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2018–19 NBA season
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2018–19 NBA season
League
National Basketball Association
Sport
Basketball
Duration
October 16, 2018 – April 10, 2019
April 13 – May 2019 (Playoffs)
May – June 2019 (Finals)
Number of games
82
Number of teams
30
TV partner(s)
ABC, TNT, ESPN, NBA TV
Draft
Top draft pick
Deandre Ayton
Picked by
Phoenix Suns
Regular season
Playoffs
Finals
NBA seasons
← 2017–18
2019–20 →
The 2018–19 NBA season is the 73rd season of the National Basketball Association (NBA). The regular season began on October 16, 2018, and will end on April 10, 2019. The playoffs will begin April 13, 2019,[1] with the NBA Finals concluding in June. The 2019 NBA All-Star Game will be played on February 17, 2019, at the Spectrum Center in Charlotte, North Carolina.
Contents
1 Transactions
1.1 Retirement
1.2 Free agency
1.3 Coaching changes
1.3.1 Off-season
1.3.2 In-season
2 Preseason
2.1 International games
3 Regular season
3.1 By conference
3.2 International games
4 Statistics
4.1 Individual statistic leaders
4.2 Individual game highs
4.3 Team statistic leaders
5 Awards
5.1 Players of the Week
5.2 Players of the Month
5.3 Rookies of the Month
5.4 Coaches of the Month
6 Arenas
7 Media
8 Uniforms
8.1 Uniform sponsorships
9 Notable occurrences
10 See also
11 References
12 External links
Transactions
Main article: List of 2018–19 NBA season transactions
Retirement
On May 10, 2018, Nick Collison announced his retirement from the NBA. Collison played all his 14 seasons with the Seattle SuperSonics/Oklahoma City Thunder franchise.[2]
On May 25, 2018, after playing 13 seasons in the NBA for seven teams, Mo Williams announced his retirement from the NBA to take the assistant head coaching job at Cal State Northridge.[3]
On July 17, 2018, after playing 9 seasons in the NBA for four teams, Roy Hibbert announced his retirement from the NBA.[4]
On August 27, 2018, Manu Ginóbili announced his retirement from the NBA. Ginóbili played all his 16 seasons with the San Antonio Spurs franchise.[5]
On August 30, 2018, after playing 15 seasons in the NBA for four teams, David West announced his retirement from the NBA.[6][7]
On September 6, 2018, Boris Diaw announced his retirement from basketball. Diaw spent 14 years in the NBA and was an NBA champion with the San Antonio Spurs in 2014.[8]
On October 13, 2018, after playing 17 seasons in the NBA for eight teams, Richard Jefferson announced his retirement from the NBA.[9]
Free agency
Free agency negotiations began on July 1. Players began signing on July 6 after the July moratorium ended.
Coaching changes
Coaching changes
Off-season
Team
2017–18 season
2018–19 season
Atlanta Hawks Mike Budenholzer Lloyd Pierce
Charlotte Hornets Steve Clifford James Borrego
Detroit Pistons Stan Van Gundy Dwane Casey
Milwaukee Bucks Joe Prunty (interim) Mike Budenholzer
New York Knicks Jeff Hornacek David Fizdale
Orlando Magic Frank Vogel Steve Clifford
Phoenix Suns Jay Triano (interim) Igor Kokoškov
Toronto Raptors Dwane Casey Nick Nurse
In-season
Team
Outgoing coach
Ingoing coach
Cleveland Cavaliers Tyronn Lue Larry Drew (interim)
Off-season
On April 12, 2018, the New York Knicks fired head coach Jeff Hornacek after the team missed the playoffs.[10] In addition, associate head coach Kurt Rambis was also fired.
On April 12, 2018, the Orlando Magic fired head coach Frank Vogel after the team missed the playoffs.[11]
On April 13, 2018, the Charlotte Hornets fired head coach Steve Clifford after the team missed the playoffs.[12]
On April 25, 2018, the Atlanta Hawks and Mike Budenholzer had mutually agreed to part ways.[13]
On May 1, 2018, the Memphis Grizzlies announced that J. B. Bickerstaff would become the new permanent head coach of the team.[14]
On May 2, 2018, the Phoenix Suns hired Igor Kokoškov as head coach.[15]
On May 7, 2018, the New York Knicks hired David Fizdale as head coach.[16]
On May 7, 2018, the Detroit Pistons fired head coach Stan Van Gundy after the team missed the playoffs for the second consecutive season.[17]
On May 10, 2018, the Charlotte Hornets hired James Borrego as head coach.[18]
On May 11, 2018, the Toronto Raptors fired Dwane Casey after the team was swept by the Cleveland Cavaliers for the second consecutive time in the postseason.[19]
On May 11, 2018, the Atlanta Hawks hired Lloyd Pierce as head coach.[20]
On May 17, 2018, the Milwaukee Bucks hired Mike Budenholzer as head coach.[21]
On May 30, 2018, the Orlando Magic hired Steve Clifford as head coach.[22]
On June 11, 2018, the Detroit Pistons hired Dwane Casey as head coach.[23]
On June 14, 2018, the Toronto Raptors promoted assistant coach Nick Nurse as their head coach.[24]
In-season
On October 28, 2018, the Cleveland Cavaliers fired head coach Tyronn Lue after a 0-6 start to the season and named Larry Drew interim head coach.[25][26]
Preseason
The preseason began on September 28 and ended on October 12.[1]
International games
The Toronto Raptors played two preseason games in Canada first beginning with the Portland Trail Blazers in Vancouver at the Rogers Arena on September 29, and second with the Brooklyn Nets in Montreal at the Bell Centre on October 10.[27]
The Dallas Mavericks and the Philadelphia 76ers played two preseason games in China, in Shanghai on October 5 and in Shenzhen on October 8.[28]
Regular season
The regular season began on October 16, 2018 and will end on April 10, 2019. The entire schedule was released on August 10, 2018.[29]
Eastern Conference
Atlantic Division
W
L
PCT
GB
Home
Road
Div
GP
Toronto Raptors 6 1 .857 0.0 5–0 1–1 1–0 7
Boston Celtics 4 2 .667 1.5 1–1 3–1 2–1 6
Philadelphia 76ers 4 3 .571 2.0 4–0 0–3 0–1 7
Brooklyn Nets 2 5 .286 4.0 1–1 1–4 1–1 7
New York Knicks 2 5 .286 4.0 2–2 0–3 1–2 7
Central Division
W
L
PCT
GB
Home
Road
Div
GP
Milwaukee Bucks 7 0 1.000 0.0 5–0 2–0 1–0 7
Detroit Pistons 4 1 .800 2.0 3–1 1–0 2–0 5
Indiana Pacers 4 3 .571 3.0 2–1 2–2 1–1 7
Chicago Bulls 2 5 .286 5.0 1–2 1–3 0–1 7
Cleveland Cavaliers 0 6 .000 6.5 0–3 0–3 0–2 6
Southeast Division
W
L
PCT
GB
Home
Road
Div
GP
Miami Heat 3 3 .500 0.0 2–2 1–1 1–2 6
Charlotte Hornets 3 4 .429 0.5 1–1 2–3 2–0 7
Orlando Magic 2 4 .333 1.0 1–2 1–2 1–1 6
Atlanta Hawks 2 4 .333 1.0 1–1 1–3 0–0 6
Washington Wizards 1 5 .167 2.0 0–2 1–3 0–1 6
Western Conference
Northwest Division
W
L
PCT
GB
Home
Road
Div
GP
Denver Nuggets 5 1 .833 0.0 4–0 1–1 0–0 6
Utah Jazz 4 2 .667 1.0 0–2 4–0 0–0 6
Portland Trail Blazers 4 2 .667 1.0 2–1 2–1 0–0 6
Minnesota Timberwolves 3 4 .429 2.5 3–1 0–3 0–0 7
Oklahoma City Thunder 1 4 .200 3.5 1–2 0–2 0–0 5
Pacific Division
W
L
PCT
GB
Home
Road
Div
GP
Golden State Warriors 7 1 .875 0.0 3–0 4–1 1–0 8
Los Angeles Clippers 4 2 .667 2.0 3–1 1–1 0–0 6
Sacramento Kings 4 3 .571 2.5 2–1 2–2 0–0 7
Los Angeles Lakers 2 5 .286 4.5 1–2 1–3 1–0 7
Phoenix Suns 1 5 .167 5.0 1–1 0–4 0–2 6
Southwest Division
W
L
PCT
GB
Home
Road
Div
GP
New Orleans Pelicans 4 2 .667 0.0 3–1 1–1 1–0 6
San Antonio Spurs 4 2 .667 0.0 3–1 1–1 1–0 6
Memphis Grizzlies 3 2 .600 0.5 2–0 1–2 0–0 5
Dallas Mavericks 2 5 .286 2.5 2–1 0–4 0–1 7
Houston Rockets 1 4 .200 2.5 0–3 1–1 0–1 5
By conference
Eastern Conference
#
Team
W
L
PCT
GB
GP
1
Milwaukee Bucks * 7 0 1.000 – 7
2
Toronto Raptors * 6 1 .857 1.0 7
3
Detroit Pistons 4 1 .800 2.0 5
4
Boston Celtics 4 2 .667 2.5 6
5
Indiana Pacers 4 3 .571 3.0 7
6
Philadelphia 76ers 4 3 .571 3.0 7
7
Miami Heat * 3 3 .500 3.5 6
8
Charlotte Hornets 3 4 .429 4.0 7
9
Orlando Magic 2 4 .333 4.5 6
10
Atlanta Hawks 2 4 .333 4.5 6
11
Brooklyn Nets 2 5 .286 5.0 7
12
New York Knicks 2 5 .286 5.0 7
13
Chicago Bulls 2 5 .286 5.0 7
14
Washington Wizards 1 5 .167 5.5 6
15
Cleveland Cavaliers 0 6 .000 6.5 6
Western Conference
#
Team
W
L
PCT
GB
GP
1
Golden State Warriors * 7 1 .875 – 8
2
Denver Nuggets * 5 1 .833 1.0 6
3
New Orleans Pelicans * 4 2 .667 2.0 6
4
Utah Jazz 4 2 .667 2.0 6
5
Los Angeles Clippers 4 2 .667 2.0 6
6
Portland Trail Blazers 4 2 .667 2.0 6
7
San Antonio Spurs 4 2 .667 2.0 6
8
Memphis Grizzlies 3 2 .600 2.5 5
9
Sacramento Kings 4 3 .571 2.5 7
10
Minnesota Timberwolves 3 4 .429 3.5 7
11
Dallas Mavericks 2 5 .286 4.5 7
12
Los Angeles Lakers 2 5 .286 4.5 7
13
Oklahoma City Thunder 1 4 .200 4.5 5
14
Houston Rockets 1 4 .200 4.5 5
15
Phoenix Suns 1 5 .167 5.0 6
Notes
* – Division leader
International games
On June 20, 2018, the NBA announced that the Washington Wizards would play the New York Knicks at the O2 Arena in London, United Kingdom on January 17, 2019.[30]
On August 7, 2018, the NBA announced that the Orlando Magic would play two games at Mexico City Arena in Mexico City. They will play the Chicago Bulls on December 13, 2018, and they will play the Utah Jazz on December 15, 2018.[31]
Statistics
Individual statistic leaders
Category
Player
Team
Statistic
Points per game Stephen Curry Golden State Warriors 32.5
Rebounds per game Hassan Whiteside Miami Heat 16.2
Assists per game Kyle Lowry Toronto Raptors 11.0
Steals per game Jimmy Butler Minnesota Timberwolves 3.0
Blocks per game Hassan Whiteside Miami Heat 3.2
Turnovers per game Devin Booker Phoenix Suns 5.5
Fouls per game Andre Drummond Detroit Pistons 4.6
Minutes per game DeMar DeRozan San Antonio Spurs 38.8
FG% Rudy Gobert Utah Jazz 75.0%
FT% 8 players 100.0%
3FG% Nikola Vučević Orlando Magic 77.8%
Efficiency per game Stephen Curry Golden State Warriors 32.7
Double-doubles Joel Embiid Philadelphia 76ers 5
Triple-doubles Ben Simmons Philadelphia 76ers 2
Individual game highs
Category
Player
Team
Statistic
Points Klay Thompson Golden State Warriors 52
Rebounds Hassan Whiteside Miami Heat 24
Assists Kyle Lowry Toronto Raptors 15
Steals P. J. Tucker Houston Rockets 7
Blocks Michael Kidd-Gilchrist Charlotte Hornets 7
Three pointers Klay Thompson Golden State Warriors 14
Team statistic leaders
Category
Team
Statistic
Points per game New Orleans Pelicans 124.8
Rebounds per game Milwaukee Bucks 55.3
Assists per game Golden State Warriors 29.2
Steals per game Los Angeles Lakers 9.8
Blocks per game Memphis Grizzlies 7.8
Turnovers per game Atlanta Hawks 18.4
FG% Golden State Warriors 52.3%
FT% Portland Trail Blazers 87.2%
3FG% Indiana Pacers 44.6%
+/− Milwaukee Bucks +16.0
Awards
Players of the Week
The following players were named the Eastern and Western Conference Players of the Week.
Week
Eastern Conference
Western Conference
Ref
October 16–21 Kemba Walker (Charlotte Hornets) (1/1) Nikola Jokić (Denver Nuggets) (1/1) [32][33]
October 22–28 Giannis Antetokounmpo (Milwaukee Bucks) (1/1) Stephen Curry (Golden State Warriors) (1/1) [34][35]
October 29 – November 3
Players of the Month
The following players were named the Eastern and Western Conference Players of the Month.
Month
Eastern Conference
Western Conference
Ref
October/November
Rookies of the Month
The following players were named the Eastern and Western Conference Rookies of the Month.
Month
Eastern Conference
Western Conference
Ref
October/November
Coaches of the Month
The following coaches were named the Eastern and Western Conference Coaches of the Month.
Month
Eastern Conference
Western Conference
Ref
October/November
Arenas
The Atlanta Hawks' home arena formerly known as Phillips Arena was renamed State Farm Arena on August 29, 2018, coinciding with a $192.5 million renovation to the arena.[36]
This is the Golden State Warriors' final season at Oracle Arena in Oakland, before moving to the new Chase Center in San Francisco. The Warriors will play their final regular season game there on April 7, 2019, against the Los Angeles Clippers.[37]
This is the Milwaukee Bucks' first season at the new Fiserv Forum after playing at the Bradley Center from 1988 to 2018. The Bucks played their first game there on October 3, 2018, in a preseason game against the Chicago Bulls, the first regular season game there was played on October 19, 2018, against the Indiana Pacers.[38][39]
The Toronto Raptors' home arena formerly known as Air Canada Centre was renamed Scotiabank Arena on July 1, 2018.[40][41]
Media
This will be the third year of the current nine-year contracts with ABC, ESPN, TNT, and NBA TV.[citation needed]
Uniforms
The NBA will allow the players to wear any color of shoes they wish; previously, they were restricted to white, black, or the colors of their teams.[42]
The "City Edition" jerseys will first be worn on November 9, 2018, while a sixth jersey apparel will first be worn on December 25, 2018.[43]
On June 6, 2018, the Denver Nuggets unveiled their new logos and uniforms to enter the 2018–19 season.[44][45]
On July 25, 2018, the Charlotte Hornets unveiled their new "Classic Edition" uniform to commemorate their 30th anniversary.[46]
On July 31, 2018, the Los Angeles Lakers unveiled their new uniforms to enter the 2018–19 season.[47]
On August 2, 2018, the Memphis Grizzlies unveiled their new logos and uniforms to enter the 2018–19 season.[48]
On August 8, 2018, the Orlando Magic unveiled their first "Classic Edition" uniform to commemorate their 30th anniversary.[49]
On August 16, 2018, the Atlanta Hawks unveiled their new "Classic Edition" uniform to commemorate their 50th anniversary since moving from St. Louis to Atlanta.[50][51]
On August 30, 2018, the Minnesota Timberwolves unveiled their first "Classic Edition" uniform to commemorate their 30th anniversary.[52]
On September 20, 2018, the Utah Jazz unveiled their first "Classic Edition" uniform to commemorate their 40th anniversary.[53]
On October 10, 2018, the Golden State Warriors unveiled their new "Classic Edition" uniform.[54][better source needed]
Uniform sponsorships
In April 2016, the NBA announced that teams would be permitted to sign a uniform sponsorship contract for the 2017–18 season.[55] Prior and during the 2017–18 season 21 teams signed a uniform sponsorship contract. Before the 2018–19 season six more teams signed such contracts:
Chicago Bulls – Zenni Optical[56]
Houston Rockets – ROKiT Phones[57]
Memphis Grizzlies – FedEx[58]
Phoenix Suns – PayPal[59]
Portland Trail Blazers – Performance Health[60]
San Antonio Spurs – Frost Bank[61]
As of October 16, 2018, three teams have not signed a uniform sponsorship contract: Indiana Pacers, Oklahoma City Thunder, and Washington Wizards.
Notable occurrences
On September 21, 2018, the NBA approved of three different rule changes affecting gameplay, starting with this season onward. These changes include shortening the game clock from the typical 24 seconds to 14 seconds during offensive rebounds, simplifying the clear path foul rule, and expanding the definition of a "hostile act" to invoke instant replays on certain events more easily.[62]
See also
National Basketball Association portal
References
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Jump up ^ "Timberwolves Unveil Classics Edition Uniforms and 30th Season Logo Design". NBA.com. August 30, 2018. Retrieved August 30, 2018.
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Real number
From Wikipedia, the free encyclopedia
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For the real numbers used in descriptive set theory, see Baire space (set theory). For the computing datatype, see Floating-point number.
number representing a continuous quantity
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A symbol of the set of real numbers (ℝ)
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2 (1.41421356..., the square root of 2, an irrational algebraic number). Included within the irrationals are the transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more.
Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one. The real line can be thought of as a part of the complex plane, and complex numbers include real numbers.
Real numbers can be thought of as points on an infinitely long number line
These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. The discovery of a suitably rigorous definition of the real numbers – indeed, the realization that a better definition was needed – was one of the most important developments of 19th-century mathematics. The current standard axiomatic definition is that real numbers form the unique Dedekind-complete ordered field (R ; + ; · ; <), up to an isomorphism,[a] whereas popular constructive definitions of real numbers include declaring them as equivalence classes of Cauchy sequences of rational numbers, Dedekind cuts, or infinite decimal representations, together with precise interpretations for the arithmetic operations and the order relation. All these definitions satisfy the axiomatic definition and are thus equivalent.
The reals are uncountable; that is: while both the set of all natural numbers and the set of all real numbers are infinite sets, there can be no one-to-one function from the real numbers to the natural numbers: the cardinality of the set of all real numbers (denoted c {\displaystyle {\mathfrak {c}}} and called cardinality of the continuum) is strictly greater than the cardinality of the set of all natural numbers (denoted ℵ 0 {\displaystyle \aleph _{0}} 'aleph-naught'). The statement that there is no subset of the reals with cardinality strictly greater than ℵ 0 {\displaystyle \aleph _{0}} and strictly smaller than c {\displaystyle {\mathfrak {c}}} is known as the continuum hypothesis (CH). It is known to be neither provable nor refutable using the axioms of Zermelo–Fraenkel set theory including the axiom of choice (ZFC), the standard foundation of modern mathematics, in the sense that some models of ZFC satisfy CH, while others violate it.
Contents
1 History
2 Definition
2.1 Axiomatic approach
2.2 Construction from the rational numbers
3 Properties
3.1 Basic properties
3.2 Completeness
3.3 "The complete ordered field"
3.4 Advanced properties
4 Applications and connections to other areas
4.1 Real numbers and logic
4.2 In physics
4.3 In computation
4.4 "Reals" in set theory
5 Vocabulary and notation
6 Generalizations and extensions
7 See also
8 Notes
9 Citations
10 References
11 External links
History[edit]
Real numbers (ℝ) include the rational numbers (ℚ), which include the integers (ℤ), which include the natural numbers (ℕ)
Simple fractions were used by the Egyptians around 1000 BC; the Vedic "Sulba Sutras" ("The rules of chords") in, c. 600 BC, include what may be the first "use" of irrational numbers. The concept of irrationality was implicitly accepted by early Indian mathematicians since Manava (c. 750–690 BC), who were aware that the square roots of certain numbers such as 2 and 61 could not be exactly determined.[1] Around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2.
The Middle Ages brought the acceptance of zero, negative, integral, and fractional numbers, first by Indian and Chinese mathematicians, and then by Arabic mathematicians, who were also the first to treat irrational numbers as algebraic objects,[2] which was made possible by the development of algebra. Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers.[3] The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850–930) was the first to accept irrational numbers as solutions to quadratic equations or as coefficients in an equation, often in the form of square roots, cube roots and fourth roots.[4]
In the 16th century, Simon Stevin created the basis for modern decimal notation, and insisted that there is no difference between rational and irrational numbers in this regard.
In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" ones.
In the 18th and 19th centuries, there was much work on irrational and transcendental numbers. Johann Heinrich Lambert (1761) gave the first flawed proof that π cannot be rational; Adrien-Marie Legendre (1794) completed the proof,[5] and showed that π is not the square root of a rational number.[6] Paolo Ruffini (1799) and Niels Henrik Abel (1842) both constructed proofs of the Abel–Ruffini theorem: that the general quintic or higher equations cannot be solved by a general formula involving only arithmetical operations and roots.
Évariste Galois (1832) developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Joseph Liouville (1840) showed that neither e nor e2 can be a root of an integer quadratic equation, and then established the existence of transcendental numbers; Georg Cantor (1873) extended and greatly simplified this proof.[7] Charles Hermite (1873) first proved that e is transcendental, and Ferdinand von Lindemann (1882), showed that π is transcendental. Lindemann's proof was much simplified by Weierstrass (1885), still further by David Hilbert (1893), and has finally been made elementary by Adolf Hurwitz[8] and Paul Gordan.[9]
The development of calculus in the 18th century used the entire set of real numbers without having defined them cleanly. The first rigorous definition was published by Georg Cantor in 1871. In 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, which he published in 1891. See Cantor's first uncountability proof.
Definition[edit]
Main article: Construction of the real numbers
The real number system ( R ; + ; ⋅ ; < ) {\displaystyle (\mathbf {R} ;{}+{};{}\cdot {};{}<{})} can be defined axiomatically up to an isomorphism, which is described hereafter. There are also many ways to construct "the" real number system, for example, starting from natural numbers, then defining rational numbers algebraically, and finally defining real numbers as equivalence classes of their Cauchy sequences or as Dedekind cuts, which are certain subsets of rational numbers. Another possibility is to start from some rigorous axiomatization of Euclidean geometry (Hilbert, Tarski, etc.) and then define the real number system geometrically. From the structuralist point of view all these constructions are on equal footing.
Axiomatic approach[edit]
Let R denote the set of all real numbers. Then:
The set R is a field, meaning that addition and multiplication are defined and have the usual properties.
The field R is ordered, meaning that there is a total order ≥ such that, for all real numbers x, y and z:
if x ≥ y then x + z ≥ y + z;
if x ≥ 0 and y ≥ 0 then xy ≥ 0.
The order is Dedekind-complete; that is: every non-empty subset S of R with an upper bound in ℝ has a least upper bound (also called supremum) in R .
The last property is what differentiates the reals from the rationals (and from other, more exotic ordered fields). For example, the set of rationals with square less than 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 is not rational.
These properties imply Archimedean property (which is not implied by other definitions of completeness). That is, the set of integers is not upper-bounded in the reals. In fact, if this were false, then the integers would have a least upper bound N; then, N – 1 would not be an upper bound, and there would be an integer n such that n > N – 1, and thus n + 1 > N, which is a contradiction with the upper-bound property of N.
The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind-complete ordered fields R1 and R2, there exists a unique field isomorphism from R1 to R2, allowing us to think of them as essentially the same mathematical object.
For another axiomatization of ℝ, see Tarski's axiomatization of the reals.
Construction from the rational numbers[edit]
The real numbers can be constructed as a completion of the rational numbers in such a way that a sequence defined by a decimal or binary expansion like (3; 3.1; 3.14; 3.141; 3.1415; ...) converges to a unique real number, in this case π. For details and other constructions of real numbers, see construction of the real numbers.
Properties[edit]
Basic properties[edit]
A real number may be either rational or irrational; either algebraic or transcendental; and either positive, negative, or zero. Real numbers are used to measure continuous quantities. They may be expressed by decimal representations that have an infinite sequence of digits to the right of the decimal point; these are often represented in the same form as 324.823122147... The ellipsis (three dots) indicates that there would still be more digits to come.
More formally, the real numbers have the two basic properties of being an ordered field, and having the least upper bound property. The first says that real numbers comprise a field, with addition and multiplication as well as division by non-zero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication. The second says that, if a non-empty set of real numbers has an upper bound, then it has a real least upper bound. The second condition distinguishes the real numbers from the rational numbers: for example, the set of rational numbers whose square is less than 2 is a set with an upper bound (e.g. 1.5) but no (rational) least upper bound: hence the rational numbers do not satisfy the least upper bound property.
Completeness[edit]
Main article: Completeness of the real numbers
A main reason for using real numbers is that the reals contain all limits. More precisely, a sequence of real numbers has a limit, which is a real number, if (and only if) its elements eventually come and remain arbitrarily close to each other. This is formally defined in the following, and means that the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section). :
A sequence (xn) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |xn − xm| is less than ε for all n and m that are both greater than N. This definition, originally provided by Cauchy, formalizes the fact that the xn eventually come and remain arbitrarily close to each other.
A sequence (xn) converges to the limit x if its elements eventually come and remain arbitrarily close to x, that is, if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |xn − x| is less than ε for n greater than N.
Every convergent sequence is a Cauchy sequence, and the converse is true for real numbers, and this means that the topological space of the real numbers is complete.
The set of rational numbers is not complete. For example, the sequence (1; 1.4; 1.41; 1.414; 1.4142; 1.41421; ...), where each term adds a digit of the decimal expansion of the positive square root of 2, is Cauchy but it does not converge to a rational number (in the real numbers, in contrast, it converges to the positive square root of 2).
The completeness property of the reals is the basis on which calculus, and, more generally mathematical analysis are built. In particular, the test that a sequence is a Cauchy sequence allows proving that a sequence has a limit, without computing it, and even without knowing it.
For example, the standard series of the exponential function
e x = ∑ n = 0 ∞ x n n ! {\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}
converges to a real number for every x, because the sums
∑ n = N M x n n ! {\displaystyle \sum _{n=N}^{M}{\frac {x^{n}}{n!}}}
can be made arbitrarily small (independently of M) by choosing N sufficiently large. This proves that the sequence is Cauchy, and thus converges, showing that e x {\displaystyle e^{x}} is well defined for every x.
"The complete ordered field"[edit]
The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.
First, an order can be lattice-complete. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element z, z + 1 is larger), so this is not the sense that is meant.
Additionally, an order can be Dedekind-complete, as defined in the section Axioms. The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way.
These two notions of completeness ignore the field structure. However, an ordered group (in this case, the additive group of the field) defines a uniform structure, and uniform structures have a notion of completeness (topology); the description in the previous section Completeness is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces, since the definition of metric space relies on already having a characterization of the real numbers.) It is not true that R is the only uniformly complete ordered field, but it is the only uniformly complete Archimedean field, and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field". Every uniformly complete Archimedean field must also be Dedekind-complete (and vice versa), justifying using "the" in the phrase "the complete Archimedean field". This sense of completeness is most closely related to the construction of the reals from Cauchy sequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) and forms the uniform completion of it in a standard way.
But the original use of the phrase "complete Archimedean field" was by David Hilbert, who meant still something else by it. He meant that the real numbers form the largest Archimedean field in the sense that every other Archimedean field is a subfield of R. Thus R is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field. This sense of completeness is most closely related to the construction of the reals from surreal numbers, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield.
Advanced properties[edit]
See also: Real line
The reals are uncountable; that is: there are strictly more real numbers than natural numbers, even though both sets are infinite. In fact, the cardinality of the reals equals that of the set of subsets (i.e. the power set) of the natural numbers, and Cantor's diagonal argument states that the latter set's cardinality is strictly greater than the cardinality of N. Since the set of algebraic numbers is countable, almost all real numbers are transcendental. The non-existence of a subset of the reals with cardinality strictly between that of the integers and the reals is known as the continuum hypothesis. The continuum hypothesis can neither be proved nor be disproved; it is independent from the axioms of set theory.
As a topological space, the real numbers are separable. This is because the set of rationals, which is countable, is dense in the real numbers. The irrational numbers are also dense in the real numbers, however they are uncountable and have the same cardinality as the reals.
The real numbers form a metric space: the distance between x and y is defined as the absolute value |x − y|. By virtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical, but yield different presentations for the topology – in the order topology as ordered intervals, in the metric topology as epsilon-balls. The Dedekind cuts construction uses the order topology presentation, while the Cauchy sequences construction uses the metric topology presentation. The reals are a contractible (hence connected and simply connected), separable and complete metric space of Hausdorff dimension 1. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to the reals.
Every nonnegative real number has a square root in R, although no negative number does. This shows that the order on R is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one real root: these two properties make R the premier example of a real closed field. Proving this is the first half of one proof of the fundamental theorem of algebra.
The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalized such that the unit interval [0;1] has measure 1. There exist sets of real numbers that are not Lebesgue measurable, e.g. Vitali sets.
The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. It is not possible to characterize the reals with first-order logic alone: the Löwenheim–Skolem theorem implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first-order logic as the real numbers themselves. The set of hyperreal numbers satisfies the same first order sentences as R. Ordered fields that satisfy the same first-order sentences as R are called nonstandard models of R. This is what makes nonstandard analysis work; by proving a first-order statement in some nonstandard model (which may be easier than proving it in R), we know that the same statement must also be true of R.
The field R of real numbers is an extension field of the field Q of rational numbers, and R can therefore be seen as a vector space over Q. Zermelo–Fraenkel set theory with the axiom of choice guarantees the existence of a basis of this vector space: there exists a set B of real numbers such that every real number can be written uniquely as a finite linear combination of elements of this set, using rational coefficients only, and such that no element of B is a rational linear combination of the others. However, this existence theorem is purely theoretical, as such a base has never been explicitly described.
The well-ordering theorem implies that the real numbers can be well-ordered if the axiom of choice is assumed: there exists a total order on R with the property that every non-empty subset of R has a least element in this ordering. (The standard ordering ≤ of the real numbers is not a well-ordering since e.g. an open interval does not contain a least element in this ordering.) Again, the existence of such a well-ordering is purely theoretical, as it has not been explicitly described. If V=L is assumed in addition to the axioms of ZF, a well ordering of the real numbers can be shown to be explicitly definable by a formula.[10]
Applications and connections to other areas[edit]
Real numbers and logic[edit]
The real numbers are most often formalized using the Zermelo–Fraenkel axiomatization of set theory, but some mathematicians study the real numbers with other logical foundations of mathematics. In particular, the real numbers are also studied in reverse mathematics and in constructive mathematics.[11]
The hyperreal numbers as developed by Edwin Hewitt, Abraham Robinson and others extend the set of the real numbers by introducing infinitesimal and infinite numbers, allowing for building infinitesimal calculus in a way closer to the original intuitions of Leibniz, Euler, Cauchy and others.
Edward Nelson's internal set theory enriches the Zermelo–Fraenkel set theory syntactically by introducing a unary predicate "standard". In this approach, infinitesimals are (non-"standard") elements of the set of the real numbers (rather than being elements of an extension thereof, as in Robinson's theory).
The continuum hypothesis posits that the cardinality of the set of the real numbers is ℵ 1 {\displaystyle \aleph _{1}} ; i.e. the smallest infinite cardinal number after ℵ 0 {\displaystyle \aleph _{0}} , the cardinality of the integers. Paul Cohen proved in 1963 that it is an axiom independent of the other axioms of set theory; that is: one may choose either the continuum hypothesis or its negation as an axiom of set theory, without contradiction.
In physics[edit]
In the physical sciences, most physical constants such as the universal gravitational constant, and physical variables, such as position, mass, speed, and electric charge, are modeled using real numbers. In fact, the fundamental physical theories such as classical mechanics, electromagnetism, quantum mechanics, general relativity and the standard model are described using mathematical structures, typically smooth manifolds or Hilbert spaces, that are based on the real numbers, although actual measurements of physical quantities are of finite accuracy and precision.
Physicists have occasionally suggested that a more fundamental theory would replace the real numbers with quantities that do not form a continuum, but such proposals remain speculative.[12]
In computation[edit]
With some exceptions, most calculators do not operate on real numbers. Instead, they work with finite-precision approximations called floating-point numbers. In fact, most scientific computation uses floating-point arithmetic. Real numbers satisfy the usual rules of arithmetic, but floating-point numbers do not.
Computers cannot directly store arbitrary real numbers with infinitely many digits. The achievable precision is limited by the number of bits allocated to store a number, whether as floating-point numbers or arbitrary-precision numbers. However, computer algebra systems can operate on irrational quantities exactly by manipulating formulas for them (such as 2 , {\displaystyle {\sqrt {2}},} arcsin  ( 2 / 23 ) , {\displaystyle \arcsin(2/23),} or ∫ 0 1 x x d x {\displaystyle \textstyle \int _{0}^{1}x^{x}\,dx} ) rather than their rational or decimal approximation.[13] It is not in general possible to determine whether two such expressions are equal (the constant problem).
A real number is called computable if there exists an algorithm that yields its digits. Because there are only countably many algorithms,[14] but an uncountable number of reals, almost all real numbers fail to be computable. Moreover, the equality of two computable numbers is an undecidable problem. Some constructivists accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable.
"Reals" in set theory[edit]
In set theory, specifically descriptive set theory, the Baire space is used as a surrogate for the real numbers since the latter have some topological properties (connectedness) that are a technical inconvenience. Elements of Baire space are referred to as "reals".
Vocabulary and notation[edit]
Mathematicians use the symbol R, or, alternatively, ℝ, the letter "R" in blackboard bold (encoded in Unicode as U+211D ℝ DOUBLE-STRUCK CAPITAL R (HTML ℝ)), to represent the set of all real numbers. As this set is naturally endowed with the structure of a field, the expression field of real numbers is frequently used when its algebraic properties are under consideration.
The sets of positive real numbers and negative real numbers are often noted R+ and R−,[15] respectively; R+ and R− are also used.[16] The non-negative real numbers can be noted R≥0 but one often sees this set noted R+ ∪ {0}.[15] In French mathematics, the positive real numbers and negative real numbers commonly include zero, and these sets are noted respectively ℝ+ and ℝ−.[16] In this understanding, the respective sets without zero are called strictly positive real numbers and strictly negative real numbers, and are noted ℝ+* and ℝ−*.[16]
The notation Rn refers to the cartesian product of n copies of R, which is an n-dimensional vector space over the field of the real numbers; this vector space may be identified to the n-dimensional space of Euclidean geometry as soon as a coordinate system has been chosen in the latter. For example, a value from R3 consists of three real numbers and specifies the coordinates of a point in 3‑dimensional space.
In mathematics, real is used as an adjective, meaning that the underlying field is the field of the real numbers (or the real field). For example, real matrix, real polynomial and real Lie algebra. The word is also used as a noun, meaning a real number (as in "the set of all reals").
Generalizations and extensions[edit]
The real numbers can be generalized and extended in several different directions:
The complex numbers contain solutions to all polynomial equations and hence are an algebraically closed field unlike the real numbers. However, the complex numbers are not an ordered field.
The affinely extended real number system adds two elements +∞ and −∞. It is a compact space. It is no longer a field, or even an additive group, but it still has a total order; moreover, it is a complete lattice.
The real projective line adds only one value ∞. It is also a compact space. Again, it is no longer a field, or even an additive group. However, it allows division of a non-zero element by zero. It has cyclic order described by a separation relation.
The long real line pastes together ℵ1* + ℵ1 copies of the real line plus a single point (here ℵ1* denotes the reversed ordering of ℵ1) to create an ordered set that is "locally" identical to the real numbers, but somehow longer; for instance, there is an order-preserving embedding of ℵ1 in the long real line but not in the real numbers. The long real line is the largest ordered set that is complete and locally Archimedean. As with the previous two examples, this set is no longer a field or additive group.
Ordered fields extending the reals are the hyperreal numbers and the surreal numbers; both of them contain infinitesimal and infinitely large numbers and are therefore non-Archimedean ordered fields.
Self-adjoint operators on a Hilbert space (for example, self-adjoint square complex matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvalues are real and they form a real associative algebra. Positive-definite operators correspond to the positive reals and normal operators correspond to the complex numbers.
See also[edit]
Mathematics portal
Algebra portal
Number theory portal
Analysis portal
Continued fraction
Real analysis
Notes[edit]
Jump up ^ More precisely, given two complete totally ordered fields, there is a unique isomorphism between them. This implies that the identity is the unique field automorphism of the reals that is compatible with the ordering.
Citations[edit]
Jump up ^ T. K. Puttaswamy, "The Accomplishments of Ancient Indian Mathematicians", pp. 410–1. In: Selin, Helaine; D'Ambrosio, Ubiratan, eds. (2000), Mathematics Across Cultures: The History of Non-western Mathematics, Springer, ISBN 1-4020-0260-2 .
Jump up ^ O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics archive, University of St Andrews .
Jump up ^ Matvievskaya, Galina (1987), "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics", Annals of the New York Academy of Sciences, 500: 253–277 [254], Bibcode:1987NYASA.500..253M, doi:10.1111/j.1749-6632.1987.tb37206.x
Jump up ^ Jacques Sesiano, "Islamic mathematics", p. 148, in Selin, Helaine; D'Ambrosio, Ubiratan (2000), Mathematics Across Cultures: The History of Non-western Mathematics, Springer, ISBN 1-4020-0260-2
Jump up ^ Beckmann, Petr (1993), A History of Pi, Dorset Classic Reprints, Barnes & Noble Publishing, p. 170, ISBN 9780880294188 .
Jump up ^ Arndt, Jörg; Haenel, Christoph (2001), Pi Unleashed, Springer, p. 192, ISBN 9783540665724 .
Jump up ^ Dunham, William (2015), The Calculus Gallery: Masterpieces from Newton to Lebesgue, Princeton University Press, p. 127, ISBN 9781400866793, Cantor found a remarkable shortcut to reach Liouville's conclusion with a fraction of the work
Jump up ^ Hurwitz, Adolf (1893). "Beweis der Transendenz der Zahl e". Mathematische Annalen. Teubner (43): 134–135.
Jump up ^ Gordan, Paul (1893). "Transcendenz von e und π". Mathematische Annalen. Teubner. 43 (2-3): 222–224.
Jump up ^ Moschovakis, Yiannis N. (1980), "Descriptive set theory", Studies in Logic and the Foundations of Mathematics, Amsterdam - New York: North-Holland Publishing Co., 100, pp. xii, 637, ISBN 0-444-85305-7 , chapter V.
Jump up ^ Bishop, Errett; Bridges, Douglas (1985), Constructive analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 279, Berlin, New York: Springer-Verlag, ISBN 978-3-540-15066-4 , chapter 2.
Jump up ^ Wheeler, John Archibald (1986). "Hermann Weyl and the Unity of Knowledge: In the linkage of four mysteries—the "how come" of existence, time, the mathematical continuum, and the discontinuous yes-or-no of quantum physics—may lie the key to deep new insight". American Scientist. 74 (4): 366–375. Bibcode:1986AmSci..74..366W. JSTOR 27854250.
Bengtsson, Ingemar (2017). "The Number Behind the Simplest SIC-POVM". Foundations of Physics. 47: 1031–1041. arXiv:1611.09087  . Bibcode:2017FoPh...47.1031B. doi:10.1007/s10701-017-0078-3.
Jump up ^ Cohen, Joel S. (2002), Computer algebra and symbolic computation: elementary algorithms, 1, A K Peters, p. 32, ISBN 978-1-56881-158-1
Jump up ^ Hein, James L. (2010), "14.1.1", Discrete Structures, Logic, and Computability (3 ed.), Sudbury, Massachusetts, USA: Jones and Bartlett Publishers
^ Jump up to: a b Schumacher 1996, pp. 114-115
^ Jump up to: a b c École Normale Supérieure of Paris, “Nombres réels” (“Real numbers”), p. 6
References[edit]
Georg Cantor, 1874, "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen", Journal für die Reine und Angewandte Mathematik, volume 77, pages 258–262.
Solomon Feferman, 1989, The Number Systems: Foundations of Algebra and Analysis, AMS Chelsea, ISBN 0-8218-2915-7.
Robert Katz, 1964, Axiomatic Analysis, D. C. Heath and Company.
Edmund Landau, 2001, ISBN 0-8218-2693-X, Foundations of Analysis, American Mathematical Society.
Howie, John M., Real Analysis, Springer, 2005, ISBN 1-85233-314-6.
Schumacher, Carol (1996), ChapterZero / Fundamental Notions of Abstract Mathematics, Addison-Wesley, ISBN 0-201-82653-4 .
External links[edit]
Hazewinkel, Michiel, ed. (2001) [1994], "Real number", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
The real numbers: Pythagoras to Stevin
The real numbers: Stevin to Hilbert
The real numbers: Attempts to understand
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values: Реален број – Macedonian

values:
interlanguage-link interwiki-mk
display: none;

values: വാസ്തവികസംഖ്യ – Malayalam

values:
interlanguage-link interwiki-ml
display: none;

values: सत् संख्या – Marathi

values:
interlanguage-link interwiki-mr
display: none;

values:
interlanguage-link interwiki-ms
display: none;

values:
interlanguage-link interwiki-nl
display: none;

values: वास्तविक सङ्ख्या – Nepali

values:
interlanguage-link interwiki-ne
display: none;

values:
interlanguage-link interwiki-ja
display: none;

values:
interlanguage-link interwiki-no
display: none;

values:
interlanguage-link interwiki-nn
display: none;

values:
interlanguage-link interwiki-uz
display: none;

values: ਵਾਸਤਵਿਕ ਨੰਬਰ – Punjabi

values:
interlanguage-link interwiki-pa
display: none;

values:
interlanguage-link interwiki-jam
display: none;

values: ចំនួនពិត – Khmer

values:
interlanguage-link interwiki-km
display: none;

values:
interlanguage-link interwiki-pms
display: none;

values:
interlanguage-link interwiki-pl
display: none;

values:
interlanguage-link interwiki-pt
display: none;

values:
interlanguage-link interwiki-ro
display: none;

values: Вещественное число – Russian

values:
interlanguage-link interwiki-ru
display: none;

values:
interlanguage-link interwiki-sco
display: none;

values:
interlanguage-link interwiki-scn
display: none;

values: තාත්වික සංඛ්‍යා – Sinhala

values:
interlanguage-link interwiki-si
display: none;

values:
interlanguage-link interwiki-simple
display: none;

values:
interlanguage-link interwiki-sk
display: none;

values:
interlanguage-link interwiki-sl
display: none;

values: ژمارەی ڕاستەقینە – Central Kurdish

values:
interlanguage-link interwiki-ckb
display: none;

values: Реалан број – Serbian

values:
interlanguage-link interwiki-sh
display: none;

values:
interlanguage-link interwiki-fi
display: none;

values:
interlanguage-link interwiki-sv
display: none;

values:
interlanguage-link interwiki-tl
display: none;

values: மெய்யெண் – Tamil

values:
interlanguage-link interwiki-ta
display: none;

values: จำนวนจริง – Thai

values:
interlanguage-link interwiki-th
display: none;

values:
interlanguage-link interwiki-tr
display: none;

values: Дійсне число – Ukrainian

values:
interlanguage-link interwiki-uk
display: none;

values: حقیقی عدد – Urdu

values:
interlanguage-link interwiki-vi
display: none;

values:
interlanguage-link interwiki-fiu-vro
display: none;

values:
interlanguage-link interwiki-wuu
display: none;

values: רעאלע צאל – Yiddish

values:
interlanguage-link interwiki-yi
display: none;

values: interlanguage-link interwiki-yo badge-Q17437798 badge-goodarticle

values:
interlanguage-link interwiki-zh-yue
display: none;

values:
interlanguage-link interwiki-diq
display: none;

values:
interlanguage-link interwiki-lfn
display: none;

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