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Hat-trick
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A hat-trick or hat trick in sports is the achievement of a positive feat three times in a game, or another achievement based on the number three in some sports.
In association and rugby football, the scoring of two goals or tries by one individual in a single match is referred to as a brace.
Contents
[hide]
1 Origin
2 Association football
3 Baseball
4 Cricket
5 Darts
6 Hockey
6.1 Variations
7 Lacrosse
8 Marbles
9 Motor racing
10 Poker
11 Rugby football
12 Scrabble
13 Team handball/Water polo
14 Video games
15 See also
16 References
Origin[edit]
The term first appeared in 1858 in cricket, to describe H. H. Stephenson's taking three wickets with three consecutive deliveries. Fans held a collection for Stephenson, and presented him with a hat bought with the proceeds.[1] The term was used in print for the first time in 1865.[2] The term was eventually adopted by many other sports including hockey, association football, water polo and team handball.
Association football[edit]
See also: Lists of hat-tricks, List of FIFA World Cup hat-tricks, and List of UEFA Champions League hat-tricks
A hat-trick occurs in association football when a player scores three goals (not necessarily consecutive) in a single game, whereas scoring two goals constitutes a brace.[3][4] In common with other official record-keeping rules, goals in a penalty shootout are excluded from the tally.[5] The extra time in a knockout cup match may also be calculated towards a player's potential hat-trick. The fastest recorded time to score a hat-trick is 70 seconds, a record set by Alex Torr in a Sunday league game in 2013.[6] The previous record of 90 seconds was held by Tommy Ross playing for Ross County against Nairn County on 28 November 1964.[7]
The first hat-trick in an international game was by Scottish player John McDougall, against England on 2 March 1878.[8] American player Bert Patenaude scored the first hat-trick in the FIFA World Cup, against Paraguay in the inaugural event. Two hat-tricks have been scored in a final, by Geoff Hurst for England in the 1966 final during extra time against West Germany,[9] and Carli Lloyd for the USA against Japan in the 2015 Women's World Cup final. Lloyd's was the fastest hat-trick scored in a World Cup final, and the fastest from kickoff in any World Cup match for either sex. However, the fastest hat-trick for either men or women, as measured by time between goals, belongs to Fabienne Humm of Switzerland, who scored in the 47th, 49th and 52nd minutes against Ecuador in the 2015 group stage.
Football has also extended the term to include the phrase perfect hat-trick, achieved when a player scores one right-footed goal, one left-footed goal and one headed goal within one match.[10][11] In Germany, the term (lupenreiner) Hattrick ("flawless hat-trick") refers to when a player scores three goals in a row in one half without the half-time break or a goal scored by another player interfering the performance.
Baseball[edit]
In the past, the term was occasionally used to describe when a player struck out three times in a baseball game, and the term golden sombrero was more commonly used when a player struck out four times in a game.
In recent years, hat trick has been more often used to describe when a player hits three home runs in a game.
For example, on 29 August 2015, Toronto Blue Jays fans celebrated Edwin Encarnación's third home run of the game by throwing hats onto the field, similar to the tradition in ice hockey.[12] The phenomenon continued during the 2016 season, and on 17 June 2016, a number of Blue Jays fans at Oriole Park at Camden Yards threw hats on to the field after Canadian Blue Jays player Michael Saunders hit his third home run of the night, and again on 28 August at Rogers Centre, when Blue Jays player Josh Donaldson hitting his third home run of the game in the eighth inning against the Minnesota Twins.
Cricket[edit]
See also: List of Test cricket hat-tricks, List of One Day International cricket hat-tricks, List of Twenty20 International cricket hat-tricks, and List of first-class cricket records § Hat-tricks
A hat-trick occurs in cricket when a bowler dismisses three batsmen with consecutive deliveries. The deliveries may be interrupted by an over bowled by another bowler from the other end of the pitch or the other team's innings, but must be three consecutive deliveries by the individual bowler in the same match. Only wickets attributed to the bowler count towards a hat-trick; run outs do not count.
In some countries, especially Australia, four wickets in four balls is sometimes referred to a double hat-trick on the basis that there are two ways of compiling the three-in-three sequence (ie wickets 1,2 and 3 or wickets 2,3 and 4). By extension, five consecutive wickets is a triple hat-trick and six consecutive wickets a quadruple hat-trick. The logic however is dubious - double centuries consist of two hundred runs, not a score of 101. Furthermore, there are no instances of this type of usage in other sports - for instances, four goals in association football is not called a double hat-trick. In England, a double hat-trick is considered to be two hat-tricks in one match.
Hat-tricks are rare, and as such are treasured by bowlers. In Test cricket history there have been just 43 hat-tricks, the first achieved by Fred Spofforth for Australia against England in 1879. In 1912, Australian Jimmy Matthews achieved the feat twice in one game against South Africa. The only other players to achieve two hat-tricks are Australia's Hugh Trumble, against England in 1902 and 1904, Pakistan's Wasim Akram, in separate games against Sri Lanka in 1999, and England's Stuart Broad.
In One Day International cricket there have been 36 hat-tricks, the first by Jalal-ud-Din for Pakistan against Australia in 1982, and the most recent by James Faulkner (cricketer) of Australia, playing in the World Cup quarter final against Sri Lanka on 18 March 2015 at the Sydney Cricket Ground.[13]
Lasith Malinga s the only bowler to take three hat-tricks in any form of international cricket. Three players have taken at least two ODI hat-tricks in their careers: Wasim Akram and Saqlain Mushtaq of Pakistan and Chaminda Vaas of Sri Lanka. (Akram therefore has four international hat-tricks in total).
Taking two wickets in two consecutive deliveries is occasionally known as a brace, or (more commonly, especially until the next delivery has been made) being on a hat-trick.
The feat of taking four wickets in four balls has occurred only once in international one-day cricket, in the 2007 World Cup, when Sri Lanka's Lasith Malinga managed the feat against South Africa by dismissing Shaun Pollock, Andrew Hall, Jacques Kallis and Makhaya Ntini, though it has occurred on other occasions in first-class cricket. Kevan James of Hampshire took four wickets in four balls and scored a century in the same county game against India in 1996. The Cricinfo report on the game claimed that this was unique in cricket.[14][15]
Nuwan Zoysa of Sri Lanka is the only bowler to achieve a hat-trick off his first three balls in a Test, dismissing Murray Goodwin, Neil Johnson and Trevor Gripper of Zimbabwe.[16] In 2006 Irfan Pathan of India achieved a hat-trick in the first over of the test match, off the last three balls, when dismissing Salman Butt, Younis Khan and Mohammad Yousuf of Pakistan. Chaminda Vaas is the only one to achieve a hat-trick of the very first deliveries in one day internationals, against Bangladesh in the tenth match of 2003 ICC World Cup at City Oval, Pietermaritzburg. He got Hannan Sarkar, Mohammad Ashraful and Ehsanul Haque out in the first three balls and took his fourth wicket in the fifth ball of the same over, just missing the double-hat-trick.
Albert Trott and Joginder Rao are the only two bowlers credited with two hat-tricks in the same innings in first-class cricket. One of Trott's two hat-tricks, for Middlesex against Somerset at Lords in 1907, was a four in four.
Some hat-tricks are particularly extraordinary. On 2 December 1988, Merv Hughes, playing for Australia, dismissed Curtly Ambrose with the last ball of his penultimate over and Patrick Patterson with the first ball of his next over, wrapping up the West Indies first innings. When Hughes returned to bowl in the West Indies second innings, he trapped Gordon Greenidge lbw with his first ball, completing a hat-trick over two different innings and becoming the only player in Test cricket history to achieve the three wickets of a hat-trick in three different overs.
In 1844, underarm bowler William Clark, playing for "England" against Kent, achieved a hat-trick spread over two innings, dismissing Kent batsman John Fagge twice within the hat-trick. Fagge batted at number 11 in the first innings and at number 3 in the second. This event is believed to be unique in first-class cricket.[17]
The most involved hat-trick was perhaps when Melbourne club cricketer Stephen Hickman, playing for Power House, achieved a hat-trick spread over three overs, two days, two innings, involving the same batsman twice, and observed by the same non-striker, with the hat-trick ball being bowled from the opposite end to the first two. In the Mercantile Cricket Association C Grade semi-final at Fawkner Park South Yarra in Melbourne, Gunbower United Cricket Club were 8 for 109 when Hickman came on to bowl his off spin. He took a wicket with the last ball of his third over and then bowled number 11 batsman Richard Higgins with the first ball of his next over to complete the Gunbower innings, leaving Chris Taylor the not out batsman. Power House scored 361 putting the game out of reach of Gunbower. In the second innings opener Taylor was joined by Higgins at the fall of the fourth wicket as Hickman returned to the attack. With his first ball, observed by an incredulous Taylor at the non-striker's end, he clean bowled Higgins leaving Higgins with a pair of golden ducks.[18]
A triple hat-trick was achieved by Scott Babot of Wainuiomata Cricket Club playing in the Senior 3 competition in New Zealand in 2008. It consisted of five wickets in five balls, across two innings and separated by seven days, as the match took place on consecutive Saturdays.[19]
Another triple hat trick was taken by Hanuman Choudhary when playing for the Cricket Association of Memphis team (CAM) against Wolfpacks in the Arkansas Tennessee Cricket League on 23 October 2016, in the semifinal of India Association of Memphis tournament in Memphis Premier League 3. His five consecutive dismissals helped close the innings from 142–5 to 142 all out.[20]
A 'perfect over' of 6 wickets taken with 6 consecutive balls was achieved by Australian Aled Carey on 21 January 2017 while bowling for his club Golden Point against East Ballarat. This very rare quadruple Hat-Trick consisted of 2 catches, an LBW and 3 bowled.[21]
Darts[edit]
A hat-trick in darts is a score of three consecutive bullseyes, also known as the "Alan Evans Shot", after Alan Evans scored three bulleyes during a match on numerous occasions.[22]
Hockey[edit]
This section uses citations without providing full publisher and source details. Please consider adding fuller citations so that the article remains verifiable and that the origin of sources becomes clearer. Several templates are available to assist in formatting. (May 2017)
Hats on the Verizon Center ice after Alex Ovechkin's hat trick, 7 February 2010.
In both field hockey and ice hockey, a hat trick occurs when a player scores three goals in a single game. A hat trick, as it is known in its current form, culminates with fans throwing hats onto the ice from the stands. The tradition is said to have begun among fans in the National Hockey League around the 1950s.[23]
There are several conflicting legends of how the "hat trick" was popularised in professional hockey. Most stories involve hats being awarded to any of the local players who scored three goals in a game. According to the NHL, in the 1940s, a Toronto haberdasher used to give free hats to players with the Toronto Maple Leafs when they scored three goals in a game, which introduced the "Hat Trick" expression into the world of hockey.[24]
Montreal hatter Henri Henri makes a similar claim, that between 1950 and 1970 they would reward any NHL player who scored three or more goals in a game at the Montreal Forum with a free hat.[25]
Finally, in the 1950s, the Guelph Biltmore Mad Hatters of the Ontario Hockey Association, who were then a farm team of the NHL's New York Rangers, were sponsored by Guelph-based Biltmore Hats, a leading manufacturer of hats with North American dominance. The sponsor would award any Madhatters player who scored three goals in a game with a new fedora.[26]
In a slightly different account, the expression originates not with any member of a team, but with a particular player. According to legend, Chicago Blackhawks forward Alex Kaleta entered the shop of Toronto businessman Sammy Taft to purchase a new hat, but did not have enough money. Taft arranged a deal with Kaleta stipulating that if Kaleta scored three goals as he played the Toronto Maple Leafs that night, Taft would give him a free hat. That night, on 26 January 1946, Kaleta scored four goals against the Maple Leafs and Taft made good on his offer.[27] This is the story accepted as the origin of the phrase by the Hockey Hall of Fame.[28]
While charming fables, all these explanations of the introduction of the term “hat trick” in hockey are too late to be true. On Dec. 8, 1933, the Winnipeg Free Press[full citation needed] describes a hockey game in which “Romeo Rivers, rugged wingman” for the Monarchs scored three goals in the same game, describing how “Romeo completed his ‘hat trick’” when he scored his third goal of the night after taking a pass from a teammate who had drawn the goalie out of position.
The Jan. 16, 1939 Lethbridge Herald (p.10)[full citation needed] describes a hockey game in which the Lethbridge Maple Leafs defeated the Calgary Stampeders and Jimmie McIndoe of the Leafs “turned the hat trick, when he converted three straight consecutive passes” from a teammate.
By 1944, the term “hat trick” was so well established in hockey that the Winnipeg Free Press (Nov. 29, 1944, p. 14)[full citation needed] reports that “hockey’s traditional ‘hat-trick’ – the feat of scoring three goals in a single game – will receive official recognition from the Amateur Hockey Association” of the US by awarding a small silver derby hat to players to mark the accomplishment. Thus, by 1944 the term "hat trick" was common enough to be termed “traditional.” Given how frequently the words “hat trick” were used in sports reporting on cricket and soccer in the early 20th century, the term was probably routinely used in hockey by the early 1930s.
Wayne Gretzky holds the NHL record for the most hat tricks in a career with 50. Harry Hyland scored the league's first hat trick, in the league's very first game on December 19, 1917, in which Hyland's Montreal Wanderers defeated the Toronto Arenas 10–9.[29]
Variations[edit]
A natural hat trick occurs when a player scores three consecutive goals, uninterrupted by any other player scoring for either team.[30]
The NHL record for the fastest natural hat trick is 21 seconds, set by Bill Mosienko in 1952 for the Chicago Blackhawks.[31]
A Gordie Howe hat trick is a tongue-in-cheek play on the feat. It is achieved by scoring a goal, getting an assist, and getting in a fight, all in the same game. Namesake Gordie Howe himself only recorded two in his NHL career, as opposed to league leader Rick Tocchet, who accrued 18 Gordie Howe hat tricks.[32]
In December 1995, Florida Panthers captain Scott Mellanby scored a rat trick, the term coined by teammate John Vanbiesbrouck. Prior to the game, Mellanby killed a rat in the Panthers' locker room with his hockey stick, and proceeded to score a pair of goals later that night.[33] When Mellanby scored a hat trick in a later game, some Florida fans threw plastic rats on the ice, a tradition that continued for all Panthers' goals throughout the 1996 playoffs. Due to the resulting game delays caused by the necessary clean-up of the plastic rats, the league eventually banned the activity and modified Rule 63 to impose a minor penalty against the home team for a violation.[34] The more traditional practice of fans throwing hats on the ice following genuine hat tricks remains exempt from this penalty.
Lacrosse[edit]
A hat-trick in lacrosse is when a player scores three goals in one game.
Marbles[edit]
In marbles, a hat-trick occurs when a player hits all marbles in a single turn.
Motor racing[edit]
In motor racing, three successive race wins, winning the same event three times in a row, or securing pole position, fastest lap and race victory in one event may all be referred to as a hat-trick.
Poker[edit]
Eliminating three players from a table with one hand in live poker play is sometimes referred to as a hat-trick and is incredibly rare. It is a much more frequent occurrence in online poker games, given the faster and greater number of hands played in online tournaments and the continuing presence of multiple "all-in" players during the early stages of tournament play as players look to build large chip stacks quickly and early.
Checking and raising an opponent three times,[35] as well as winning the European Poker Tour (EPT), World Series of Poker (WSOP), and World Poker Tour (WPT) in the same year,[36] are also called hat tricks in poker.
Rugby football[edit]
See also: List of Rugby World Cup hat-tricks
In both codes of rugby football (rugby union and rugby league) a hat-trick is when a player scores three or more tries in a game. In rugby union, a related concept is that of a "full house" (scoring a try, conversion, penalty goal, and drop goal) in a single game. When a player scored two tries, this is often referred to as a brace. As with association football, it is common to award the match ball to a player who scores a hat-trick.
Ken Irvine and Frank Burge both scored 16 hat-tricks in Australian first grade rugby league[37]
Shaun Johnson scored a hat-trick in under 6 minutes against the Canberra Raiders in 2013.
Scrabble[edit]
In Scrabble, a hat-trick occurs when a player plays three consecutive bingos.
Team handball/Water polo[edit]
In team handball or water polo, if a player scores three times in a game, a hat trick is made.
Video games[edit]
In many online video games, mostly first-person shooters, the term has entered the players' vocabulary and occurs when a player achieves 3 difficult kills in a row (commonly headshots). Examples include Team Fortress 2, Quake, and Unreal Tournament, among others.
See also[edit]
Look up hat trick in Wiktionary, the free dictionary.
Hitting for the cycle
Nap hand
Trifecta
Triple Crown (disambiguation)
Turkey (bowling)
Triple double
References[edit]
Jump up ^ Extended Oxford English Dictionary 1999 Edition : "It came into use after HH Stephenson took three wickets in three balls for the all-England eleven against the twenty-two of Hallam at the Hyde Park ground, Sheffield in 1858. A collection was held for Stephenson (as was customary for outstanding feats by professionals) and he was presented with a cap or hat bought with the proceeds."
Jump up ^ A report of a match between Grays and Romford in "The Chelmsford Chronicle". 23 June 1865. OCLC 866859233. OCLC 17645885, 702688846, 42349342. (Subscription required (help)). Mr Biddell...with his second ball bowled the Romford leviathan Mr Beauchamp and afterwards effected the hat-trick by getting three wickets in the over. [verification needed]
Jump up ^ Soccer Definitions & Slang Terms soccer-training-info.com
Jump up ^ "Brace" means two of a kind or a pair of something Harper Collins
Jump up ^ Kicks from the penalty mark (aka penalty shootout) are not part of the match. IFAB (July 2009). "Laws of the Game 2009/2010" (PDF). Zurich: FIFA. p. 130. Retrieved 9 April 2010.
Jump up ^ Steve White (3 May 2013). "Sunday league footballer scores hat-trick in record-breaking 70 seconds". mirror. Retrieved 6 July 2015.
Jump up ^ "Fastest time to score a hat-trick, Football". Guinness World Records. Retrieved 18 August 2011.
Jump up ^ Scotland – International Matches 1872–1880 Rec.Sport.Soccer Statistics Foundation. Retrieved 11 September 2015.
Jump up ^ "ON THIS DAY 30 July – 1966: Football glory for England". BBC. 30 July 1966. Retrieved 18 August 2011.
Jump up ^ Michel Platini: Uefa chief has his critics but is used to success, Ben Smith, 9 December 2012
Jump up ^ Alberto Bueno scored four La Liga goals in ..16 .. minutes , BBC News, 28 February 2015
Jump up ^ "Hat trick! Edwin Encarnación hits three homers, drives in nine in Jays win". Yahoo Sports. 29 August 2015.
Jump up ^ "Duminy hat-trick blows quarter-final wide open". cricket.com.au. 18 March 2015.
Jump up ^ "Hampshire v Indians, Match Report.". CricInfo. Retrieved 11 April 2007.
Jump up ^ "Hampshire v Indians at Southampton, 29 June-1 July 1996". CricInfo. Retrieved 11 April 2007.
Jump up ^ "Cricket: Zoysa performs opening hat-trick". London: The Independent. 27 November 1999. Retrieved 14 November 2012.
Jump up ^ "Stirling's gold, and the not-so-roaring forties". CricInfo. Retrieved 21 September 2010.
Jump up ^ Warwick Franks, Australian Wisden Cricketers' Almanack 2002. Hardie Grant Books.
Jump up ^ "Cricket Wellington – Babot: Triple Hat-trick". cricketwellington.co.nz. Archived from the original on 5 December 2008. Retrieved 25 October 2016.
Jump up ^ "CricHQ – Making cricket even better". crichq.com. Retrieved 25 October 2016.
Jump up ^ "Six of best for Aussie club bowler Aled Carey". New Zealand Herald. 26 January 2017. Retrieved 27 June 2017.
Jump up ^ Edwards, Dave (6 January 2011). "Darts: Sporting giants face-off". Wales Online. Archived from the original on 23 February 2014. Retrieved 4 January 2014.
Jump up ^ "Guelph's tricky claim". 'Guelph Mercury'. 21 March 2009. Archived from the original on 25 March 2009. Retrieved 6 April 2009.
Jump up ^ "Miscellaneous Trivia". NHL.
Jump up ^ "About Henri Henri". Henri Henri. Retrieved 13 March 2012.
Jump up ^ "The Hat Trick Returns to Its Hockey Birthplace". GuelphStorm.com. Retrieved 21 October 2014.
Jump up ^ "Toronto Invents: The Hat Trick". Torontoist. Retrieved 21 October 2014.
Jump up ^ "Frequently Asked Questions". Hockey Hall of Fame Resource Centre.
Jump up ^ Boswell, Randy (April 16, 2017). "Solving the mystery of the NHL's 1st game". CBC News. Retrieved April 16, 2017.
Jump up ^ Adam Kimmelman, Jonah Bruce (15 November 2010). "Hat tricks coming at a fairly frequent pace". NHL Insider. NHL. Retrieved 13 March 2012.
Jump up ^ "Ryan's natural hat trick not enough as Ducks fall to Kings". The Sports Network. Archived from the original on 22 December 2011. Retrieved 18 August 2011.
Jump up ^ John McGourty (25 January 2010). "Going inside the "Gordie Howe Hat Trick"". Inside NHL. NHL. Retrieved 13 March 2012.
Jump up ^ Montville, Leigh (10 June 1996). "Rat Pack". Sports Illustrated. Archived from the original on 6 December 2008. Retrieved 29 January 2009.
Jump up ^ "OFFICIAL NHL RULES – Rule 63, Delaying the Game". Retrieved 13 June 2010.
Jump up ^ "Fast 2014-09-16: Poker’s hat trick". Poker1.com | Mike Caro. 2014-09-16. Retrieved 2016-12-16.
Jump up ^ "Top 5 UK Poker Pros: Their Stories & Career Earnings". www.pokersites.co.uk. Retrieved 2016-12-16.
Jump up ^ "Rugby League Tables / Scoring Records". afltables.com. Retrieved 6 July 2015.
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Homogeneous and heterogeneous mixtures
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A homogeneous mixture is a solid, liquid or gaseous mixture that has the same proportions of its components throughout a given sample (or multiple samples of different proportion). Homogeneous mixtures can have a variable composition. Conversely, a heterogeneous mixture is not uniform in composition, but proportions of its components vary throughout the sample. "Homogeneous" and "heterogeneous" are not absolute terms, but depend on context and size of the sample. In chemistry, a homogeneous suspension of material means that when dividing the volume in half, the same amount of material is suspended in both halves of the substance. However, it might be possible to see the particles under the microscope. An example of a homogeneous mixture is air.
In physical chemistry and materials science that refers to substances and mixtures which are in a single phase. This is in contrast to a substance that is heterogeneous.[1]
A figure reprinting at the atomic level the differences between homogeneous mixtures, heterogeneous mixtures, compounds, and elements.
Contents
[hide]
1 Homogeneous mixture
1.1 Solutions
1.2 Gaseous
1.3 Solids
2 Metrics
3 Homogenization
4 References
Homogeneous mixture[edit]
Solutions[edit]
A solution is a special type of homogeneous mixture. Solutions are homogeneous because the ratio of solute to solvent remains the same throughout the solution even if homogenized with multiple sources, and stable because the solute will not settle out after any period of time, and it cannot be removed by a filter or by centrifuge.[2] This type of mixture is very stable, i.e., its particles do not settle, or separate. As a homogeneous mixture, a solution has one phase (liquid) although the solute and solvent can vary: for example, salt water.
Gaseous[edit]
Air can be more specifically described as a gaseous solution (oxygen and other gases dissolved in the major component, nitrogen). Since interactions between molecules play almost no role, dilute gases form rather trivial solutions. In part of the literature, they are not even classified as solutions.
Solids[edit]
In chemistry, a mixture is a substance containing two or more elements or compounds that are not covalently bound to each other and which retain their own chemical and physical identities; – a substance which has two or more constituent physical substances. Mixtures, in the broader sense, are two or more substances physically in the same place, but these are not chemically combined, and therefore ratios are not necessarily considered.[3]
Metrics[edit]
Homogeneous mixtures have the same proportions of the various components throughout a given sample (or multiple samples of different proportion), creating a consistent mixture. However, two homogeneous mixtures of the same pair of substances may differ widely from each other and can be homogenized to make a constant. Homogeneous mixtures always have the same composition. Mixtures can be characterized by being separable by mechanical means e.g. heat, filtration, gravitational sorting, etc.[3]
During the sampling of heterogeneous mixtures of particles, the variance of the sampling error is generally non-zero. Gy's sampling theory [4] quantitatively defines the heterogeneity of a particle as:
h i = ( c i − c batch ) m i c batch m aver . {\displaystyle h_{i}={\frac {(c_{i}-c_{\text{batch}})m_{i}}{c_{\text{batch}}m_{\text{aver}}}}.}
h i {\displaystyle h_{i}} , the heterogeneity of the i {\displaystyle i} th particle of the population
c i {\displaystyle c_{i}} , the mass concentration of the property of interest in the i {\displaystyle i} th particle of the population
c batch {\displaystyle c_{\text{batch}}} ,the mass concentration of the property of interest in the population
m i {\displaystyle m_{i}} , the mass of the i {\displaystyle i} th particle in the population
m aver {\displaystyle m_{\text{aver}}} , the average mass of a particle in the population
Homogenization[edit]
Main articles: Homogenization (chemistry) and Mixing (process engineering)
References[edit]
Jump up ^ Lew, Kristi (2009). "Homogeneous". Acids and Bases, Essential Chemistry. New York: Chelsea House Publishing. Online publisher: Science Online. Facts On File, Inc. ISBN 978-0-7910-9783-0. access date: 2010-01-01
Jump up ^ "Solution (chemistry)" (authors: William Ashworth and Charles E. Little)|chapter-format= requires |chapter-url= (help). Encyclopedia of Environmental Studies, New Edition. Online publisher:Science Online. Facts On File, Inc. 2001. access date: 2010-01-01
^ Jump up to: a b "Mixture" (authors: William Ashworth and Charles E. Little)|chapter-format= requires |chapter-url= (help). Encyclopedia of Environmental Studies. Online publisher:Science Online. Facts On File, Inc. 2001. access date: 2010-01-01
Jump up ^ Gy, P. (1979), Sampling of Particulate Materials: Theory and Practice, Elsevier: Amsterdam, 431 pp.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Homogeneous_and_heterogeneous_mixtures&oldid=815323946"
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Number line
From Wikipedia, the free encyclopedia
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This article is about the mathematical concept. For the printer's key, see Printer's key.
This article covers basic topics. For advanced topics, see Real line.
In basic mathematics, a number line is a picture of a graduated straight line that serves as abstraction for real numbers, denoted by R {\displaystyle \mathbb {R} } . Every point of a number line is assumed to correspond to a real number, and every real number to a point.[1]
The integers are often shown as specially-marked points evenly spaced on the line. Although this image only shows the integers from −9 to 9, the line includes all real numbers, continuing forever in each direction, and also numbers not marked that are between the integers. It is often used as an aid in teaching simple addition and subtraction, especially involving negative numbers.
In advanced mathematics, the expressions real number line, or real line are typically used to indicate the above-mentioned concept that every point on a straight line corresponds to a single real number, and vice versa.
Contents
[hide]
1 Drawing the number line
2 Comparing numbers
3 Portions of the number line
4 Extensions of the concept
4.1 Scaling
4.2 Combining number lines
5 See also
6 References
Drawing the number line[edit]
A number line is usually represented as being horizontal, but in a Cartesian coordinate plane the vertical axis (y-axis) is also a number line.[2] According to one convention, positive numbers always lie on the right side of zero, negative numbers always lie on the left side of zero, and arrowheads on both ends of the line are meant to suggest that the line continues indefinitely in the positive and negative directions. Another convention uses only one arrowhead which indicates the direction in which numbers grow.[2] The line continues indefinitely in the positive and negative directions according to the rules of geometry which define a line without endpoints as an infinite line, a line with one endpoint as a ray, and a line with two endpoints as a line segment.
Comparing numbers[edit]
If a particular number is farther to the right on the number line than is another number, then the first number is greater than the second (equivalently, the second is less than the first). The distance between them is the magnitude of their difference—that is, it measures the first number minus the second one, or equivalently the absolute value of the second number minus the first one. Taking this difference is the process of subtraction.
Thus, for example, the length of a line segment between 0 and some other number represents the magnitude of the latter number.
Two numbers can be added by "picking up" the length from 0 to one of the numbers, and putting it down again with the end that was 0 placed on top of the other number.
Two numbers can be multiplied as in this example: To multiply 5 × 3, note that this is the same as 5 + 5 + 5, so pick up the length from 0 to 5 and place it to the right of 5, and then pick up that length again and place it to the right of the previous result. This gives a result that is 3 combined lengths of 5 each; since the process ends at 15, we find that 5 × 3 = 15.
Division can be performed as in the following example: To divide 6 by 2—that is, to find out how many times 2 goes into 6—note that the length from 0 to 2 lies at the beginning of the length from 0 to 6; pick up the former length and put it down again to the right of its original position, with the end formerly at 0 now placed at 2, and then move the length to the right of its latest position again. This puts the right end of the length 2 at the right end of the length from 0 to 6. Since three lengths of 2 filled the length 6, 2 goes into 6 three times (that is, 6 ÷ 2 = 3).
The ordering on the number line: Greater elements are in direction of the arrow.
The difference 3-2=3+(-2) on the real number line.
The addition 1+2 on the real number line
The absolute difference.
The multiplication 2 times 1.5
The division 3÷2 on the real number line
Portions of the number line[edit]
The closed interval [a,b].
The section of the number line between two numbers is called an interval. If the section includes both numbers it is said to be a closed interval, while if it excludes both numbers it is called an open interval. If it includes one of the numbers but not the other one, it is called a half-open interval.
All the points extending forever in one direction from a particular point are together known as a ray. If the ray includes the particular point, it is a closed ray; otherwise it is an open ray.
Extensions of the concept[edit]
Scaling[edit]
A log-log plot of y = x (blue), y = x2 (green), and y = x3 (red).
Note the logarithmic scale markings on each of the axes, and that the log x and log y axes (where the logarithms are 0) are where x and y themselves are 1.
It is sometimes convenient to scale the numbers on the number line with a logarithmic scale, using scientific notation. For example, the number one inch to the right of 0 might be 1, then the number one inch farther to the right would be 101 (=10), then one inch to the right of that would be 102 (=100), then 1000, then 10,000, etc. This approach is useful, for example, in illustrating a sequence of events in the history of the universe or of evolution, or in comparing distances to various stars.
Combining number lines[edit]
A line drawn through the origin at right angles to the real number line can be used to represent the imaginary numbers. This line, called imaginary line, extends the number line to a complex number plane, with points representing complex numbers.
Alternatively, one real number line can be drawn horizontally to denote potential values of one real number, commonly called x, and another real number line can be drawn vertically to denote potential values of another real number, commonly called y. Together these lines form what is known as the Cartesian coordinate system, and any point in the Cartesian plane denotes the values of a pair of real numbers. Further, the Cartesian coordinate system can itself be extended by visualizing a third number line "coming out of the screen (or page)", measuring a third variable called z. Positive numbers are closer to the viewer's eyes than the screen is, while negative numbers are "behind the screen"; larger number are farther from the screen. Then any point in the three-dimensional space that we live in represents the values of a trio of real numbers.
See also[edit]
Chronology
Complex plane
Cuisenaire rods
Extended real number line
Hyperreal number line
Number form (neurological phenomenon)
The construction of a decimal number
References[edit]
Jump up ^ Stewart, James B.; Redlin, Lothar; Watson, Saleem (2008). College Algebra (5th ed.). Brooks Cole. pp. 13–19. ISBN 0-495-56521-0.
^ Jump up to: a b Introduction to the x,y-plane "Purplemath" Retrieved 2015-11-13
Retrieved from "https://en.wikipedia.org/w/index.php?title=Number_line&oldid=794501358"
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