then the energy in an oscillator contributing to the n-th harmonic is nħω/2. So using the divergent series, the sum over all harmonics is −ħω(D − 2)/24. Ultimately it is this fact, combined with the Goddard–Thorn theorem, which leads to bosonic string theory failing to be consistent in dimensions other than 26.


1 + 2 + 3 + 4 + ⋯, Physics
The regularization of 1 + 2 + 3 + 4 + ⋯ is also involved in computing the Casimir force for a scalar field in one dimension. An exponential cutoff function suffices to smooth the series, representing the fact that arbitrarily high-energy modes are not blocked by the conducting plates. The spatial symmetry of the problem is responsible for canceling the quadratic term of the expansion. All that is left is the constant term −1/12, and the negative sign of this result reflects the fact that the Casimir force is attractive.


1 + 2 + 3 + 4 + ⋯, Physics
A similar calculation is involved in three dimensions, using the Epstein zeta-function in place of the Riemann zeta function.


1 + 2 + 3 + 4 + ⋯, History
It is unclear whether Leonhard Euler summed the series to −+1/12. According to Morris Kline, Euler's early work on divergent series relied on function expansions, from which he concluded 1 + 2 + 3 + 4 + ⋯ = ∞. According to Raymond Ayoub, the fact that the divergent zeta series is not Abel-summable prevented Euler from using the zeta function as freely as the eta function, and he "could not have attached a meaning" to the series.


1 + 2 + 3 + 4 + ⋯, History
Other authors have credited Euler with the sum, suggesting that Euler would have extended the relationship between the zeta and eta functions to negative integers.


1 + 2 + 3 + 4 + ⋯, History
In the primary literature, the series 1 + 2 + 3 + 4 + ⋯ is mentioned in Euler's 1760 publication De seriebus divergentibus alongside the divergent geometric series 1 + 2 + 4 + 8 + ⋯. Euler hints that series of this type have finite, negative sums, and he explains what this means for geometric series, but he does not return to discuss 1 + 2 + 3 + 4 + ⋯. In the same publication, Euler writes that the sum of 1 + 1 + 1 + 1 + ⋯ is infinite.


1 + 2 + 3 + 4 + ⋯, In popular media
David Leavitt's 2007 novel The Indian Clerk includes a scene where Hardy and Littlewood discuss the meaning of this series. They conclude that Ramanujan has rediscovered ζ(−1), and they take the "lunatic asylum" line in his second letter as a sign that Ramanujan is toying with them.


1 + 2 + 3 + 4 + ⋯, In popular media
Simon McBurney's 2007 play A Disappearing Number focuses on the series in the opening scene. The main character, Ruth, walks into a lecture hall and introduces the idea of a divergent series before proclaiming, "I'm going to show you something really thrilling", namely 1 + 2 + 3 + 4 + ⋯ = −+1/12. As Ruth launches into a derivation of the functional equation of the zeta function, another actor addresses the audience, admitting that they are actors: "But the mathematics is real. It's terrifying, but it's real."


1 + 2 + 3 + 4 + ⋯, In popular media
In January 2014, Numberphile produced a YouTube video on the series, which gathered over 1.5 million views in its first month. The 8-minute video is narrated by Tony Padilla, a physicist at the University of Nottingham. Padilla begins with 1 − 1 + 1 − 1 + ⋯ and 1 − 2 + 3 − 4 + ⋯ and relates the latter to 1 + 2 + 3 + 4 + ⋯ using a term-by-term subtraction similar to Ramanujan's argument.


1 + 2 + 3 + 4 + ⋯, In popular media
Numberphile also released a 21-minute version of the video featuring Nottingham physicist Ed Copeland, who describes in more detail how 1 − 2 + 3 − 4 + ⋯ = 1/4 as an Abel sum, and 1 + 2 + 3 + 4 + ⋯ = −+1/12 as ζ(−1). After receiving complaints about the lack of rigour in the first video, Padilla also wrote an explanation on his webpage relating the manipulations in the video to identities between the analytic continuations of the relevant Dirichlet series.


1 + 2 + 3 + 4 + ⋯, In popular media
In The New York Times coverage of the Numberphile video, mathematician Edward Frenkel commented: "This calculation is one of the best-kept secrets in math. No one on the outside knows about it."


1 + 2 + 3 + 4 + ⋯, In popular media
Coverage of this topic in Smithsonian magazine describes the Numberphile video as misleading and notes that the interpretation of the sum as −+1/12 relies on a specialized meaning for the equals sign, from the techniques of analytic continuation, in which equals means is associated with.


1 + 2 + 4 + 8 + ⋯
In mathematics, 1 + 2 + 4 + 8 + ⋯ is the infinite series whose terms are the successive powers of two. As a geometric series, it is characterized by its first term, 1, and its common ratio, 2. As a series of real numbers it diverges to infinity, so in the usual sense it has no sum. In a much broader sense, the series is associated with another value besides ∞, namely −1, which is the limit of the series using the 2-adic metric.


1 + 2 + 4 + 8 + ⋯, Summation
The partial sums of 1+2+4+8+⋯{\displaystyle 1+2+4+8+\cdots } are 1,3,7,15,…;{\displaystyle 1,3,7,15,\ldots ;} since these diverge to infinity, so does the series. 20+21+⋯+2k=2k+1−1{\displaystyle 2^{0}+2^{1}+\cdots +2^{k}=2^{k+1}-1}


1 + 2 + 4 + 8 + ⋯, Summation
Therefore, any totally regular summation method gives a sum of infinity, including the Cesàro sum and Abel sum. On the other hand, there is at least one generally useful method that sums 1+2+4+8+⋯{\displaystyle 1+2+4+8+\cdots } to the finite value of −1. The associated power seriesf(x)=1+2x+4x2+8x3+⋯+2nxn+⋯=11−2x{\displaystyle f(x)=1+2x+4x^{2}+8x^{3}+\cdots +2^{n}{}x^{n}+\cdots ={\frac {1}{1-2x}}} has a radius of convergence around 0 of only 12{\displaystyle {\frac {1}{2}}} so it does not converge at x=1.


1 + 2 + 4 + 8 + ⋯, Summation
{\displaystyle x=1.} Nonetheless, the so-defined function f{\displaystyle f} has a unique analytic continuation to the complex plane with the point x=12{\displaystyle x={\frac {1}{2}}} deleted, and it is given by the same rule f(x)=11−2x. {\displaystyle f(x)={\frac {1}{1-2x}}.} Since f(1)=−1,{\displaystyle f(1)=-1,} the original series 1+2+4+8+⋯{\displaystyle 1+2+4+8+\cdots } is said to be summable (E) to −1, and −1 is the (E) sum of the series. (The notation is due to G. H. Hardy in reference to Leonhard Euler's approach to divergent series).


1 + 2 + 4 + 8 + ⋯, Summation
An almost identical approach (the one taken by Euler himself) is to consider the power series whose coefficients are all 1, that is,1+y+y2+y3+⋯=11−y{\displaystyle 1+y+y^{2}+y^{3}+\cdots ={\frac {1}{1- y}}}and plugging in y=2. {\displaystyle y=2.} These two series are related by the substitution y=2x. {\displaystyle y=2x.}


1 + 2 + 4 + 8 + ⋯, Summation
The fact that (E) summation assigns a finite value to 1+2+4+8+⋯{\displaystyle 1+2+4+8+\cdots } shows that the general method is not totally regular. On the other hand, it possesses some other desirable qualities for a summation method, including stability and linearity. These latter two axioms actually force the sum to be −1, since they make the following manipulation valid:


1 + 2 + 4 + 8 + ⋯, Summation
In a useful sense, s=∞{\displaystyle s=\infty } is a root of the equation s=1+2s. {\displaystyle s=1+2s.} (For example, ∞{\displaystyle \infty } is one of the two fixed points of the Möbius transformation z↦1+ 2z{\displaystyle z\mapsto 1+2z} on the Riemann sphere). If some summation method is known to return an ordinary number for s{\displaystyle s}; that is, not ∞,{\displaystyle \infty ,} then it is easily determined. In this case s{\displaystyle s} may be subtracted from both sides of the equation, yielding 0=1+s,{\displaystyle 0=1+s,} so s=−1. {\displaystyle s=-1.}


1 + 2 + 4 + 8 + ⋯, Summation
The above manipulation might be called on to produce −1 outside the context of a sufficiently powerful summation procedure. For the most well-known and straightforward sum concepts, including the fundamental convergent one, it is absurd that a series of positive terms could have a negative value. A similar phenomenon occurs with the divergent geometric series 1−1+1−1+⋯{\displaystyle 1-1+1-1+\cdots } (Grandi's series), where a series of integers appears to have the non-integer sum 12. {\displaystyle {\frac {1}{2}}.} These examples illustrate the potential danger in applying similar arguments to the series implied by such recurring decimals as 0.111… {\displaystyle 0.111\ldots } and most notably 0.999… {\displaystyle 0.999\ldots }. The arguments are ultimately justified for these convergent series, implying that 0.111… =19{\displaystyle 0.111\ldots ={\frac {1}{9}}} and 0.999… =1,{\displaystyle 0.999\ldots =1,} but the underlying proofs demand careful thinking about the interpretation of endless sums.


1 + 2 + 4 + 8 + ⋯, Summation
It is also possible to view this series as convergent in a number system different from the real numbers, namely, the 2-adic numbers. As a series of 2-adic numbers this series converges to the same sum, −1, as was derived above by analytic continuation.


1 + 3
1 + 3 is a live album by bassist Ron Carter which was recorded in Tokyo in 1978 and released on the Japanese JVC label the following year.


1 + 3, Reception
The AllMusic review by Ron Wynn stated "Exactly the kind of impressive, high level playing and interaction you'd expect from this trio. ... While it's Carter's session, there's really no leader or followers, just three wonderful musicians fully attuned to each other".


1 2 3 (Moneybagg Yo song)
"1 2 3" is a song by American rapper Moneybagg Yo featuring American rapper Blac Youngsta, from the former's third studio album Time Served (2020). It was sent to rhythmic contemporary radio on April 14, 2020, as the third single of the album. The music video was released on the same day.


1 2 3 (Moneybagg Yo song), Chart performance
123 missed the US Billboard Hot 100 chart, peaking at number one on the US Bubbling Under Hot 100 chart. On August 18, 2020, the album was certified gold by the Recording Industry Association of America (RIAA) for combined sales and streaming data of over 500,000 units in the United States.


1 April attacks (Cyprus)
The 1 April Attacks were a series of attacks across Cyprus in 1955 by the EOKA which led to the start of the Cyprus Emergency. Multiple British locations were attacked after midnight by EOKA members. This attack was accompanied by the distribution of leaflets across Cyprus.


1 April attacks (Cyprus), Attack
On the night of 31 March–1 April 1955, simultaneous attacks were launched across the island after midnight at various British or British-linked facilities, including:


1 April attacks (Cyprus), Attack
During this attack, Modestos Panteli of the EOKA died from electric shock trying to cut a cable, the first casualty of the Emergency. The attacks were accompanied by the distribution of leaflets from EOKA claiming responsibility and calling on Cypriots to participate in the struggle. Several EOKA officers were arrested in connection to the attacks and sentenced to long prison terms.


1 Aquarii
1 Aquarii is a binary star system in the zodiac constellation of Aquarius, about 257 light years away from the Sun. 1 Aquarii is the Flamsteed designation. It is visible to the naked eye as a faint, orange-hued star with an apparent visual magnitude of 5.151, located a degree north of the celestial equator. The system is moving closer to the Earth with a heliocentric radial velocity of −41 km/s.


1 Aquarii
Systematic observation for determining the orbit of this system began in 2002, some eighty years following the first radial velocity measurements. It is a single-lined spectroscopic binary with an orbital period of 5.385 yr and an eccentricity of 0.368. The visible component is an aging giant star with a stellar classification of K1III. At the age of 1.26 billion years old it is a red clump giant, which indicates it is on the horizontal branch and is generating energy through helium fusion at its core. The star has 1.5 times the mass of the Sun and has expanded to 11 times the Sun's radius. It is radiating 53.7 times the Sun's luminosity from its enlarged photosphere at an effective temperature of 4,715 K.


1 Aquarii
The mass of the companion appears small, suggesting a red dwarf no higher than class M5. In addition to the spectroscopic companion there are two faint optical companions that have no physical relation to 1 Aqr.


1 Aquarii, Etymology
1 Aquarii was known to the ancients as al-sa'd al-malik, or "the lucky star of the king." Interpreting the unexpressed Arabic vowels, al-sa'd al-mulk, gives an alternate translation of  "the lucky star of the kingdom." In English, the name is Sadalmelik (or Sadalmelek), although rarely used today.


1 Area Support Group
1 Area Support Group was a Canadian Army formation that provided support to other Canadian Army formations and units in western Canada


1 Area Support Group, Former Constituent bases and units
1 Area Support Group consists of 12 bases and units:


1 Area Support Group, Restructuring
With the re-organization and re-structuring of the Canadian Army the 1st Area Support Group was merged with Area Support Unit Edmonton to organize the 3rd Canadian Division Support Group.


1 Arietis
1 Arietis is a double star in the northern constellation of Aries. 1 Arietis is the Flamsteed designation. The pair have a combined visual magnitude of 5.86, making it faintly visible to the naked eye. Based upon an annual parallax shift of 5.57 mas, the distance to the two stars is approximately 590 light-years (180 parsecs). As of 2016, the secondary had an angular separation of 2.90″ along a position angle of 165° from the primary. They are moving further from the Earth with a heliocentric radial velocity of +7 km/s.


1 Arietis
The brighter star, designated component A, is a magnitude 6.40 giant star with a stellar classification of K1 III. The companion star, component B, is a magnitude 7.20 A-type main sequence star with a classification of A6 V. Helmut Abt (1985) had this star classified as A3 IV, matching a more evolved subgiant.


1 August 2007 Baghdad bombings
The 1 August 2007 Baghdad bombings occurred on 1 August 2007, when several suicide bombings and car bombings occurred in Baghdad, the capital of Iraq. In one of the attacks, a fuel tanker packed with explosives exploded near a petrol station in the suburb of Mansour. The bombing set fire to ten vehicles at the petrol station killing 50 and wounding 60 in an around the petrol station.


1 Aurigae
1 Aurigae is the original name for a star now in the constellation Perseus. It was the first entry in John Flamsteed's catalogue of stars in Auriga. When Eugène Joseph Delporte drew up simplified boundaries for the constellations on behalf of the International Astronomical Union in 1930, 1 Aurigae ended up over the border in Perseus. To avoid confusion, the star may instead be referred to by its Harvard Revised catalogue number, HR 1533.


1 Aurigae
Based upon its parallax measurement of 6.48 mas, this star is located approximately 520 light years from Earth. It is visible to the naked eye as a faint, orange-hued star with an apparent visual magnitude of 4.89. 1 Aurigae is moving closer to the Earth with a heliocentric radial velocity of −25 km/s.


1 Aurigae
This is a possible binary star system, based upon the status of the visible component as a mild barium star. The primary is an aging giant star with a stellar classification of K3.5 III Ba0.2:. It is 3.9 billion years old with 1.49 times the mass of the Sun and around 44 times the Sun's radius. This star is radiating 561 times the luminosity of the Sun from its enlarged photosphere at an effective temperature of 4,102 K. The suspected companion star should be a white dwarf that previously transferred s-process elements to the visible member.


1 BC
Year 1 BC was a common year starting on Friday or Saturday in the Julian calendar (the sources differ; see leap year error for further information) and a leap year starting on Thursday in the Proleptic Julian calendar. It is also a leap year starting on Saturday in the Proleptic Gregorian calendar. At the time, it was known as the Year of the Consulship of Lentulus and Piso (or, less frequently, year 753 Ab urbe condita). The denomination 1 BC for this year has been used since the early medieval period when the Anno Domini calendar era became the prevalent method in Europe for naming years. The following year is 1 AD in the widely used Julian calendar, which does not have a "year zero".


1 Bentinck Street
1 Bentinck Street is a grade II listed house in Bentinck Street, in the City of Westminster, London. The house was completed around 1800. It is on the corner with Welbeck Street.


1 Billion Views
1 Billion Views is the first studio album by duo Exo-SC, the second official sub-unit of South Korean–Chinese boy group Exo. It was released on July 13, 2020, by SM Entertainment, and features nine tracks. The album debuted at number one on the Gaon Album Chart, and was the duo's first top-ten entry on the Oricon Albums Chart, peaking at number ten in its first week.


1 Billion Views
To promote the album, "Telephone" was released as a pre-release single on July 7, 2020, while the eponymous title track served as the second single on July 13, 2020. The latter earned the duo their first top-thirty entry on the Gaon Digital Chart since their debut.


1 Billion Views, Background and release
On June 9, SM confirmed that EXO-SC were preparing to release a new album in July. On June 23, the album's release date and title were announced along with a teaser image of the digital cover.


1 Billion Views, Background and release
On July 1, a teaser image of the album's track list and schedule was released. On July 2, five teaser images of Chanyeol were released, with another five images of Sehun released the next day. On July 3, it was reported that the duo participated in writing the lyrics of every song on the album and helped compose the tracks "Telephone", "Fly Away" and "On Me". On July 4, five teaser images of both Sehun and Chanyeol were released. On July 6, teaser images of the duo both together and individually were released.


1 Billion Views, Background and release
On the same day, it was reported that the album features two solo tracks: "Nothin'", sung by Chanyeol and "On Me", sung by Sehun. On July 7, two teaser images of the duo were released. On the same day, "Telephone" featuring 10cm was released digitally along with a music video. On July 8, teaser images of both members were released, along with the music video for "Nothin'", a solo track by Chanyeol. On July 9, the music video for Sehun's solo track "On Me" was released. On July 12, "1 Billion Views" music video teaser was released. On July 13, the album was officially released along with "1 Billion Views" music video.


1 Billion Views, Composition and production
The album opens with the titular "1 Billion Views", a trendy hip hop song with a funky guitar sound and addictive disco rhythm. Featuring fellow singer Moon, the song finds the duo "wanting to see one's loved one" by playing their video repeatedly. The second track "Say It", featuring Penomeco (who also co-wrote the lyrics), is described as a hip hop song that combines a heavy 808 base and bossa nova rhythm in order to "feel the summer vibe".


1 Billion Views, Composition and production
"Telephone", featuring 10cm, is described as hip hop song with a cheerful piano riff and a heavy bass beat.


1 Billion Views, Composition and production
"Fly away", featuring Gaeko who also participated in composing the song, is described as an r&b hip-hop song based on sentimental lyrical band sounds.


1 Billion Views, Composition and production
"Nothin'", Chanyeol's solo song, in which he participated in both writing and composing, is described as a hip hop R&B song with a harmony between the dreamy vibe electric guitar sounds and the heavy beats. The lyrics are about one's determination to go on their own way silently and without paying attention to the surroundings.


1 Billion Views, Composition and production
"On Me", Sehun's solo, in which he participated in both writing and composing, is described as a trap hip hop song with a rhythmical bass & strong synthesizer. The lyrics contain a message of "I'll do my best in every present moment".


1 Billion Views, Composition and production
"Rodeo Station" is a hip hop song which combines simple guitar riffs and casual beats. In the lyrics, Sehun and Chanyeol look back on their past and present lives and recalling the scenery around Apgujeong Rodeo Station during their trainee days.


1 Billion Views, Composition and production
"Jet Lag" is a charming R&B hip hop song with a lyrical guitar performance with lyrics about the main character who's in a relationship where they can't meet their loved one easily, and both parts feel like their love is growing apart because of the time they spend away from each other.


1 Bligh Street
1 Bligh Street is a skyscraper in Sydney, New South Wales, Australia.


1 Bligh Street
The modern style office building is located in the Sydney central business district overlooking Circular Quay, the Sydney Harbour and the Sydney Harbour Bridge.


1 Bligh Street, Design
The premium grade office tower was designed by Ingenhoven Architects of Germany and Architectus of Australia.


1 Bligh Street, Design
It is an ecologically sustainable development and was awarded six-star green status by the Green Building Council of Australia. Green features include a basement sewage plant that recycles 90 percent of the building waste water, solar panels on the roof and air conditioning by chilled beams. It is Australia's first major high-rise building with a full double-skin façade with external louvres. These conserve energy, eliminate sky glare and optimise user comfort. The angle of the louvre blades is automatically adjusted according to their orientation to the sun. A naturally ventilated, full height atrium, on the southern side of the building, maximises natural light to each office level.


1 Bligh Street, Design
The building also houses a childcare centre, two cafés and a basement car park for 96 cars.


1 Bligh Street, Design
The large-scale aluminium sculpture at the top of the curving steps at the entrance on the corner of Bligh and O'Connell streets is by New York-based Australian James Angus. The developers describe it as "a complex network of three-dimensional ellipsoidal surfaces drawn from shapes expressed in the design of the building", adding that its brightly painted colour scheme traces the underlying geometry of the sculpture.


1 Bligh Street, Design
The building was named the Best Tall Building Award in Asia & Australasia for 2012 in the Council on Tall Buildings and Urban Habitat's Skyscraper Awards and also won the International Highrise Award 2012.


1 Boötis
1 Boötis (1 Boo) is a binary star system in the northern constellation of Boötes, located 318 light years away from the Sun. It is visible to the naked eye as a dim, white-hued star with a combined apparent visual magnitude of 5.71. The pair had an angular separation of 4.660″ as of 2008. It is moving closer to the Earth with a heliocentric radial velocity of −26 km/s.


1 Boötis
The magnitude 5.78 primary component is an A-type main-sequence star with a stellar classification of A1 V. This star has 2.5 times the mass of the Sun and is radiating 56 times the Sun's luminosity from its photosphere at an effective temperature of 9,863 K. It is 323 million years old and is spinning with a projected rotational velocity of 60 km/s.


1 Boötis
The system is a source for X-ray emission, which is most likely coming from the companion star. This magnitude 9.60 component is a possible pre-main sequence star with a mass similar to the Sun. It is radiating 76% of the Sun's luminosity at an effective temperature of 6,370 K.


1 Bridge Street, Chester
1 Bridge Street, Chester, is located at the junction of Bridge Street and Eastgate Street at Chester Cross in the centre of the city of Chester, Cheshire, England. Its architecture is that of the black-and-white revival, it incorporates part of the Chester Rows, and is recorded in the National Heritage List for England as a designated Grade II* listed building. Because of its prominent position and its black-and-white architecture, the historian Simon Ward has described it as an "iconic" building.


1 Bridge Street, Chester, History
The building was designed by Thomas Lockwood, and built in 1888 for the 1st Duke of Westminster, although by 1889 it was owned by Chester City Council. The building is now occupied by shops. The citation in the National Heritage List describes the building as "the best liked of T. M. Lockwood's buildings in Chester, well executed in his most flamboyant style".


1 Bridge Street, Chester, Architecture
The building is in four storeys, including an undercroft, whose floor is below street level, and an attic. It extends for one bay down Bridge Street and for one bay along Eastgate Street. At its corner is a flight of seven steps leading from the street to the row level above which is an octagonal turret. Above the steps is a canted, mullioned and transomed, oriel window. Over this are three pargetted panels under a four-light canted casement window. The roof is curved and surmounted by a weather vane.


1 Bridge Street, Chester, Architecture
On each side of the steps at street level are modern shop fronts. Above these, at the front of the Rows, are balustrades behind which are sloping stall boards, then the walkway and shop fronts. On Bridge Street at the third storey level are decorative panels and a three-light window. Above this is a gabled dormer with more panels and another three-light window. On Eastgate Street at the third storey is a six-light window with panels above and below. There is again a dormer with panels, one of which includes the arms of the Grosvenor family.


1 Cabot Square
1 Cabot Square (also known as the Credit Suisse building) is a 21-floor office building occupied by Credit Suisse in Cabot Square, Canary Wharf, London, England.


1 Cabot Square, History of Project
Original plans called for a skyscraper on this site for Credit Suisse First Boston, however the slump at the end of the 1980s saw these plans scaled back.


1 Cabot Square, Architect
The architect on the project was Pei Cobb Freed & Partners, and the building was completed in 1991. The project had two management contractors Ellis Don and Sir Robert McAlpine.


1 Cabot Square, Architecture
It is 89 metres tall (292 feet), with a floorspace of 50,166 square metres (164,587 square feet).


1 Cabot Square, Architecture
The building has large, open plates on the floor, that range in size from 64,500 square foot to 24,000 square foot in the executive offices.


1 Cabot Square, Architecture
It is internally connected to the west, to 20 Columbus Courtyard, which is also connected to a full-height internal link to the north, 17 Columbus Courtyard.


1 Cabot Square, Architecture
The building is the second largest building located at Canary Wharf, behind the Canary Wharf Tower. The Credit Suisse building has 18 stories with an additional two stories of plant at the top. The building also has a two story arcade that is located at the base of the building.


1 Cabot Square, Ownership
In early 2012 it was purchased by Qatar’s sovereign wealth fund Qatar Investment Authority (QIA)  Subsequently QIA considered selling it multiple times before eventually selling to KB Securities of South Korea.


1 Cabot Square, Ownership
The building is leased to Credit Suisse for around 20 years and part of Canary Wharf, of which QIA bought a majority in a joint venture with Brookfield Properties in 2015.


1 Cabot Square, Gallery
1 Cabot Square with One Canada Square from Westfery Circus


1 Camelopardalis
1 Camelopardalis (1 Cam) is a double star system in the constellation Camelopardalis. Its combined apparent magnitude is 5.56 and it is approximately 800 parsecs (2,600 ly) away.


1 Camelopardalis
The 1 Camelopardalis system is part of the Camelopardalis OB1 stellar association, which is 820 pc away. 1 Camelopardalis A is a hot massive star which has evolved away from the main sequence to become a giant. 1 Camelopardalis B is 10" away and is probably an early B class subgiant.


1 Camelopardalis
There is an 11th magnitude star 150" away. It is unclear whether it is a member of a triple system, or if the fainter star is only a foreground object lying in the same direction.


1 Camelopardalis
1 Camelopardalis A is a variable star with a small amplitude. It has a likely period of 0.22132 days and is thought to be a β Cephei variable or slowly pulsating B-type star. Hipparcos photometry shows an amplitude of 0.035 magnitudes. It has a rotational velocity of 275 km/s, one of the highest known.


1 Canadian Air Division
1 Canadian Air Division (1 Cdn Air Div) (French: 1re Division aérienne du Canada) is the operational-level command and control formation of the Royal Canadian Air Force (RCAF). Prior to 2006 the official abbreviation for the division was 1 CAD. It is commanded by an air force major-general.


1 Canadian Air Division, History, Formation (1952–1967)
The division traces its origins to the activation of Headquarters No. 1 Air Division, Royal Canadian Air Force in Paris, France, on 1 October 1952. Air Division headquarters relocated to  Metz, France in April 1953. No. 1 Air Division was established to meet Canada's NATO air defence commitments in Europe. It consisted of four wings of twelve fighter squadrons located at four bases.


1 Canadian Air Division, History, Formation (1952–1967)
Two bases were located in France (RCAF Station Marville (No. 1 Wing) and RCAF Station Grostenquin (No. 2 Wing)) and two were located in West Germany (RCAF Station Zweibrücken (No. 3 Wing) and RCAF Station Baden-Soellingen (No. 4 Wing)). ( No. 1 Wing was first located at RCAF North Luffenham, England and was moved to Marville some time after October, 1954).


1 Canadian Air Division, History, Formation (1952–1967)
These wings were part of a group of bases which also included U.S. and French installations, all of which came under the jurisdiction of NATO's Fourth Allied Tactical Air Force (4 ATAF) which, in turn, was commanded by Allied Air Forces Central Europe (AAFCE). Components located in Metz included  Air Division Headquarters, an air traffic control centre, a telecommunications centre, a combat operations centre, and a support unit. From 1952 to 1963 the RCAF operated the 30 Air Materiel Base, at RCAF Langar (RAF Langar) in Nottinghamshire. RCAF Langar was Canada's last base in the U.K. and served as a primary supply station for No. 1 Air Division RCAF in Europe.


1 Canadian Air Division, History, Formation (1952–1967)
Canadian squadrons were originally equipped with Canadair Sabre day fighters. One squadron of each wing, however, would be replaced by the all-weather CF-100 in 1956. The Sabre squadrons were replaced by (nuclear) strike/reconnaissance CF-104 Starfighters in 1962.


1 Canadian Air Division, History, Reorganization (1967–1993)
After the RCAF left France in 1967 and after the RCAF was reorganized and consolidated with Canada's other two services, No. 1 Air Division was replaced by No. 1 Canadian Air Group (1 CAG) with headquarters at CFB Lahr, West Germany.


1 Canadian Air Division, History, Reorganization (1967–1993)
As an austerity measure, in 1968, No. 3 Wing Zweibrücken was closed and its two squadrons were moved to Nos. 1 and 4 Wing. 1969 brought the announcement that the amalgamation of the Canadian Forces in Europe to one command and two bases, and that the Canadian army in northern Germany (Soest area) would be moving south to Nos. 1 and 4 Wings.


1 Canadian Air Division, History, Reorganization (1967–1993)
This meant that No. 1 Wing Lahr would close its doors and the air force in Europe would be reduced in strength (from 6 to 3 squadrons) and concentrated at Baden-Soellingen; the new name would be 1 Canadian Air Group (CAG). The close out parade was held at Baden in the arena on June 29, 1970. This was the change date from 4 Wing to 1 CAG. It was also the disbanding date for two of the squadrons.


1 Canadian Air Division, History, Reorganization (1967–1993)
The Group remained until 1988 when Canada increased its commitment to NATO (three squadrons in theatre and two squadrons in Canada) and No. 1 Canadian Air Division stood-up again. However, shortly after this, relations with the east started to warm and Canada made another announcement: Canada would withdraw its forces stationed in Europe and close the doors on its two bases by 1994. The Air Division, reduced to three squadrons then to two and finally one, ceased flying operations 1 January 1993. This ended a major era of Canada's Air Force.


1 Canadian Air Division, History, Recent history (1997–present)
In the summer of 1997, the functional groups (Air Transport Group, Fighter Group, Maritime Air Group, Air Reserve Group, and 10 Tactical Air Group) were dissolved, and 14 Training Group was absorbed within Air Command Headquarters. 1 Canadian Air Division was stood up in Winnipeg to exercise operational command of all CF air assets.


1 Canadian Air Division, History, Recent history (1997–present)
Today based in Winnipeg, Manitoba, the division is also the headquarters for the Canadian NORAD Region (CANR), commands 11 of the RCAF's 13 wings, and oversees the monitoring of Canada's airspace in support of the nation's commitments to the North American Aerospace Defence Command (NORAD). The division is staffed by 600 regular and reserve force members. In addition to military personnel the headquarters is also assisted by civilian personnel in the Operational Research and Analysis Directorate (ORAD).


1 Canadian Forces Flying Training School
While the lineage of 1 Canadian Forces Flying Training School (1 CFFTS; French: 1re École de pilotage des Forces canadiennes) can be traced back through previous similarly named Canadian military pilot flying schools, such as No. 1 Service Flying Training School and 1 Flying Training School, the unit located at CFB Winnipeg and currently bearing the name does not participate in pilot training.


1 Canadian Forces Flying Training School, History of advanced pilot training in the RCAF/CAF
Although military pilot training in Canada dates back to the early years of the 20th century, the modern Royal Canadian Air Force (RCAF) pilot training organization got its start in the British Commonwealth Air Training Plan (BCATP) with the formation of service flying training schools one of which, No. 1 Service Flying Training School (No. 1 SFTS)  at Camp Borden, Ontario, specialized in advanced pilot training and eventually evolved into 1 Canadian Forces Flying Training School in 1965. Under this system pilot candidates could expect to advance from 3 CFFTS to 2 CFFTS, then finish at 1 CFFTS, mimicking the Royal Air Force pilot training philosophy adopted by the RCAF of three separate stages of pilot training – elementary, intermediate and advanced.


1 Canadian Forces Flying Training School, History of advanced pilot training in the RCAF/CAF
1 Flying Training School (FTS) was formed on 16 September 1945 at Centralia, Ontario, as a pilot training school which, on July 11, 1952, started training pilots using the deHavilland Chipmunk aircraft. The school trained RCAF and North Atlantic Treaty Organization (NATO) pilots until the school disbanded on March 6, 1957.


1 Canadian Forces Flying Training School, History of advanced pilot training in the RCAF/CAF
1 FTS was reactivated on 31 August 1965. On 23 July 1970 it combined with the Flight Instructor School (FIS) to become 1 Canadian Forces Flying Training School (1 CFFTS) with the role of advanced pilot training. On 9 November 1970, 1 CFFTS flying instructors converted to the CF-5 tactical fighter trainer and on July 22 1 CFFTS moved to Cold Lake, Alberta. Advanced pilot training became a post-wings course, and on 10 September 1977 1 CFFTS was renamed 419 Tactical Fighter Training Squadron.


1 Canadian Forces Flying Training School, Advanced pilot training today
419 (Tactical) Fighter Training Squadron is carrying on 1 CFFTS's historic role of advanced pilot training for future fighter pilots.


1 Canadian Forces Flying Training School, Current use of the name 1 CFFTS
On 26 June 2009, the Canadian Forces Air Navigation School adopted the name of 1 CFFTS. 1 CFFTS currently trains air combat systems officers (previously called navigators) and airborne electronic sensor operators. In the past, other aircrew trades used No. 1 in their school names, but these school names were distinguished by their particular trade, for example No. 1 Air Observer School and No. 1 Air Navigation School. Pilot training schools were distinguished by the use of the word flying in the school's name. The use of the word flying in 1 CFFTS's name was a break with this tradition that changed again in 2014 when 1 CFFTS was amalgamated into 402 Squadron.


1 Canadian Mechanized Brigade Group
1 Canadian Mechanized Brigade Group (1 CMBG; French: 1er Groupe-brigade mécanisé du Canada, 1 GBMC) is a Canadian Forces brigade group that is part of the 3rd Canadian Division of the Canadian Army. Originally headquartered at CFB Calgary, it is currently based in CFB Edmonton in Alberta with two major units at CFB Shilo in Manitoba, and consists of eight Regular Force units.


1 Canadian Mechanized Brigade Group, History
The brigade was established on 14 October 1953 in Europe.


1 Canadian Mechanized Brigade Group, History
In 1989 at the height of the Cold War the 1st Canadian Mechanized Infantry Division consisted of three brigades, one of them being the 1st Canadian Brigade Group. It had the following structure:


1 Canadian Mechanized Brigade Group, Composition
Three regular force units of other formations, 408 Tactical Helicopter Squadron of 1 Wing, 1 Field Ambulance of 1 Area Support Group, and 1 Military Police Platoon/1 Military Police Regiment are co-located with 1 CMBG at CFB Edmonton and work closely with the brigade group, but are not officially a part of it.


1 Canadian Mechanized Brigade Group, Composition
1 CMBG sent a contingent of approximately 200 personnel for Task Force Kandahar Headquarters for a nine-month rotation starting February 2009 and sent approximately 2500 personnel for the task force in September 2009 for a six-month tour.


1 Canadian Mechanized Brigade Group Headquarters and Signal Squadron
1 Canadian Mechanized Brigade Group Headquarters and Signal Squadron is a Regular Force Army unit of the Canadian Armed Forces at CFB Edmonton. The Squadron is part of 3rd Canadian Division's 1 Canadian Mechanized Brigade Group (1 CMBG).


1 Canadian Mechanized Brigade Group Headquarters and Signal Squadron, History
1 Canadian Mechanized Brigade Group Headquarters and Signal Squadron traces its heritage back to the formation of the Royal Canadian Corps of Signals on 24 Oct 1903. Before the end of the Second World War, small signal sections provided tactical communications for the Canadian Army's various Brigade Headquarters. After World War II, Brigade communication requirements expanded significantly, and it became necessary for the Army to expand these sections into Squadrons. In 1958, as part of the Army reorganization the 1st Canadian Infantry Brigade Group relocated its headquarters to Calgary from Edmonton.


1 Canadian Mechanized Brigade Group Headquarters and Signal Squadron, History
On 15 May 1958 as part of this reorganization, 1 Signal Squadron was formed from elements of the 1 Canadian Divisional Signal Regiment in Borden, Ontario. The Squadron was garrisoned in Calgary. Initially, 1 Signal Squadron was a separate unit from the Brigade Group Headquarters. In 1968, for reasons of efficiency and cost effectiveness, 1st Canadian Infantry Brigade Group Headquarters and 1 Signal Squadron were combined into 1 Canadian Brigade Group Headquarters and Signal Squadron. The role of the Squadron was and continues to be the provision of communications, security and service support to the Commander of 1 Canadian Mechanized Brigade Group and his staff.


1 Canadian Ranger Patrol Group
No. 1 Canadian Ranger Patrol Group is a patrol group of the Canadian Rangers.


1 Canadian Ranger Patrol Group
The Rangers have provided a visible military presence in isolated, coastal and northern communities for over 65 years and continue to serve as the military's "eyes, ears and voice" of the North. 1 CRPG's motto "Vigilans," which translated means "The Watchers," reflects how the Rangers continue to guard their local areas and provide support to the Canadian Armed Forces during domestic operations across the North.


1 Canadian Ranger Patrol Group, History
No. 1 Canadian Ranger Patrol Group (1 CRPG) traces its origin to the Pacific Coast Militia Rangers (PCMR), formed as coast watchers in 1942 and stood down at the end of the Second World War.


1 Canadian Ranger Patrol Group, History
Soon after the war ended, senior defence officials decided that the Ranger concept was still valid and decided to expand it to include all stretches of Canada's coastline:  the Pacific, the Arctic, and the Atlantic. Accordingly, the Canadian Rangers were officially formed as a Corps of the Reserve Militia by an Order-in-Council in 1947. Army Headquarters authorized the first two Ranger companies on September 4, 1947. The first Canadian Ranger (CR) patrol was stood up in Dawson on September 22, 1947. The second CR patrol was stood up in Whitehorse soon thereafter.


1 Canadian Ranger Patrol Group, History
The role of the CRs, as stated in 1947, was to provide a military presence in those sparsely settled northern, coastal and isolated areas of Canada which cannot conveniently or economically be covered by other elements of the Canadian Armed Forces (CAF). This role is still valid today.


1 Canadian Ranger Patrol Group, History
Members were generally outdoor workers who would perform military surveillance while going about their daily jobs. In the late 1940s and early 1950s rifles were issued to reliable individuals in communities in the high arctic along with instructions to report any unusual activity in their areas. By the late 1950s Ranger Platoons and Companies had been established throughout the North.


1 Canadian Ranger Patrol Group, History
Over the years, there have been several organizational accommodations to decentralize command and control of the Rangers. This culminated in November 1997 with the signing of a Ministerial Organizational Order to create five distinct and individual units Canadian Ranger Patrol Groups. The 1 Canadian Ranger Patrol Group (1 CRPG), established on 2 April 1998, consist of Rangers from Nunavut, the Northwest Territory (NWT), Yukon Territory and Atlin, British Columbia, with the CRPG headquarters in Yellowknife, NT.


1 Canadian Ranger Patrol Group, 1 CRPG Headquarters
1 CRPG is a unique Army unit due to its large area of responsibility (AOR), diversity, culture, size (largest military unit in Canada), mandate, and command relationships. It is under direct command (OPCOM) of 3rd Canadian Division but is under Operational Control (OPCON) to Joint Task Force (North) for Force Employment, while also receiving policy direction from the Canadian Ranger National Authority (CRNA), and the National Cadet and Junior Canadian Ranger Support Group. Its primary mission is to prepared to respond to emergencies, conduct sovereignty and nation building activities, support other CAF assets and other  government departments (OGDs), while delivering a successful Junior Canadian Ranger (JCR) program throughout the North.


1 Canadian Ranger Patrol Group, 1 CRPG Headquarters
The HQ has a manning strength of 128 Regular and Reserve Force personnel who are responsible for the administration, training and operations of 60 Ranger patrols and 44 JCR patrols. Of the 128 HQ staff, approximately 21 are Ranger Instructors (RIs) and deploy on a monthly basis for up to 10 days at a time. The remaining HQ staff deploy less frequently, but have a very demanding workload throughout the year supporting the activities associated with Rangers, JCRs and RIs.


1 Canadian Ranger Patrol Group, Activities
As of January 2016, 1 CRPG has over 2200 Rangers in 60 patrols across the three Territories and Atlin, BC. Its area of responsibility encompasses approximately four million square kilometers (40 per cent) of Canada's land mass, 75 per cent of its coastal regions, and run 8000 km (5000 mi) east to west. The members of 1CRPG carry out over 200 exercises and operations a year.


1 Canadian Ranger Patrol Group, Activities
The Canadian Rangers across the North are organized into individual patrols, identified by the name of their local community, and come under the command of 1 CRPG. There are 25 patrols in Nunavut, 22 patrols in the Northwest Territories, 12 in the Yukon Territory, and one in Atlin, British Columbia.


1 Canadian Ranger Patrol Group, Activities
Most of the patrols in the NWT and Nunavut can only be reached by air. Commercial airlines, charter or Twin Otter aircraft of 440 Transport Squadron facilitate travel to these isolated patrols. The exceptions to this situation are the NWT patrols of Behchokǫ̀, Fort Simpson, Fort Resolution, and Fort Smith; these communities can be reached by an all-weather, year-round road. During the long winter season, travel to the villages of Whatì, Gamèti, Wekweeti and Trout Lake can be accomplished by ice road. All the patrols in the Yukon Territory, less the community of Old Crow, can be reached by an all-weather road system.


1 Canadian Ranger Patrol Group, Activities
The Canadian Ranger patrols reflect the demographics of their communities and range in size from 12 to 60 Rangers. The majority of the Canadian Rangers are Aboriginal. Many are unilingual in their native language; however there are always enough English-speaking Canadian Rangers to translate conversations.


1 Canadian Ranger Patrol Group, Activities
Enrollment in the Canadian Rangers is open to both males and females over the age of 18. Patrol leadership consists of a Ranger Sergeant, assisted by a Ranger Master Corporal, and several Corporals depending on the patrol size. Unlike the CAF, local leadership positions are elected by the Rangers in the patrol, based on the leadership candidates’ traditional knowledge of living on the land and the respect that the community has for them.


1 Canadian Ranger Patrol Group, Activities
The Rangers, upon enrolment are issued with their uniform that consist of red sweatshirt with the Ranger crest, red baseball cap, camouflage pants, boots, and Lee-Enfield .303 caliber rifle. Canadian Rangers are also provided up to 100 rounds of ammunition per Ranger per year.


1 Canadian Ranger Patrol Group, Activities
The Junior Canadian Rangers, the largest youth program in the North, are an integral part of the Ranger family. In 1 CRPG, there are over 1600 Junior Rangers in 41 communities co-located with Ranger Patrols. The JCR program is open to all youth ages 12 to 18 years old and promotes traditional cultures and lifestyles, in remote and isolated communities of the North. JCRs make a valuable contribution to their communities and become active, responsible citizens.


1 Canadian Ranger Patrol Group, Training
Canadian Rangers undergo six days of basic training upon enrollment. In all territories, the objective is for all patrols to conduct an annual refresher training patrol which consists of up to 10 days classroom and training on the land.


1 Canadian Ranger Patrol Group, Training
The Ranger concept takes advantage of the Northerners special skills. The CAF relies on their unsurpassed knowledge of the local environment, their uncanny ability to navigate and to survive on the land, and depend on them to help the military operate in the harsh and often extreme environment of the North. They are more than the eyes and ears of Canada; they represent the CAF in a vast part of the country.


1 Canadian Ranger Patrol Group, Training
Ranger training is based on the specific level of training and experience of each patrol and is broken down into individual, collective or leadership training. Key subjects that can be taught include:	a.	first aid;	b.	in-service weapons;	c.	guiding or scouting;	d.	navigation using map and compass, as well as GPS;	e.	traditional and survival skills;	f.	operations in support of Regular Force units;	g.	basic military drill;	h	surveillance;	i.	search and rescue;	j. North Warning System patrolling; and	k.	patrol leadership and administration.


1 Canadian Ranger Patrol Group, Training
Ultimately, the CAF maintains the responsibility to respond to any kind of threat to Canada's north. The CAF prepares for this commitment by undertaking three sovereignty exercises per year to maintain the skills required to operate in weather extremes at great distances from support bases. Ranger support includes 1 CRPG staff and Rangers for each exercise.


1 Canadian Ranger Patrol Group, Training
Ranger Patrols are additionally tasked with sovereignty patrols in the area of 300 km radius around their communities, with patrol members taking as much as a week to exercise a military presence, at specific locations around the North.


1 Cancri
1 Cancri is a single star in the zodiac constellation of Cancer, positioned near the border with Gemini at a distance of around 470 light years from the Sun. It is barely visible to the naked eye as a dim, orange-hued star with an apparent visual magnitude of 5.97. The object is moving further from the Earth with a heliocentric radial velocity of +14 km/s.


1 Cancri
This is an evolved giant star with a stellar classification of K3− III, having exhausted the hydrogen at its core and expanded. It is specified as a spectral standard for that type. The angular diameter of the star measured from a lunar occultation is 2.1±0.6 mas, which, at its estimated distance, equates to a physical radius of about 31 times the radius of the Sun. It is radiating 137 times the Sun's luminosity from its enlarged photosphere at an effective temperature of 4,487 K.


1 Canis Minoris
1 Canis Minoris is a single star in the equatorial constellation of Canis Minor, located about 287 light years away from the Sun. It is visible to the naked eye as a faint, white-hued star with an apparent visual magnitude of 5.37. The radial velocity of this object is poorly constrained at −1.0±4.2 km/s.


1 Canis Minoris
Cowley et al. (1969) listed a stellar classification of A5 IV for 1 Canis Minoris, matching an A-type subgiant star that has exhausted the hydrogen at its core and is evolving into a giant. However, Gray and Garrison (1989) catalogued it as an A-type main-sequence star with a class of A4 V. The Hipparcos team used a class of A3 Vn, where the 'n' indicates "nebulous" lines due to rapid rotation.


1 Canis Minoris
This star is estimated to be 716 million years old and is at or near the end of its main sequence lifetime. It has a high rate of spin, showing a projected rotational velocity of 159 km/s. The star has more than double the mass of the Sun with about 4.6 times the Sun's radius. It is radiating 66 times the Sun's luminosity from its photosphere at an effective temperature of 8,374 K.


1 Cassiopeiae
1 Cassiopeiae is a single star in the northern constellation of Cassiopeia, located around 1,130 light years from the Sun. It is visible to the naked eye as a faint, blue-white hued star with an apparent visual magnitude of +4.84. This object is moving closer to the Earth with a heliocentric radial velocity of −9 km/s.


1 Cassiopeiae
The stellar classification of 1 Cassiopeiae is B0.5 III, matching an evolved B-type giant star. It is 5.7 million years old with a projected rotational velocity of 31. The star has 13.1 times the mass of the Sun and 10.2 times the Sun's radius. It is radiating 18,200 times the Sun's luminosity from its photosphere at an effective temperature of 27,200 K.


1 Centauri
1 Centauri, or i Centauri, is a yellow-white hued binary star system in the southern constellation Centaurus. It can be seen with the naked eye, having an apparent visual magnitude of +4.23. Based upon an annual parallax shift of 51.54 mas as seen from Earth's orbit, it is located 51.5 light years from the Sun. The system is moving closer to the Sun with a radial velocity of −21.5 km/s.


1 Centauri
Spectrographic images taken at the Cape Observatory between 1921 and 1923 showed this star has a variable radial velocity, which indicated this is a single-lined spectroscopic binary star system. The pair have an orbital period of 9.94 days and an eccentricity of about 0.2.


1 Centauri
The primary component has received a number of different stellar classifications. For example, Jaschek et al. (1964) lists F0V, F2III, F4III, and F4IV, thus ranging in evolutionary state from an ordinary F-type main-sequence star to a giant star. More recently, Houk (1982) listed a class of F3 V, matching an ordinary main-sequence star that is generating energy through hydrogen fusion at its core. The NStars project gives it a classification of F2 V.


1 Ceti
1 Ceti is a star in the constellation of Cetus. With an apparent magnitude of about 6.2, the star is barely visible to the naked eye (see Bortle scale). Parallax estimates put it at a distance of about 535 light-years (164 parsecs) away from the Earth. It is moving further from the Sun with a heliocentric radial velocity of 4 km/s.


1 Ceti
This star has a spectral type of K1III, implying a K-type giant. These types of stars are generally reddish-colored stars with spectral types from K to M, with radii that are 10 to 100 times larger than the Sun. The "CNII" in its spectral type indicates strong cyanogen signature in its outer atmosphere. The star is radiating 144 times the Sun's luminosity from its enlarged photosphere.


1 Chor 2 Mastikhor
1 Chor 2 Mastikhor, is a 2017 Indian-Costa Rican comedy-drama film directed by Prabhakar Sharan and produced under the banner of Pacific Investment Corporation. The film stars Nancy Dobles, Prabhakar Sharan, Mario Chacón, José Castro and Scott Steiner. The movie was released on 15 December 2017.


1 Chor 2 Mastikhor
The film was filmed in Costa Rica and released in Spanish as Enredados: La Confusión.


1 Chor 2 Mastikhor, Plot
Leo, played by Sharan, is in a deep dilemma to choose between love and money. The movie focuses on a robbery takes a U-turn after an accident after which Leo chooses love over money.


1 Chronicles 1
1 Chronicles 1 is the first chapter of the Books of Chronicles in the Hebrew Bible or the First Book of Chronicles in the Old Testament of the Christian Bible. The book is compiled from older sources by an unknown person or group, designated by modern scholars as "the Chronicler", and had the final shape established in late fifth or fourth century BCE.


1 Chronicles 1
The content of this chapter is the genealogy list from Adam to Israel (=Jacob) in the following structure: Adam to Noah (verses 1–4); Noah's descendants from his three sons Shem, Ham, and Japheth: the Japhethites (verses 5–7), Hamites (verses 8–23), Semites (verses 24–27); the sons of Abraham (verses 28–34a); the sons of Isaac (34b–54; continued to 2:2 for Israel's sons). This chapter belongs to the section focusing on the list of genealogies from Adam to the lists of the people returning from exile in Babylon (1 Chronicles 1:1 to 9:34).


1 Chronicles 1, Text
This chapter was originally written in the Hebrew language and is divided into 54 verses.


1 Chronicles 1, Text, Textual witnesses
Some early manuscripts containing the text of this chapter in Hebrew are of the Masoretic Text tradition, which includes the Aleppo Codex (10th century), and Codex Leningradensis (1008).


1 Chronicles 1, Text, Textual witnesses
There is also a translation into Koine Greek known as the Septuagint, made in the last few centuries BCE. Extant ancient manuscripts of the Septuagint version include Codex Vaticanus (B; G{\displaystyle {\mathfrak {G}}}B; 4th century) and Codex Alexandrinus (A; G{\displaystyle {\mathfrak {G}}}A; 5th century).


1 Chronicles 1, From Adam to Abraham (1:1–27)
The list of names is taken exclusively from the Book of Genesis and reduced to a 'skeletal framework', with some omissions of those 'whose lines ended with their deaths', such as Cain's descendants and Abraham's brothers. It links the origin of Israel to the origin of all people – Abraham's ancestry in Adam and Noah's – and thus, within the whole human history.


1 Chronicles 1, From Adam to Abraham (1:1–27)
Verses 1–4 (from Adam to Noah) match closely to the genealogy in Genesis 5:1–32; verses 5–12 (the genealogy of Noah's sons) match that in Genesis 10:1–32; verses 13–27 (Shem's descendants until Abraham) parallel the genealogy in Genesis 11:10–26. Verse 27 contains "Abram, that is, Abraham" (the name first given by God in Genesis 17:1), representing a jump from Genesis 11 to Genesis 17.


1 Chronicles 1, From Adam to Abraham (1:1–27), Verse 1
Noah was the immediate descendant of Seth, so it is not necessary to mention Cain and Abel, or any of the other sons of Adam.


1 Chronicles 1, The descendants of Abraham (1:28–54)
This section focuses on the offsprings of Abraham (but none of his brothers'). Verses 32–40 lists Abraham's sons other than Isaac and Ishmael with the direct connection to verse 28 and has been more extensively reworked than other genealogies in this chapter, whereas verses 43–54 contain an extensive reworking of Genesis 36 to list the descendants of Edom who are Judah's neighbors with 'the closest ties through the best and worst of times'.


1 Chronicles 10
1 Chronicles 10 is the tenth chapter of the Books of Chronicles in the Hebrew Bible or the First Book of Chronicles in the Old Testament of the Christian Bible. The book is compiled from older sources by an unknown person or group, designated by modern scholars as "the Chronicler", and had the final shape established in late fifth or fourth century BCE. This chapter describes Saul's downfall and the reasons of his rejection by God. The whole chapter belongs to the section focusing on the kingship of David (1 Chronicles 9:35 to 29:30).


1 Chronicles 10, Text
This chapter was originally written in the Hebrew language. It is divided into 14 verses.


1 Chronicles 10, Text, Textual witnesses
Some early manuscripts containing the text of this chapter in Hebrew are of the Masoretic Text tradition, which includes the Aleppo Codex (10th century), and Codex Leningradensis (1008).


1 Chronicles 10, Text, Textual witnesses
There is also a translation into Koine Greek known as the Septuagint, made in the last few centuries BCE. Extant ancient manuscripts of the Septuagint version include Codex Vaticanus (B; G{\displaystyle {\mathfrak {G}}}B; 4th century), Codex Sinaiticus (S; BHK: G{\displaystyle {\mathfrak {G}}}S; 4th century), Codex Alexandrinus (A; G{\displaystyle {\mathfrak {G}}}A; 5th century) and Codex Marchalianus (Q; G{\displaystyle {\mathfrak {G}}}Q; 6th century).


1 Chronicles 10, Death of Saul and his sons (10:1–10)
This section marks the change of form in the Books of Chronicles changes from a list-based text to a more narrative description based on the historical documents such as the books of Samuel and books of Kings, and additional materials to provide information on the legitimate Davidic kingdom. It begins with Saul's downfall to theologically link the whole exposition with the Babylonian Exile at the end.


1 Chronicles 10, Death of Saul and his sons (10:1–10), Verse 6
and all his men that same day' in 1 Samuel 31:6 to underline the fact that with Saul's death, his kingdom basically ended. The episode of Ishbaal's brief rule in 2 Samuel 2–4 is considered irrelevant to the Chronicler, although the name is mentioned in the genealogy of Saul (1 Chronicles 8:33–40; 9:39–40).


1 Chronicles 10, Death of Saul and his sons (10:1–10), Verse 10
According to 1 Samuel 31:10 Saul's armour was placed in the temple of Ashtaroth (Astarte) and his body fastened to the walls of Beth-shan. The Chronicler avoids naming foreign gods, with few exception, such as Dagon.


1 Chronicles 10, Burial of Saul (10:11–14)
The narrative of Saul's burial is shorter than the account in 1 Samuel 31:11–13, omitting details such as the all-night walk of the valiant men from Jabesh Gilead to fetch Saul's body and the hanging of the corpses on the city walls of Beth-shan. The Chronicler focuses more on Saul's rejection by God, giving no less than four reasons:


1 Chronicles 11
1 Chronicles 11 is the eleventh chapter of the Books of Chronicles in the Hebrew Bible or the First Book of Chronicles in the Old Testament of the Christian Bible. The book is compiled from older sources by an unknown person or group, designated by modern scholars as "the Chronicler", and had the final shape established in late fifth or fourth century BCE. This chapter contains the accounts of David's installation as the king of Israel, the conquest of Jerusalem, and a list of David's heroes. The whole chapter belongs to the section focusing on the kingship of David (1 Chronicles 9:35 to 29:30).


1 Chronicles 11, Text
This chapter was originally written in the Hebrew language. It is divided into 47 verses.


1 Chronicles 11, Text, Textual witnesses
Some early manuscripts containing the text of this chapter in Hebrew are of the Masoretic Text tradition, which includes the Aleppo Codex (10th century), and Codex Leningradensis (1008).


1 Chronicles 11, Text, Textual witnesses
There is also a translation into Koine Greek known as the Septuagint, made in the last few centuries BCE. Extant ancient manuscripts of the Septuagint version include Codex Vaticanus (B; G{\displaystyle {\mathfrak {G}}}B; 4th century), Codex Sinaiticus (S; BHK: G{\displaystyle {\mathfrak {G}}}S; 4th century), Codex Alexandrinus (A; G{\displaystyle {\mathfrak {G}}}A; 5th century) and Codex Marchalianus (Q; G{\displaystyle {\mathfrak {G}}}Q; 6th century).


1 Chronicles 11, Structure
1 Chronicles 11 and 12 combine a 'variety of chronologically and geographically disparate lists' to establish the unity of "all Israel" (north and south), with their unanimous recognition of David's kingship. The outer framework consists of David's anointing at Hebron (1 Chronicles 11:1–3; 12:38–40) to enclose the lists of the warriors who attended the festivities (11:10–47; 12:23–38). The inner framework comprises the lists of David's forces while at Ziklag (12:1–7; 12:19–22) to enclose the warriros who joined him at “the stronghold” (12:8–18).


1 Chronicles 11, David, king of Israel (11:1–3)
The report concerning David's crowning in Hebron can be found in the books of Samuel, but the Chronicler also add some notes.


1 Chronicles 11, David, king of Israel (11:1–3), Verse 2
This is the only place in the Chronicles that Saul was stated as king.


1 Chronicles 11, David conquers Jerusalem (11:4–9)
The section is a rework of the report in 2 Samuel 5:6–10, with the removal of obscure and unclear terms and insertion of unique details, such as the role of Joab in Jerusalem's capture.


1 Chronicles 11, David conquers Jerusalem (11:4–9), Verse 6
This verse contains a play on words: whoever attack "first" (Hebrew: רִאשׁוֹן, ) will be "chief" (Hebrew: רֹאשׁ, ), Joab went up "first" and became "chief", although he was not listed among David's mighty men (1 Chronicles 11:10–(12:40).


1 Chronicles 11, David's mighty men (11:10–47)
Verses 10–41 conform with 2 Samuel 23:8–39 (with some spelling differences), whereas verses 42–47 are unique to the Chronicles. Without clear historical context, it is unclear whether the list refers to the period before or after David's accession to the throne. This passage consists of three parts (similar to the list in 2 Samuel):


1 Chronicles 11, David's mighty men (11:10–47)
The purpose of the list is to portray David as a 'divinely chosen leader' with strong support from various groups in northern and southern Israel.


1 Chronicles 11, David's mighty men (11:10–47), Verse 10
Here and in 1 Chronicles 12:23 the Chronicler underlines that David's kingdom encompasses all Israel as a fulfillment of YHWH's pledge to Israel, although this promise is not directly cited,


1 Chronicles 12
1 Chronicles 12 is the twelfth chapter of the Books of Chronicles in the Hebrew Bible or the First Book of Chronicles in the Old Testament of the Christian Bible. The book is compiled from older sources by an unknown person or group, designated by modern scholars as "the Chronicler", and had the final shape established in late fifth or fourth century BCE. This chapter contains the list of people who joined David: before his coronation (verses 1–22) and after he was made king in Hebron (verses 23–40). The whole chapter belongs to the section focusing on the kingship of David (1 Chronicles 9:35 to 29:30).


1 Chronicles 12, Text
This chapter was originally written in the Hebrew language. It is divided into 40 verses.


1 Chronicles 12, Text, Textual witnesses
Some early manuscripts containing the text of this chapter in Hebrew are of the Masoretic Text tradition, which includes the Aleppo Codex (10th century), and Codex Leningradensis (1008).


1 Chronicles 12, Text, Textual witnesses
There is also a translation into Koine Greek known as the Septuagint, made in the last few centuries BCE. Extant ancient manuscripts of the Septuagint version include Codex Vaticanus (B; G{\displaystyle {\mathfrak {G}}}B; 4th century), Codex Sinaiticus (S; BHK: G{\displaystyle {\mathfrak {G}}}S; 4th century), Codex Alexandrinus (A; G{\displaystyle {\mathfrak {G}}}A; 5th century) and Codex Marchalianus (Q; G{\displaystyle {\mathfrak {G}}}Q; 6th century).


1 Chronicles 12, Structure
1 Chronicles 11 and 12 combine a 'variety of chronologically and geographically disparate lists' to establish the unity of "all Israel" (north and south), with their unanimous recognition of David's kingship. The outer framework consists of David's anointing at Hebron (1 Chronicles 11:1–3; 12:38–40) to enclose the lists of the warriors who attended the festivities (11:10–47; 12:23–38). The inner framework comprises the lists of David's forces while at Ziklag (12:1–7; 12:19–22) to enclose the warriros who joined him at “the stronghold” (12:8–18).


1 Chronicles 12, The mighty men join David (12:1–22)
The first set of lists of the chapter contains the warriors joining David before he became king. It divides into four sections: the Benjaminites who came to David in Ziklag (verses 1–8); the Gadites who came toDavid's mountain stronghold (verses 9–16), as well as the people of Benjamin and Judah (verses 17–19); the people of Manasseh came to David in Ziklag (verses 20–22). Although only four tribes were mentioned, the structure clearly points to the conclusion in verse 22, that David got much support. The account of Manassites summarizes some incidents in 1 Samuel 28–30.


1 Chronicles 12, The mighty men join David (12:1–22), Verses 1–2
This passage suggests that some people of Benjamin defected to David while Saul was still reigning (cf. 2 Samuel 2:12–32).


1 Chronicles 12, The mighty men join David (12:1–22), Verse 18
Amasai's prophetic words, with "peace" (Hebrew: שָׁל֜וֹם, ) mentioned three times, speak of David's closeness with his supporters at the beginning of his rise to power. This relationship would be dissolved when the kingdom was divided in the time of Rehoboam's reign (2 Chronicles 10:16).


1 Chronicles 12, The mighty men join David (12:1–22), Verse 19
The battle between Saul and the Philistines was mentioned in chapter 10 and the case of David not involved in that battle was summarized from 1 Samuel 29:1–30.


1 Chronicles 12, David’s army at Hebron (12:23–40)
The subsequent list is bracketed by brief accounts of David's coronation in Hebron (verses 23, 38–40); structured as a kind of military census. David was accepted as king by all people with all their hearts (verse 38), followed by great feasts of joy, unique to the Chronicles (cf. e.g. 1 Chronicles 29:22; 2 Chronicles 30:21–6). The three-day celebration involves many meals brought by donkeys, mules, camels and oxen from three norther tribes: Issachar, Zebulun and Naphtali. The mood is describe in the word "joy" (or "rejoicing"; verse 40) which later appears in Hezekiah's Passover festival (2 Chronicles 30:23, 26) and post-exilic worship festivals (Nehemiah 8:12, 17; Nehemiah 12:43).


1 Chronicles 12, David’s army at Hebron (12:23–40), Verse 23
Saul's kingdom was passed on peacefully to David in Hebron (cf. 1 Chronicles 10:14–11:3).


1 Chronicles 13
1 Chronicles 13 is the thirteenth chapter of the Books of Chronicles in the Hebrew Bible or the First Book of Chronicles in the Old Testament of the Christian Bible. The book is compiled from older sources by an unknown person or group, designated by modern scholars as "the Chronicler", and had the final shape established in late fifth or fourth century BCE. This chapter contains the account of an unsuccessful attempt to bring the ark of the covenant to Jerusalem by David. The whole chapter belongs to the section focusing on the kingship of David (1 Chronicles 9:35 to 29:30).


1 Chronicles 13, Text
This chapter was originally written in the Hebrew language. It is divided into 14 verses.


1 Chronicles 13, Text, Textual witnesses
Some early manuscripts containing the text of this chapter in Hebrew are of the Masoretic Text tradition, which includes the Aleppo Codex (10th century), and Codex Leningradensis (1008).


1 Chronicles 13, Text, Textual witnesses
There is also a translation into Koine Greek known as the Septuagint, made in the last few centuries BCE. Extant ancient manuscripts of the Septuagint version include Codex Vaticanus (B; G{\displaystyle {\mathfrak {G}}}B; 4th century), Codex Sinaiticus (S; BHK: G{\displaystyle {\mathfrak {G}}}S; 4th century), Codex Alexandrinus (A; G{\displaystyle {\mathfrak {G}}}A; 5th century) and Codex Marchalianus (Q; G{\displaystyle {\mathfrak {G}}}Q; 6th century).


1 Chronicles 13, The ark brought from Kiriath-Jearim (13:1–4)
Verses 1–4 details the preparation by David involving all Israel in the first attempt to bring the ark into Jerusalem, more than the parallel account in 2 Samuel 6:1–2. The ark is a national symbol of Israel's religion and important for David as he had been firmly and unanimously established as the king of all Israel. Consistent with the earlier chapters, David consulted his military leaders and then the whole congregation (verse 2) to achieve two conditions for the execution of the effort: the willingness of the participants and God's acceptance of the plan. It was later revealed that the plan lacks God's acceptance as it was done without the significant collaboration of the priests and Levites.


1 Chronicles 13, The ark brought from Kiriath-Jearim (13:1–4), Verse 2
The priests and Levites lived within the territories of Israe's tribes (1 Chronicles 6:54–81.


1 Chronicles 13, Uzza and the ark (13:5–14)
Verses 5–14 follows closely to the report in 2 Samuel 6:3–11 (without verse 12). The boundaries of Israel were expanded in Chronicles from the usual phrase "from Beersheba to Dan" to be between "the Shihor river in Egypt and Lebo-hamath"; the area achieved after David's spectacular victories (2 Chronicles 7:8; cf. Joshua 13:3, 5 although the extended regions were not conquered in the time of Joshua).


1 Chronicles 13, Uzza and the ark (13:5–14), Verse 13
"The City of David": refers to a section in southern Jerusalem fortified by David and named after him (1 Chronicles 11:7), also may refer to "Mount Zion"..


1 Chronicles 14
1 Chronicles 14 is the fourteenth chapter of the Books of Chronicles in the Hebrew Bible or the First Book of Chronicles in the Old Testament of the Christian Bible. The book is compiled from older sources by an unknown person or group, designated by modern scholars as "the Chronicler", and had the final shape established in late fifth or fourth century BCE. This chapter contains the successes of David as he established himself in Jerusalem and defeated the Philistines. The whole chapter belongs to the section focusing on the kingship of David (1 Chronicles 9:35 to 29:30).


1 Chronicles 14, Text
This chapter was originally written in the Hebrew language. It is divided into 17 verses.


1 Chronicles 14, Text, Textual witnesses
Some early manuscripts containing the text of this chapter in Hebrew are of the Masoretic Text tradition, which includes the Aleppo Codex (10th century), and Codex Leningradensis (1008).


1 Chronicles 14, Text, Textual witnesses
There is also a translation into Koine Greek known as the Septuagint, made in the last few centuries BCE. Extant ancient manuscripts of the Septuagint version include Codex Vaticanus (B; G{\displaystyle {\mathfrak {G}}}B; 4th century), Codex Sinaiticus (S; BHK: G{\displaystyle {\mathfrak {G}}}S; 4th century), Codex Alexandrinus (A; G{\displaystyle {\mathfrak {G}}}A; 5th century) and Codex Marchalianus (Q; G{\displaystyle {\mathfrak {G}}}Q; 6th century).


1 Chronicles 14, David Established at Jerusalem (14:1–7)
This passage emphasizes the greatness of David's reign for the sake of Israel after the transportation of the ark (whereas in 2 Samuel 5, the account was placed after the conquest of Jerusalem). The accumulation of wives and sons is seen as a 'positive sign of stature' in the books of Chronicles (1 Chronicles 25:5; 26:4–5; 2 Chronicles 11:18–23; 13:21; 14:3–7).


1 Chronicles 14, David Defeats the Philistines (14:8–17)
The passage has similar structures as 2 Samuel 5:17–25 ('the advance of the Philistines, an enquiry to God with a positive response and the Philistines' defeat'), with a change of place-name "Geba" to "Gibeon" (verse 16) apparently to fit the perspective of Isaiah 28:21 (which refers to the battles in 2 Samuel 5:25 and Joshua 10:10). The military successes had an astonishing effect of increasing David's fame (and name) internationally, denoting divine blessings for David.


1 Chronicles 15
1 Chronicles 15 is the fifteenth chapter of the Books of Chronicles in the Hebrew Bible or the First Book of Chronicles in the Old Testament of the Christian Bible. The book is compiled from older sources by an unknown person or group, designated by modern scholars as "the Chronicler", and had the final shape established in late fifth or fourth century BCE. This chapter contains the account of successful transportation of the Ark of the Covenant to the City of David in Jerusalem. The whole chapter belongs to the section focusing on the kingship of David (1 Chronicles 9:35 to 29:30).


1 Chronicles 15, Text
This chapter was originally written in the Hebrew language. It is divided into 29 verses.