−1

In mathematics, −1 is the additive inverse of 1, that is, the number that when added to 1 gives the additive identity element, 0. It is the negative integer greater than negative two (−2) and less than 0.

Negative one bears relation to Euler's identity since eiπ = −1.

In software development, −1 is a common initial value for integers and is also used to show that a variable contains no useful information.

In programming languages, −1 can be used to index the last (or 2nd last) item of an array, depending on whether 0 or 1 represents the first item.

Negative one has some similar but slightly different properties to positive one.[1]

← −2 −1 0 →
-1 0 1 2 3 4 5 6 7 8 9
Cardinal−1, minus one, negative one
Ordinal−1st (negative first)
Arabic١
Chinese numeral负一,负弌,负壹
Bengali
Binary (byte)
S&M: 1000000012
2sC: 111111112
Hex (byte)
S&M: 0x10116
2sC: 0xFF16

Algebraic properties

Multiplying a number by −1 is equivalent to changing the sign on the number. This can be proved using the distributive law and the axiom that 1 is the multiplicative identity: for x real, we have

where we used the fact that any real x times 0 equals 0, implied by cancellation from the equation

ImaginaryUnit5
0, 1, −1, i, and −i in the complex or cartesian plane

In other words,

so (−1) · x, or −x, is the arithmetic inverse of x.

Square of −1

The square of −1, i.e. −1 multiplied by −1, equals 1. As a consequence, a product of two negative real numbers is positive.

For an algebraic proof of this result, start with the equation

The first equality follows from the above result. The second follows from the definition of −1 as additive inverse of 1: it is precisely that number that when added to 1 gives 0. Now, using the distributive law, we see that

The second equality follows from the fact that 1 is a multiplicative identity. But now adding 1 to both sides of this last equation implies

The above arguments hold in any ring, a concept of abstract algebra generalizing integers and real numbers.

Square roots of −1

Although there are no real square roots of -1, the complex number i satisfies i2 = −1, and as such can be considered as a square root of −1. The only other complex number whose square is 1 is −i.[2] In the algebra of quaternions, which contain the complex plane, the equation x2 = −1 has infinitely many solutions.

Exponentiation to negative integers

Exponentiation of a non-zero real number can be extended to negative integers. We make the definition that x−1 = 1/x, meaning that we define raising a number to the power −1 to have the same effect as taking its reciprocal. This definition is then extended to negative integers preserves the exponential law xaxb = x(a + b) for real numbers a and b.

Exponentiation to negative integers can be extended to invertible elements of a ring, by defining x−1 as the multiplicative inverse of x.

−1 that appears next to functions or matrices does not mean raising them to the power −1 but their inverse functions or inverse matrices. For example, f−1(x) is the inverse of f(x), or sin−1(x) is a notation of arcsine function.

Computer representation

Most computer systems represent negative integers using two's complement. In such systems, −1 is represented using a bit pattern of all ones. For example, an 8-bit signed integer using two's complement would represent −1 as the bitstring "11111111", or "FF" in hexadecimal (base 16). If interpreted as an unsigned integer, the same bitstring of n ones represents 2n − 1, the largest possible value that n bits can hold. For example, the 8-bit string "11111111" above represents 28 − 1 = 255.

Programming languages

In some programming languages, when used to index some data types (such as an array), then −1 can be used to identify the very last (or 2nd last) item, depending on whether 0 or 1 represents the first item. If the first item is indexed by 0, then −1 identifies the last item. If the first item is indexed by 1, then −1 identifies the second-to-last item.

See also

References

  1. ^ Mathematical analysis and applications By Jayant V. Deshpande, ISBN 1-84265-189-7
  2. ^ "Ask Dr. Math". Math Forum. Retrieved 2012-10-14.
1

1 (one, also called unit, unity, and (multiplicative) identity) is a number, numeral, and glyph. It represents a single entity, the unit of counting or measurement. For example, a line segment of unit length is a line segment of length 1. It is also the first of the infinite sequence of natural numbers, followed by 2.

2014–15 UEFA Champions League

The 2014–15 UEFA Champions League was the 60th season of Europe's premier club football tournament organised by UEFA, and the 23rd season since it was renamed from the European Champion Clubs' Cup to the UEFA Champions League.

The 2015 UEFA Champions League Final was played at the Olympiastadion in Berlin, Germany, with Spanish side Barcelona defeating Italian side Juventus by 3–1 to win their fifth title and complete their treble. Real Madrid were the title holders, but they were eliminated by Juventus in the semi-finals.

This season was the first where clubs must comply with UEFA Financial Fair Play Regulations in order to participate. Moreover, this season was the first where a club from Gibraltar competed in the tournament, after the Gibraltar Football Association was accepted as the 54th UEFA member at the UEFA Congress in May 2013. They were granted one spot in the Champions League, which was taken by Lincoln Red Imps, the champions of the 2013–14 Gibraltar Premier Division.On 17 July 2014, the UEFA emergency panel ruled that Ukrainian and Russian clubs would not be drawn against each other "until further notice" due to the political unrest between the countries. Another ruling centred in regional instability was also made where Israeli teams were prohibited from hosting any UEFA competitions due to the 2014 Israel–Gaza conflict. The rules regarding suspension due to yellow card accumulation were also changed such that all bookings expired on completion of the quarter-finals and were not carried forward to the semi-finals. Moreover, this was the first season in which vanishing spray was used.

2015–16 Premier League

The 2015–16 Premier League (known as the Barclays Premier League for sponsorship reasons) was the 24th season of the Premier League, the top English professional league for association football clubs, since its establishment in 1992. The season began on 8 August 2015, and was scheduled to conclude on 15 May 2016. However, the Manchester United vs Bournemouth fixture was postponed to 17 May 2016 on the final day due to a suspicious package inside Old Trafford.Chelsea began the season as defending champions of the 2014–15 season. Bournemouth, Watford and Norwich City entered as the three promoted teams from the 2014–15 Football League Championship.

Leicester City won the championship for the first time in their 132-year history, becoming the 24th club to become English football champions, and the sixth club to win the Premier League. Many commentators consider this to be one of the greatest sports shocks in history, especially considering that Leicester spent half of the previous season at the bottom of the table before finishing 14th.

2015–16 UEFA Champions League

The 2015–16 UEFA Champions League was the 61st season of Europe's premier club football tournament organised by UEFA, and the 24th season since it was renamed from the European Champion Clubs' Cup to the UEFA Champions League. Barcelona were the title holders, but were eliminated by Atlético Madrid in the quarter-finals.

The 2016 UEFA Champions League Final was played between Real Madrid and Atlético Madrid at the San Siro in Milan, Italy. It was the second time in the tournament's history that both finalists were from the same city, after the same clubs faced each other in the 2014 final. Real Madrid defeated Atlético Madrid 5–3 on penalties (1–1 after extra time) in the final to win a record-extending eleventh European Cup/Champions League title.

As the winners of the 2015–16 UEFA Champions League, Real Madrid qualified as the UEFA representative at the 2016 FIFA Club World Cup in Japan (their third Club World Cup appearance), and also earned the right to play against the winners of the 2015–16 UEFA Europa League, Sevilla, in the 2016 UEFA Super Cup.

2016–17 UEFA Champions League

The 2016–17 UEFA Champions League was the 62nd season of Europe's premier club football tournament organised by UEFA, and the 25th season since it was renamed from the European Champion Clubs' Cup to the UEFA Champions League.

The final was played between Juventus and Real Madrid at the Millennium Stadium in Cardiff, Wales. It was the second time that the two teams faced each other in the competition's decisive match, having previously met in the 1998 final. Real Madrid, the defending champions, beat Juventus 4–1 to win a record-extending 12th title. With this victory, Real Madrid became the first team to successfully defend their title in the Champions League era, and the first to successfully defend a European Cup since Milan in 1990.

As winners, Real Madrid qualified as the UEFA representative for the 2017 FIFA Club World Cup in the United Arab Emirates, and also earned the right to play against the winners of the 2016–17 UEFA Europa League, Manchester United, in the 2017 UEFA Super Cup.

2017–18 Premier League

The 2017–18 Premier League was the 26th season of the Premier League, the top English professional league for association football clubs, since its establishment in 1992. The season started on 11 August 2017 and concluded on 13 May 2018. Fixtures for the 2017–18 season were announced on 14 June 2017. Chelsea were the defending champions, while Newcastle United, Brighton & Hove Albion and Huddersfield Town entered as the promoted teams from the 2016–17 EFL Championship.

Manchester City won their third Premier League title, and fifth English top-flight title overall, with five games to spare. The team broke numerous Premier League records over the course of the season, including: most points (100), most wins (32), most away wins (16), most goals (106), most consecutive league wins (18), highest goal difference (+79), most passes in a game (903), fewest minutes behind in matches (153 minutes) and biggest winning points margin (19). All three promoted clubs avoided relegation for the first time since the 2011–12 campaign, and for only the third time in Premier League history.

2017–18 UEFA Champions League

The 2017–18 UEFA Champions League was the 63rd season of Europe's premier club football tournament organised by UEFA, the 26th season since it was renamed from the European Champion Clubs' Cup to the UEFA Champions League.

The final was played between Real Madrid and Liverpool at the NSC Olimpiyskiy Stadium in Kiev, Ukraine. Real Madrid, the defending champions, beat Liverpool 3–1 to win a record-extending 13th title and their third title in a row.

As winners, Real Madrid qualified as the UEFA representative for the 2018 FIFA Club World Cup in the United Arab Emirates, and also earned the right to play against the winners of the 2017–18 UEFA Europa League, Atlético Madrid, in the 2018 UEFA Super Cup. Moreover, they would also have been automatically qualified for the 2018–19 UEFA Champions League group stage, but since they had already qualified through their league performance, the berth reserved was given to the champions of the 2017–18 Czech First League, the 11th-ranked association according to the 2018–19 access list.

2018–19 Premier League

The 2018–19 Premier League was the 27th season of the Premier League, the top English professional league for association football clubs, since its establishment in 1992. The season started on 10 August 2018 and concluded on 12 May 2019. Fixtures for the 2018–19 season were announced on 14 June 2018. The league was contested by the top 17 teams from the 2017–18 season as well as Wolverhampton Wanderers, Cardiff City and Fulham, who joined as the promoted clubs from the 2017–18 EFL Championship. They replaced West Bromwich Albion, Swansea City and Stoke City who were relegated to the 2018–19 EFL Championship.Defending champions Manchester City won their fourth Premier League title, and sixth English top-flight title overall. They won their last 14 league games and retained the league title on the final day of the season, finishing on 98 points. Liverpool finished runners-up with 97 points – the highest total in English top-flight history for a second-placed team. This was Manchester City’s second step in becoming the first English team to complete a domestic treble, as they earlier won the 2018–19 EFL Cup and would later win the 2018–19 FA Cup.

2018–19 UEFA Champions League

The 2018–19 UEFA Champions League is the 64th season of Europe's premier club football tournament organised by UEFA, and the 27th season since it was renamed from the European Champion Clubs' Cup to the UEFA Champions League.

The final will be played at the Wanda Metropolitano in Madrid, Spain, between Tottenham Hotspur and Liverpool. It will be the second all-English final after the 2008 final, which was contested between Manchester United and Chelsea in Moscow. The winners of the 2018–19 UEFA Champions League will earn the right to play against the winners of the 2018–19 UEFA Europa League in the 2019 UEFA Super Cup. They will also automatically qualify for the 2019–20 UEFA Champions League group stage. As both finalists have already qualified for the group stage through their league performance, the berth reserved will be given to the champions of the 2018–19 Austrian Bundesliga, the 11th-ranked association according to next season's access list.For the first time, the video assistant referee (VAR) system was used in the competition from the round of 16 onward.Real Madrid were the defending champions, having won the title for three successive seasons in 2015–16, 2016–17 and 2017–18. They were eliminated by Ajax in the round of 16.

2018–19 UEFA Europa League

The 2018–19 UEFA Europa League is the 48th season of Europe's secondary club football tournament organised by UEFA, and the 10th season since it was renamed from the UEFA Cup to the UEFA Europa League.

The final will be played at the Olympic Stadium in Baku, Azerbaijan, between English sides Arsenal and Chelsea — which it marked for the first time ever in UEFA history that both the finals of UEFA Champions League and UEFA Europa League this season will be all played by teams from one country. The winners of the 2018–19 UEFA Europa League will earn the right to play against the winners of the 2018–19 UEFA Champions League in the 2019 UEFA Super Cup. They will also automatically qualify for the 2019–20 UEFA Champions League group stage, and if they have already qualified through their league performance, the berth reserved will be given to the third-placed team of the 2018–19 Ligue 1, the 5th-ranked association according to next season's access list.For the first time, the video assistant referee (VAR) system will be used in the competition, where it will be implemented in the final.As the title holders of Europa League, Atlético Madrid qualified for the 2018–19 UEFA Champions League, although they had already qualified before the final through their league performance. They were unable to defend their title as they advanced to the Champions League knockout stage.

EFL League One

The English Football League One (often referred to as League One for short or Sky Bet League One for sponsorship reasons) is the second-highest division of the English Football League and the third tier overall in the entire English football league system.

League One was introduced for the 2004–05 season. It was previously known briefly as the Football League Second Division and for much longer, prior to the advent of the Premier League, as the Football League Third Division.

At present (2018–19 season), Walsall hold the longest tenure in League One, last being out of the division in the 2006–07 season when they were promoted from League Two. There are currently seven former Premier League clubs competing in League One, namely Barnsley (1997–98), Blackpool (2010–11), Bradford City (2000–01), Charlton Athletic (2006–07), Coventry City (2000–01), Portsmouth (2009–10) and Sunderland (2016–17).

Fibonacci number

In mathematics, the Fibonacci numbers, commonly denoted Fn form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,

and

for n > 1.

One has F2 = 1. In some books, and particularly in old ones, F0, the "0" is omitted, and the Fibonacci sequence starts with F1 = F2 = 1. The beginning of the sequence is thus:

Fibonacci numbers are strongly related to the golden ratio: Binet's formula expresses the nth Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases.

Fibonacci numbers are named after Italian mathematician Leonardo of Pisa, later known as Fibonacci. They appear to have first arisen as early as 200 BC in work by Pingala on enumerating possible patterns of poetry formed from syllables of two lengths. In his 1202 book Liber Abaci, Fibonacci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics.

Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. They also appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, an uncurling fern and the arrangement of a pine cone's bracts.

Fibonacci numbers are also closely related to Lucas numbers in that they form a complementary pair of Lucas sequences and . Lucas numbers are also intimately connected with the golden ratio.

Hypertext Transfer Protocol

The Hypertext Transfer Protocol (HTTP) is an application protocol for distributed, collaborative, hypermedia information systems. HTTP is the foundation of data communication for the World Wide Web, where hypertext documents include hyperlinks to other resources that the user can easily access, for example by a mouse click or by tapping the screen in a web browser. HTTP was developed to facilitate hypertext and the World Wide Web.

Development of HTTP was initiated by Tim Berners-Lee at CERN in 1989. Development of HTTP standards was coordinated by the Internet Engineering Task Force (IETF) and the World Wide Web Consortium (W3C), culminating in the publication of a series of Requests for Comments (RFCs). The first definition of HTTP/1.1, the version of HTTP in common use, occurred in RFC 2068 in 1997, although this was made obsolete by RFC 2616 in 1999 and then again by the RFC 7230 family of RFCs in 2014.

A later version, the successor HTTP/2, was standardized in 2015 (and HTTP/3 is its proposed successor (Internet Draft), that builds on HTTP/2), and is now supported by major web servers and browsers over Transport Layer Security (TLS) using Application-Layer Protocol Negotiation (ALPN) extension where TLS 1.2 or newer is required.

Ligue 1

Ligue 1, also called Ligue 1 Conforama for sponsorship reasons with Conforama, is a French professional league for men's association football clubs. At the top of the French football league system, it is the country's primary football competition. Administrated by the Ligue de Football Professionnel, Ligue 1 is contested by 20 clubs and operates on a system of promotion and relegation with Ligue 2.

Seasons run from August to May. Clubs play two matches against each of the other teams in the league – once home and once away – totalling to 38 matches over the course of the season. Most games are played on Saturdays and Sundays, with a few games played during weekday evenings. Play is regularly suspended the last weekend before Christmas for two weeks before returning in the second week of January. Ligue 1 is one of the top national leagues, currently ranked fifth in Europe behind Spain's La Liga, England's Premier League, Italy's Serie A and Germany's Bundesliga.Ligue 1 was inaugurated on 11 September 1932 under the name National before switching to Division 1 after a year of existence. The name lasted until 2002 before switching to its current name. AS Saint-Étienne is the most successful club with ten league titles in France while Olympique Lyonnais is the club that has won the most consecutive titles (seven between 2002 and 2008). With the presence of 69 seasons in Ligue 1, Olympique de Marseille hold the record for most seasons among the elite, while Paris Saint-Germain hold the league record for longevity with 45 consecutive seasons (from 1974 until at least 2019). The current champions are Paris Saint-Germain, who won their eighth title in the 2018–19 season. The league has been won on multiple occasions by foreign-based club AS Monaco, which makes the league a cross-border competition.

List of NBA champions

The National Basketball Association (NBA) (formerly Basketball Association of America (BAA) from 1946 to 1949) Finals is the championship series for the NBA and the conclusion of the NBA's postseason. All Finals have been played in a best-of-seven format, and are contested between the winners of the Eastern Conference and the Western Conference (formerly Divisions before 1970), except in 1950 when the Eastern Division champion faced the winner between the Western and Central Division champions. Prior to 1949, the playoffs were a three-stage tournament where the two semifinal winners played each other in the finals. The winning team of the series receives the Larry O'Brien Championship Trophy.

The current home-and-away format in the NBA Finals is in a 2–2–1–1–1 format (the team with the better regular season record plays on their home court in Games 1, 2, 5 and 7), which has been used in 1947–1948, 1950–1952, 1957–1970, 1972–1974, 1976–1977, 1979–1984 and 2014–present. It was previously in a 2–3–2 format (the team with the better regular season record plays on their home court in Games 1, 2, 6 and 7) during 1949, 1953–1955 and 1985–2013, in a 1–1–1–1–1–1–1 format during 1956 and 1971 and in a 1–2–2–1–1 format during 1975 and 1978.The Eastern Conference/Division leads the Western Conference/Division in series won (38–33). The defunct Central Division, in existence during the 1949–50 NBA season when the NBA was divided into three divisions and different from the current Central Division created in 1970 when the then existing Eastern Division was upgraded as a conference, won one championship. The Boston Celtics and the Minneapolis/Los Angeles Lakers alone own almost half of the titles, having won a combined 33 of 72 championships. As of 2018, the defending champions are the Golden State Warriors.

List of Stanley Cup champions

The Stanley Cup is a trophy awarded annually to the playoff champion club of the National Hockey League (NHL) ice hockey league. It was donated by the Governor General of Canada Lord Stanley of Preston in 1892, and is the oldest professional sports trophy in North America. Inscribed the Dominion Hockey Challenge Cup, the trophy was first awarded to Canada's amateur ice hockey clubs who won the trophy as the result of challenge games and league play. Professional clubs came to dominate the competition in the early years of the twentieth century, and in 1913 the two major professional ice hockey organizations, the National Hockey Association (NHA) (forerunner of the NHL) and the Pacific Coast Hockey Association (PCHA), reached a gentlemen's agreement in which their respective champions would face each other in an annual series for the Stanley Cup. After a series of league mergers and folds, it became the de facto championship trophy of the NHL in 1926, though it was nominally still subject to external challenge. After 1947, the Cup became the de jure NHL championship prize.

From 1914 to the end of the 2018 season, the trophy has been won 100 times. 24 teams have won the cup, 19 of which are still active in the NHL. Prior to that, the challenge cup was held by nine teams. The Montreal Canadiens have won the Stanley Cup 24 times and made the finals an additional ten times. There were two years when the Stanley Cup was not awarded: 1919, because of the Spanish flu epidemic, and 2005, because of the NHL lockout.

List of countries and dependencies by population

This is a list of countries and dependent territories by population. It includes sovereign states, inhabited dependent territories and, in some cases, constituent countries of sovereign states, with inclusion within the list being primarily based on the ISO standard ISO 3166-1. For instance, the United Kingdom is considered as a single entity, while the constituent countries of the Kingdom of the Netherlands are considered separately. In addition, this list includes certain states with limited recognition not found in ISO 3166-1.

Also given in percent is each country's population compared with the population of the world, which the United Nations estimates at 7.71 billion as of today.

Normal distribution

In probability theory, the normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.

The normal distribution is useful because of the central limit theorem. In its most general form, under some conditions (which include finite variance), it states that averages of samples of observations of random variables independently drawn from independent distributions converge in distribution to the normal, that is, they become normally distributed when the number of observations is sufficiently large. Physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have distributions that are nearly normal. Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed.

The normal distribution is sometimes informally called the bell curve. However, many other distributions are bell-shaped (such as the Cauchy, Student's t-, and logistic distributions).

The probability density of the normal distribution is

where

Standard deviation

In statistics, the standard deviation (SD, also represented by the lower case Greek letter sigma σ or the Latin letter s) is a measure that is used to quantify the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the data points tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.

The standard deviation of a random variable, statistical population, data set, or probability distribution is the square root of its variance. It is algebraically simpler, though in practice less robust, than the average absolute deviation.

A useful property of the standard deviation is that, unlike the variance, it is expressed in the same units as the data.

In addition to expressing the variability of a population, the standard deviation is commonly used to measure confidence in statistical conclusions. For example, the margin of error in polling data is determined by calculating the expected standard deviation in the results if the same poll were to be conducted multiple times. This derivation of a standard deviation is often called the "standard error" of the estimate or "standard error of the mean" when referring to a mean. It is computed as the standard deviation of all the means that would be computed from that population if an infinite number of samples were drawn and a mean for each sample were computed.

It is very important to note that the standard deviation of a population and the standard error of a statistic derived from that population (such as the mean) are quite different but related (related by the inverse of the square root of the number of observations). The reported margin of error of a poll is computed from the standard error of the mean (or alternatively from the product of the standard deviation of the population and the inverse of the square root of the sample size, which is the same thing) and is typically about twice the standard deviation—the half-width of a 95 percent confidence interval.

In science, many researchers report the standard deviation of experimental data, and only effects that fall much farther than two standard deviations away from what would have been expected are considered statistically significant—normal random error or variation in the measurements is in this way distinguished from likely genuine effects or associations. The standard deviation is also important in finance, where the standard deviation on the rate of return on an investment is a measure of the volatility of the investment.

When only a sample of data from a population is available, the term standard deviation of the sample or sample standard deviation can refer to either the above-mentioned quantity as applied to those data or to a modified quantity that is an unbiased estimate of the population standard deviation (the standard deviation of the entire population).

USB

Universal Serial Bus (USB) is an industry standard that establishes specifications for cables and connectors and protocols for connection, communication and power supply between computers, peripheral devices and other computers. Released in 1996, the USB standard is currently maintained by the USB Implementers Forum (USB IF). There have been three generations of USB specifications: USB 1.x, USB 2.0 and USB 3.x; the fourth called USB4 is scheduled to be published in the middle of 2019.

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