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Carrie Brady
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Carrie Brady
Christie Clark as Carrie Brady
Days of Our Lives character
Portrayed by
Andrea Barber (1982–86)
Christie Clark (1986–91, 1992–2018)
Tracy Middendorf (1992)
Duration
1982–99
2005–06
2010–12
2017–18
First appearance
August 4, 1982
Last appearance
February 12, 2018
Created by
Margaret DePriest and Sheri Anderson
Introduced by
Betty Corday and Al Rabin (1982)
Ken Corday and Stephen Wyman (2005)
Ken Corday and Gary Tomlin (2010)
Ken Corday, Greg Meng, and Albert Alarr (2017)
Classification
Past; guest
Profile
Other names
Carrie Roberts
Carrie Reed
Caroline Anna Brady
Occupation
Attorney
Businesswoman
Model
Residence
Zurich, Switzerland
showFamily
Family
Brady
Father
Roman Brady
Mother
Anna DiMera
Stepfather
Tony DiMera (1983–85, 2008–09)
Stepmother
Marlena Evans (1983–94)
Kate Roberts (2003–05)
Half-brothers
Eric Brady
Rex Brady
Half-sisters
Sami Brady
Cassie Brady
Spouse
Austin Reed (1997–99, 2006–)
Lucas Roberts (2006)
Sons
Noah Reed[1][2]
Grandfathers
Shawn Brady
Grandmothers
Caroline Brady
Uncles
Bo Brady
Frankie Brady (adoptive)
Max Brady (adoptive)
Brett Fredericks
Aunts
Kimberly Brady
Kayla Brady
Nephews
Will Horton
Johnny DiMera
Nieces
Allie Horton
Sydney DiMera
Grace Brady (legal)
"Baby Bonnie" Lockhart
First cousins
Shawn-Douglas Brady
Chelsea Brady
Zack Brady
Ciara Brady
Andrew Donovan IV
Theresa Donovan
Stephanie Johnson
Joey Johnson
Carrie Brady is a fictional character from the NBC soap opera, Days of Our Lives, a long-running serial set in the fictional town of Salem. Created by head writer Margaret DePriest, the role of Carrie has been most notably portrayed by Christie Clark, who took over the role in 1986 as a teenager. Clark remained with the series off-and-on until leaving in October 2006, however, returned again as a series regular in September 2011 as part of a reboot of the series.[3] Her return was short lived as she left the show the following July. On September 15, 2016, it was announced that Clark would again reprise her portrayal as Carrie in 2017.[4] Carrie, a lawyer, also returned when her mother Anna was accused of a murder for several weeks in early 2018.
Throughout the show, Carrie has been involved in classic soap opera storylines. Her romantic pairing with Austin Reed (Austin Peck; Patrick Muldoon) led them to be labeled a supercouple. She is the daughter of Anna DiMera (Leann Hunley) and Roman Brady (at the time Wayne Northrop, later Josh Taylor), though as a child she was mainly raised by her step-parents, Marlena Evans (Deidre Hall) and John Black (Drake Hogestyn). Carrie is a member of the Brady family, one of the show's two core families. She has a long-standing rivalry with her sister Sami Brady (Alison Sweeney), as Sami has routinely tried to steal Austin from Carrie. In 2011, Carrie and Sami started to get along better, but their closeness fell apart when Carrie fell in love with Sami's then-husband, Rafe Hernandez, in 2012. The affection was returned by Rafe.
Clark's portrayal of Carrie has garnered attention for Clark who was nominated for Outstanding Younger Actress at the Daytime Emmy's in 1997, and again nominated in 1998. She won a Soap Opera Digest Award with co-star Austin Peck for Hottest Romance in 1997. She was also nominated for Best Young Actress in a Daytime Drama at the Young Artist Awards in 1990, and Nominated for Best Young Actress in a Daytime Drama at the Young Artist Awards in 1989.
Contents
1 Casting and creation
2 Storylines
2.1 1982–99
2.2 2005–06
2.3 2011–12
3 See also
4 References
5 External links
Casting and creation[edit]
The role of Carrie was originated by child actress Andrea Barber.[5] She was succeeded by Clark, who was only twelve years old at the time, on April 14, 1986.[6] Clark remained with the serial until January 14, 1991, when producers wanted to age the character.[6] She went on to appear on other shows such as General Hospital, and appeared in movies such as A Nightmare on Elm Street Part 2: Freddy's Revenge.[6] Meanwhile, the show briefly re-cast Carrie with actress Tracy Middendorf from January 10, 1992 to December 14, 1992. Producers asked Clark to return on December 18, 1992. In June 1999, it was reported by Soap Opera Digest that Clark would exit that fall,[7] and she left on November 19, 1999.[6]
"Definitely. People have watched me grow up as Carrie on DAYS. I also feel like I've never wanted to play Carrie more. So yes, I think there is more story to tell. I just don't think it's over"
— Clark on her character Carrie.[8]
In 2005, it was announced that Carrie and Austin would return to the series. Austin, played by Austin Peck returned in July 2005 while Clark reprised the role as Carrie on December 2, 2005. Clark left the show a year later on October 26, 2006.[6] The matriarch of the show, Alice Horton (Frances Reid), died in 2010. The show planned a two-week tribute and brought back longtime fan favorites to the show to pay their respects. Clark was included in the tribute and returned from June 23 to 28, 2010 as Carrie.[9] She popped up again on a phone call with Sami on February 9, 2011. In 2011, it was announced that Carrie and Austin (Austin played by Patrick Muldoon) and several other characters would return permanently. Clark and Muldoon appeared on September 26, 2011. Their return was brought about to please fans and in an attempt to lure old fans of the show back.[3][10] Clark and Muldoon appeared on September 26, 2011.
They remained with the show for less than a year as producers again were re-directing the show. Carrie and Austin last aired on July 24, 2012.[11] On September 15, 2016, it was announced that Clark would return as Carrie, alongside Peck as Austin, on January 11, 2017.[4] Clark and Peck departed on February 16, 2017.[12]
"Luckily, one of my best friends in San Francisco is a top defense attorney. I went to court with her and saw that it was actually something I could do. These people were just like me."
— Clark on being told Carrie would be returning as a lawyer.[8]
The role has garnered attention for Clark who was nominated for Outstanding Younger Actress at the Daytime Emmy's in 1997, and again in 1998.[6] She won a Soap Opera Digest Award with co-star Austin Peck for Hottest Romance in 1997. She was also nominated for Best Young Actress in a Daytime Drama at the Young Artist Awards in 1990, and Nominated for Best Young Actress in a Daytime Drama at the Young Artist Awards in 1989.[6]
Storylines[edit]
1982–99[edit]
Carrie is the daughter of Roman Brady and Anna Fredericks. Her mother leaves Roman while pregnant with Carrie; Roman doesn't know his daughter exists until Anna returns with a four-year-old Carrie and leaves the girl in her father's and Marlena's care.[13] As a child, Carrie is at one point kidnapped and held captive by Stefano DiMera, who has kidnapped many members of the Brady family at various points. She also nearly drowns during a storm and falls into a coma after a car crash.[13] When Austin Reed moved to Salem in July 1992, Carrie is attracted to him and also becomes the roommate and close friend of Austin's sister, Billie Reed. Carrie and Austin begin dating, but Sami's interfering and Austin's career as a prizefighter causes some strain in their relationship. Carrie becomes a finalist in Bella magazine's "Face of the 90’s" modeling competition.[13]
In December 1992, when Austin fails to throw a fight, Carrie accidentally becomes the victim of an acid attack that was intended for Austin. The incident causes serious scarring to her face. Her father Roman, who does not like Austin, tells Carrie that if she would break up with him, he will not pursue any charges against Austin stemming from his involvement with the underworld of boxing. Carrie also feels that Austin will no longer love her, due to her scars. As a result of both factors, Carrie pushes Austin out of her life. This gives her sister Sami, who was attracted to Austin, an opportunity to pursue a relationship with him. Sami Brady teams up with Lucas Roberts. Lucas is Austin's half-brother. Lucas is attracted to Carrie and Sami is attracted to Austin. Sami and Lucas scheme to keep Carrie and Austin from reuniting, so Lucas can be with Carrie and Sami can be with Austin. Meanwhile, Carrie has successful plastic surgery on her face and is willing to give her relationship with Austin another chance. During a romantic ski weekend in February 1994, Carrie loses her virginity to Austin, after almost being raped by Sami's boyfriend Alan, who goes on to rape Sami instead.
Although Carrie and Austin are a couple, Sami continues to try to break them up and pursue Austin herself. Sami even twistedly blames Carrie for the rape Sami suffered from Alan. In March 1995, Sami drugs Austin and in his delusional state, he believes Carrie is there with him (although it's Sami) and Sami sleeps with him. Nevertheless, Carrie and Austin eventually stay together and begin planning their wedding. Sami is able to stop the ceremony by revealing that she is pregnant with Austin’s child. At the time, Sami believes that Austin was the father, but she later learns that Lucas was the true father. When she finds this out, she continues to conceal the truth from everyone, even Lucas, for several years. Since Austin believes the baby is his, he feels he had no choice but to build a life with Sami and baby Will instead of marrying Carrie. However, Austin continues to be in love with Carrie. Unfortunately, the kidnapping of Will by Sami's deranged neighbor caused a situation where Sami and Austin, in order to regain custody of Will from the French couple to whom he had been sold, are forced to marry in France to satisfy the authorities there.[13] Carrie, meanwhile, becomes good friends with Dr. Mike Horton. She also uncovered information that Austin is not Will's father. When Austin and Sami plan to wed in America, this time Carrie is able to interrupt the wedding ceremony by revealing that Lucas, not Austin, is Will's father, and that Sami knew it all along. Austin marries Carrie instead. Some time later, Austin becomes increasingly concerned about baby Will's well-being as Will's father Lucas has a drinking problem. Austin begins spending more time around Will, which means more time around Sami; much to Sami's delight and Carrie's dismay. With Austin not being home as much as he spends time with Will, Carrie begins working more hours at the hospital in which she is employed. Over time, Dr Mike Horton develops feelings for her and they slowly grow closer together. Carrie and Mike eventually have an affair, and Austin and Carrie divorce. Austin leaves Salem. Carrie accepts Mike's marriage proposal, and they move to Israel.
2005–06[edit]
A few years later, Carrie returns to the U.S. and settles in Los Angeles where she is CEO of a company called High Style. Carrie tells no one she is back in America but she is discovered by Lucas, who was attempting to take over High Style, not knowing it was headed by Carrie. He halts his takeover attempt and convinces Carrie to return to Salem as his feelings for her are reignited. Carrie returns to town and is surprised to see Austin in Salem as well. Although things look positive for Austin and Carrie's future, her sister Sami sees otherwise and, once again, schemes to break them apart and keep Austin for herself. First she is able to have Austin and his company buy out Carrie's company High Style, Austin not being aware that Carrie has any association with the small business. Carrie, feeling betrayed that Austin took over her company, grows closer to Austin's half brother Lucas. Sami, meanwhile, catches Dr. Lexie Carver having an affair with detective Tek Kramer. To ensure that Carrie will no longer want to be with Austin, Sami threatens to tell Lexie's husband Abe about Tek, unless Lexie helps her. Doctor Lexie Carver is blackmailed into telling Carrie untrue information: that Austin and Carrie both share very rare genetic markers and if they ever had children, the child will suffer extreme birth defects that could result in the baby's death.
Due to this bogus news from Lexie, Carrie decides it's best to move on with Lucas. Since she and Austin could never have children, she becomes engaged to Lucas.[14] She soon believes she is pregnant thanks to a false pregnancy test, leading her and Lucas to rush their wedding date. Carrie finds out she is actually not pregnant and still has feelings for Austin. Carrie and Austin make love on the roof of their apartment building, although Carrie is engaged to Lucas and Austin is engaged to Sami. Austin does not marry Sami and Carrie shows signs of relief. Prior to the ceremony, Carrie expresses to Marlena that she is still in love with Austin. Although she cares about Lucas, she is not in love with him. After she secretly meets with Austin on the roof of their building to make love, Lucas, with the help of new neighbor E.J., catches them. After a bitter tirade from Lucas, Carrie is kicked out of the apartment she and Lucas share. Soon after, Carrie hands Lucas annulment papers and he signs them. However, minutes later, the Gloved Hand slides a note under Sami's apartment door, and Lucas, Carrie, and Austin discover the truth behind Sami's blackmailing of Dr. Lexie Carver to keep Carrie and Austin apart. After Sami finally admits the truth about her misdeeds, a furious Carrie attacks her. The two sisters fight, and afterwards Carrie tells her half-sister that she will tell everyone in Salem what Sami did.[13]
Later, Carrie runs into Dr. Lexie Carver and reveals to her that she knows all about the bogus information Lexie was giving her due to Sami's blackmailing. Lexie tries to plead her case and apologize, but Carrie doesn't care and promises to pay Lexie back by reporting her to the hospital board and the AMA, upon which Lexie is subsequently fired. During dinner one night, Austin suggests to Carrie that they move to Switzerland, where he can work at the Mythic Communications division there and Carrie can reclaim High Style, her former company. Carrie agrees on one condition—that they get married first.[14] The two immediately go to the Justice of the Peace with their signed marriage license and get married, but not before getting briefly interrupted by a drunk Sami; and Carrie disowning her. After saying their goodbyes to John, Kate, Marlena and Roman, Austin and Carrie take a plane and leave Salem.
2011–12[edit]
In September 2011, Carrie and Austin returned to Salem for the tribute to the Horton Center. Carrie mentioned that she had become a lawyer after her company Highstyle, which was division of the now defunct Mythic Communications, was forced to close down. Carrie then takes a job, as John's defense attorney. She works with brother-in-law Rafe to help find evidence that can exonerate John from the charges against him. In December, John is set free and is cleared of all charges. Carrie and Rafe continue to work together, eventually opening their own law/detective firm. Carrie and Rafe grow closer and are caught kissing by Sami and Austin. Carrie tells Rafe that she is in love with him and had separated from Austin, only to reunite following Abigail's confession that she did not truly sleep with Austin. Despite their reunion, Carrie can't shake her feelings of love towards Rafe.
After the lives of Marlena, John, Hope and Bo are threatened by Stefano, Carrie and Rafe are become trapped inside of the safe house, which is alarmed with a bomb. While inside, she and Rafe confess their love for each other, nearly having sex, Rafe admits that EJ is the father of Nicole's child, not him. After they are safe from harm, Carrie begins to feel weak and sick. She confesses to Marlena that she was late, which arose suspicions of a possible pregnancy. After taking a take home test, it confirms her suspicion. A suspicious Austin sees the test and assumes it is his baby. Austin is thrilled about the news, but Carrie isn't as much since she loves Rafe. Rafe tells Carrie that she belongs with Austin and her child, and that they won't work out. Carrie decides to return to Switzerland with Austin, and says an emotional farewell to Rafe. But before leaving, Carrie urges Rafe to reconcile with Sami, because Carrie knows that they still love each other deep down.
In 2014, Carrie supports Sami and Kate's decision to control DiMera's company.
See also[edit]
Brady family
References[edit]
Jump up ^ Ken Corday, Albert Alarr and Greg Meng (Executive producers); Dena Higley and Ryan Quan (Head writers) (January 23, 2017). Days of Our Lives. Season 52. Episode 13008. Days of Our Lives Transcript Monday 1/23/17. NBC.
Jump up ^ Dena Higley and Ryan Quan (Head writers); Sonia Blangiardo (Director); David Cherrill (Script writer) (January 23, 2017). Days of Our Lives. Season 52. Episode 13008. NBC.
^ Jump up to: a b Logan, Michael (July 8, 2011). "Patrick Muldoon and Christie Clark Return to Days of our Lives". TV Guide. Retrieved 24 July 2007.
^ Jump up to: a b SOD (September 15, 2016). "EXCLUSIVE! DAYS STARS RETURN!". Soap Opera Digest. United States. American Media, Inc. Retrieved September 15, 2016.
Jump up ^ "Andre Barber Bio on TV.com". TV.com. Archived from the original on 2010-07-04. Retrieved 2010-06-05.
^ Jump up to: a b c d e f g "Christie Clark Bio on TV.com". TV.com. Retrieved 2010-06-05.
Jump up ^ "Comings and Goings". Soap Opera Digest. June 29, 1999. p. 12.
^ Jump up to: a b "Christie, Clear". Soap Opera Digest. 37. June 11, 2012. p. 41.
Jump up ^ "Atteendees Named for Days Memorial". SoapNet. Archived from the original on 2010-06-08. Retrieved 2010-06-05.
Jump up ^ "DAYS Stars Return". Soap Opera Digest. Retrieved 2010-06-05.
Jump up ^ Soap Opera Digest (2012-04-10). "DAYS Bloodbath". Retrieved April 10, 2012.
Jump up ^ "Who's Going". Comings and Goings. Soap Opera Digest. American Media, Inc. 42 (8): 12–13. February 20, 2017.
^ Jump up to: a b c d e "Character Bio: Days Of Our Lives". Soapscafe.com. 2006-05-14. Archived from the original on 2011-07-16. Retrieved 2010-06-05.
^ Jump up to: a b "Carrie Brady Bio". TV Fanatic. 2009-11-06. Retrieved 2010-06-05.
External links[edit]
Carrie at soapcentral.com
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Susan Banks
Belle Black
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John Black
Ciara Brady
Claire Brady
Eric Brady
Hope Brady
Kayla Brady
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Deimos Kiriakis
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Paige Larson
Bonnie Lockhart
Mimi Lockhart
Patrick Lockhart
Carly Manning
Alex Marshall
Serena Mason
Colin Murphy
Tori Narita
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Austin Reed
Billie Reed
Curtis Reed
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45th parallel north
From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
45°
45th parallel north
Map all coordinates using: OpenStreetMap
Download coordinates as: KML · GPX
The 45th parallel north is a circle of latitude that is 45 degrees north of Earth's equator. It crosses Europe, Asia, the Pacific Ocean, North America, and the Atlantic Ocean. The 45th parallel north is often called the halfway point between the Equator and the North Pole, but the true halfway point is actually 16.2 km (10.1 mi) north of the 45th parallel because Earth is an oblate spheroid; that is, it bulges at the equator and is flattened at the poles.[1]
At this latitude, the Sun is visible for 15 hours 37 minutes during the summer solstice, and 8 hours 46 minutes during the winter solstice. The midday Sun stands 21.6° above the southern horizon at the December solstice, 68.4° at the June solstice, and exactly 45.0° at either equinox.[2]
Contents
1 Around the world
2 North America
3 Asia
4 Europe
5 See also
6 References
7 External links
Around the world[edit]
Starting at the Prime Meridian and heading eastwards, the parallel 45° north passes through:
Co-ordinates
Country, territory
or sea
Notes
45°0′N 0°0′E / 45.000°N 0.000°E / 45.000; 0.000 (Prime Meridian)
France
Aquitaine
Midi-Pyrénées
Limousin
Auvergne
Rhône-Alpes – passing just south of Grenoble
Provence-Alpes-Côte d'Azur
45°0′N 6°44′E / 45.000°N 6.733°E / 45.000; 6.733 (Italy)
Italy
Piedmont – passing just south of Turin
Lombardy - passing by Voghera
Emilia-Romagna – passing just south of Piacenza
Lombardy
Veneto – passing just south of Rovigo
45°0′N 12°27′E / 45.000°N 12.450°E / 45.000; 12.450 (Adriatic Sea)
Adriatic Sea
Gulf of Venice
45°0′N 13°44′E / 45.000°N 13.733°E / 45.000; 13.733 (Croatia)
Croatia
Istrian Peninsula, islands of Cres and Krk, and the mainland again
45°0′N 15°46′E / 45.000°N 15.767°E / 45.000; 15.767 (Bosnia and Herzegovina)
Bosnia and Herzegovina
45°0′N 18°44′E / 45.000°N 18.733°E / 45.000; 18.733 (Croatia)
Croatia
45°0′N 19°6′E / 45.000°N 19.100°E / 45.000; 19.100 (Serbia)
Serbia
Passing through the centre of Ruma and through northern part of Sremska Mitrovica
Passing through the northern edge of Stara Pazova, 30 kilometres NW of Belgrade
45°0′N 21°24′E / 45.000°N 21.400°E / 45.000; 21.400 (Romania)
Romania
Passing just north of Ploieşti and just south of Târgu Jiu
45°0′N 29°39′E / 45.000°N 29.650°E / 45.000; 29.650 (Black Sea)
Black Sea
45°0′N 33°36′E / 45.000°N 33.600°E / 45.000; 33.600 (Ukraine)
Ukraine
Crimea (claimed and controlled by Russia) – passing just north of Simferopol, and just south of Feodosiya
45°0′N 35°24′E / 45.000°N 35.400°E / 45.000; 35.400 (Black Sea)
Black Sea
45°0′N 37°13′E / 45.000°N 37.217°E / 45.000; 37.217 (Russia)
Russia
Passing just south of Krasnodar, and just south of Stavropol
45°0′N 47°15′E / 45.000°N 47.250°E / 45.000; 47.250 (Caspian Sea)
Caspian Sea
45°0′N 51°4′E / 45.000°N 51.067°E / 45.000; 51.067 (Kazakhstan)
Kazakhstan
Mangystau Province
45°0′N 56°0′E / 45.000°N 56.000°E / 45.000; 56.000 (Uzbekistan)
Uzbekistan
Karakalpakstan (autonomous republic) – including Vozrozhdeniya Island in the Aral Sea
45°0′N 59°50′E / 45.000°N 59.833°E / 45.000; 59.833 (Kazakhstan)
Kazakhstan
Kyzylorda Province
South Kazakhstan Province
Zhambyl Province
Almaty Province
45°0′N 80°2′E / 45.000°N 80.033°E / 45.000; 80.033 (China)
People's Republic of China
Xinjiang
45°0′N 93°13′E / 45.000°N 93.217°E / 45.000; 93.217 (Mongolia)
Mongolia
Govi-Altai Province
Bayankhongor Province
Övörkhangai Province
Dundgovi Province
Dornogovi Province
45°0′N 111°46′E / 45.000°N 111.767°E / 45.000; 111.767 (China)
People's Republic of China
Inner Mongolia
45°0′N 112°31′E / 45.000°N 112.517°E / 45.000; 112.517 (Mongolia)
Mongolia
Sükhbaatar Province
45°0′N 114°7′E / 45.000°N 114.117°E / 45.000; 114.117 (China)
People's Republic of China
Inner Mongolia
Jilin
Heilongjiang
45°0′N 131°30′E / 45.000°N 131.500°E / 45.000; 131.500 (Russia)
Russia
Primorsky Krai
45°0′N 136°37′E / 45.000°N 136.617°E / 45.000; 136.617 (Sea of Japan)
Sea of Japan
Passing just south of Rishiri Island, Japan
45°0′N 141°41′E / 45.000°N 141.683°E / 45.000; 141.683 (Japan)
Japan
Hokkaidō
45°0′N 142°32′E / 45.000°N 142.533°E / 45.000; 142.533 (Sea of Okhotsk)
Sea of Okhotsk
45°0′N 147°31′E / 45.000°N 147.517°E / 45.000; 147.517 (Kuril Islands)
Kuril Islands
Island of Iturup, administered by Russia, claimed by Japan
45°0′N 147°59′E / 45.000°N 147.983°E / 45.000; 147.983 (Pacific Ocean)
Pacific Ocean
45°0′N 124°1′W / 45.000°N 124.017°W / 45.000; -124.017 (United States)
United States
Oregon
Idaho
Montana
Montana / Wyoming border (approximate)
South Dakota
Minnesota – passing through Minneapolis
Wisconsin – passing through a 45×90 point (halfway between the North Pole and the Equator, and halfway between the Prime Meridian and the 180th meridian)
45°0′N 87°37′W / 45.000°N 87.617°W / 45.000; -87.617 (Lake Michigan)
Lake Michigan
Green Bay – territorial waters of the United States
45°0′N 87°21′W / 45.000°N 87.350°W / 45.000; -87.350 (United States)
United States
Wisconsin – Door Peninsula
45°0′N 87°9′W / 45.000°N 87.150°W / 45.000; -87.150 (Lake Michigan)
Lake Michigan
Territorial waters of the United States
45°0′N 86°9′W / 45.000°N 86.150°W / 45.000; -86.150 (United States)
United States
Michigan – South Manitou Island
45°0′N 86°5′W / 45.000°N 86.083°W / 45.000; -86.083 (Lake Michigan)
Lake Michigan
Territorial waters of the United States
45°0′N 85°46′W / 45.000°N 85.767°W / 45.000; -85.767 (United States)
United States
Michigan – Leelanau Peninsula (Leelanau County)
45°0′N 86°5′W / 45.000°N 86.083°W / 45.000; -86.083 (Lake Michigan)
Lake Michigan
Grand Traverse Bay – territorial waters of the United States
45°0′N 85°47′W / 45.000°N 85.783°W / 45.000; -85.783 (United States)
United States
Michigan (Antrim County, Otsego County, Montmorency County, and Alpena County)
45°0′N 83°26′W / 45.000°N 83.433°W / 45.000; -83.433 (Lake Huron)
Lake Huron
Territorial waters of the United States and Canada
45°0′N 81°27′W / 45.000°N 81.450°W / 45.000; -81.450 (Canada)
Canada
Ontario – Bruce Peninsula
45°0′N 81°11′W / 45.000°N 81.183°W / 45.000; -81.183 (Lake Huron)
Lake Huron
Georgian Bay – territorial waters of Canada
45°0′N 79°59′W / 45.000°N 79.983°W / 45.000; -79.983 (Canada)
Canada
Ontario
45°0′N 74°54′W / 45.000°N 74.900°W / 45.000; -74.900 (United States)
United States
New York – for about 10 km
45°0′N 74°46′W / 45.000°N 74.767°W / 45.000; -74.767 (Canada)
Canada
Ontario – Cornwall Island
Quebec – running just north of the border with New York, United States
45°0′N 73°54′W / 45.000°N 73.900°W / 45.000; -73.900 (United States)
United States
New York – running just south of the border with Quebec, Canada
Vermont – running just south of the border with Quebec, Canada
New Hampshire
Maine
45°0′N 67°4′W / 45.000°N 67.067°W / 45.000; -67.067 (Passamaquoddy Bay)
Passamaquoddy Bay
45°0′N 67°0′W / 45.000°N 67.000°W / 45.000; -67.000 (Canada)
Canada
Deer Island, New Brunswick
45°0′N 66°57′W / 45.000°N 66.950°W / 45.000; -66.950 (Bay of Fundy)
Bay of Fundy
Territorial waters of Canada
45°0′N 65°10′W / 45.000°N 65.167°W / 45.000; -65.167 (Canada)
Canada
Nova Scotia
45°0′N 61°57′W / 45.000°N 61.950°W / 45.000; -61.950 (Atlantic Ocean)
Atlantic Ocean
45°0′N 1°12′W / 45.000°N 1.200°W / 45.000; -1.200 (France)
France
Aquitaine – passing just north of Bordeaux
North America[edit]
The 45th parallel approximately defines most of the border between Montana and Wyoming in the United States.
Historical marker marking the 45th parallel in New Hampshire.
The 45th parallel approximately defines part of the international border between Quebec, Canada, and New York and Vermont, United States.
Marker on Interstate 5 near Keizer, Oregon.
45th parallel marker in Roseville, Minnesota
In North America, the 45th parallel forms some boundaries, and passes through the U.S. states of Oregon, Idaho, South Dakota, Minnesota, Wisconsin, Michigan, New York, Vermont, New Hampshire and Maine and through the Canadian provinces of Ontario, Quebec, New Brunswick and Nova Scotia.
All of mainland New Brunswick lies north of the 45th parallel. The southernmost point in mainland New Brunswick, just north of the 45th parallel, is Greens Point, approximately 90 kilometres (56 mi) west of Saint John. Approximately two-thirds of Deer Island, plus all of Campobello and Grand Manan islands, are south of the 45th parallel.
The parallel roughly bisects mainland Nova Scotia. Halifax is approximately 40 km (25 mi) south of the parallel.
The 45th parallel roughly marks the national border between the United States and Canada between the St. Lawrence and Connecticut rivers (the northern borders of the U.S. states of New York and Vermont with the Canadian province of Quebec), where the parallel is sometimes called the "Canada line". The actual boundary of Vermont lies approximately 1 kilometre (3,300 ft) north of the parallel due to an error in the 1772 survey.[3]
In Michigan, the Old Mission Peninsula in Grand Traverse Bay ends just shy of the 45th parallel. Many guidebooks and signs at the Mission Point Lighthouse describe it as being halfway between the equator and north pole. When the Grand Traverse Bay recedes below normal level, it is possible to walk out to the exact line.
Farther west, the line roughly bisects the metropolitan area of Minneapolis-St. Paul. In the Western United States, the parallel passes through the Great Plains and Rocky Mountains, intersecting the Pacific coast in Oregon.
The 45th parallel makes up most of the boundary between Montana and Wyoming.
Throughout the United States the parallel is marked in many places on highways by a sign proclaiming that the location is halfway between the North Pole and the equator.[4][5]
Asia[edit]
Continuing west, the parallel passes over vast stretches of the North Pacific Ocean before passing through Rishiri-Rebun-Sarobetsu National Park and the adjacent town of Horonobe on the northern tip of Hokkaidō, the northernmost of Japan's main islands. It continues through the northern part of the Sea of Japan, entering the Asian mainland on the coast of Primorsky Krai in Russia, north of Vladivostok.
At Khanka Lake it enters northeast China, cutting across Heilongjiang and continuing through part of Jilin and eastern Inner Mongolia. Transecting southern Mongolia it passes through the provinces of Sükhbaatar, Dornogovi (and its capital Sainshand), Dundgovi, Övörkhangai, Bayankhongor, Govi-Altai, and Khovd. In northwest China it passes through the Ili Kazakh Autonomous Prefecture in Xinjiang and the oil city of Karamay.
The parallel passes through southern Kazakhstan, intersecting the city of Burylbaytal at the southern tip of Lake Balkhash and the city of Qyzylorda further west. At the border with Uzbekistan it bisects the Aral Sea and its toxic Vozrozhdeniya Island peninsula, site of an abandoned Soviet bioweapons laboratory. It skirts the northern edge of the Ustyurt Plateau, entering back into Kazakhstan before cutting across the northern tip of the Caspian Sea and into Europe and Russia.
Europe[edit]
Sundial at the 45th parallel in Senj, Croatia.
In Europe the 45th parallel stretches between the Caspian Sea coast of the Russian Caucasus in the east and Bay of Biscay coast of France in the west. In Russia it runs from the west coast of the Caspian Sea to the east coast of the Black Sea, through the Republic of Kalmykia, Stavropol Krai and its capital Stavropol, and Krasnodar Krai and its capital Krasnodar. In Ukraine it crosses the Crimea and its capital Simferopol.
Further west it passes through the Balkans: Romania (just north of Ploieşti, and through Târgu Jiu), the Serbian autonomous province of Vojvodina, the eastern tip of Croatia, the northern edge of Bosnia and Herzegovina, and a section of Adriatic Croatia. The capital city of Serbia – Belgrade is just south of the parallel.
In northern Italy it parallels the river Po, near Rovigo, passing by Voghera, then just south of Turin before passing into France in the Cottian Alps.
In France, it crosses the river Rhône at Pont-de-l'Isère, just north of Valence, Drôme and through Grenoble. It then continues across the Massif Central and into the Aquitaine region. The city of Bordeaux is just south of the parallel.
See also[edit]
45th parallel south
44th parallel north
46th parallel north
45×90 points
References[edit]
Jump up ^ "The Half-Way to the Pole Line". 2 May 2006. Retrieved 16 July 2018.
Jump up ^ "Duration of Daylight/Darkness Table for One Year". aa.usno.navy.mil. Retrieved 16 July 2018.
Jump up ^ "The Canada-U.S. Border - Townships Heritage WebMagazine". townshipsheritage.com. Retrieved 19 April 2018.
Jump up ^ "Concord Monitor: "New Hampshire is halfway between the equator and the North Pole – just don't ask where"". concordmonitor.com. Retrieved 19 April 2018.
Jump up ^ Brown, K. Williams (July 10, 2011). "The beauty of halfwayism". Statesman Journal. Salem, Oregon. p. 5D. Retrieved July 9, 2018 – via Newspapers.com.
External links[edit]
Wikimedia Commons has media related to 45th parallel north.
The 45th Parallel at the Minnesota Museum of the Mississippi and Other Natural Wonders
From the Minas Astronomy Group: Midway from the Equator to the North Pole
45th Parallel Markers
Degree Confluence Project
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Multiplicative inverse
From Wikipedia, the free encyclopedia
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The reciprocal function: y = 1/x. For every x except 0, y represents its multiplicative inverse. The graph forms a rectangular hyperbola.
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the function f(x) that maps x to 1/x, is one of the simplest examples of a function which is its own inverse (an involution).
The term reciprocal was in common use at least as far back as the third edition of Encyclopædia Britannica (1797) to describe two numbers whose product is 1; geometrical quantities in inverse proportion are described as reciprocall in a 1570 translation of Euclid's Elements.[1]
In the phrase multiplicative inverse, the qualifier multiplicative is often omitted and then tacitly understood (in contrast to the additive inverse). Multiplicative inverses can be defined over many mathematical domains as well as numbers. In these cases it can happen that ab ≠ ba; then "inverse" typically implies that an element is both a left and right inverse.
The notation f −1 is sometimes also used for the inverse function of the function f, which is not in general equal to the multiplicative inverse. For example, the multiplicative inverse 1/(sin x) = (sin x)−1 is the cosecant of x, and not the inverse sine of x denoted by sin−1 x or arcsin x. Only for linear maps are they strongly related (see below). The terminology difference reciprocal versus inverse is not sufficient to make this distinction, since many authors prefer the opposite naming convention, probably for historical reasons (for example in French, the inverse function is preferably called bijection réciproque).
Contents
[hide]
1 Examples and counterexamples
2 Complex numbers
3 Calculus
4 Algorithms
5 Reciprocals of irrational numbers
6 Further remarks
7 Applications
8 See also
9 Notes
10 References
Examples and counterexamples[edit]
In the real numbers, zero does not have a reciprocal because no real number multiplied by 0 produces 1 (the product of any number with zero is zero). With the exception of zero, reciprocals of every real number are real, reciprocals of every rational number are rational, and reciprocals of every complex number are complex. The property that every element other than zero has a multiplicative inverse is part of the definition of a field, of which these are all examples. On the other hand, no integer other than 1 and −1 has an integer reciprocal, and so the integers are not a field.
In modular arithmetic, the modular multiplicative inverse of a is also defined: it is the number x such that ax ≡ 1 (mod n). This multiplicative inverse exists if and only if a and n are coprime. For example, the inverse of 3 modulo 11 is 4 because 4 · 3 ≡ 1 (mod 11). The extended Euclidean algorithm may be used to compute it.
The sedenions are an algebra in which every nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, i.e. nonzero elements x, y such that xy = 0.
A square matrix has an inverse if and only if its determinant has an inverse in the coefficient ring. The linear map that has the matrix A−1 with respect to some base is then the reciprocal function of the map having A as matrix in the same base. Thus, the two distinct notions of the inverse of a function are strongly related in this case, while they must be carefully distinguished in the general case (as noted above).
The trigonometric functions are related by the reciprocal identity: the cotangent is the reciprocal of the tangent; the secant is the reciprocal of the cosine; the cosecant is the reciprocal of the sine.
A ring in which every nonzero element has a multiplicative inverse is a division ring; likewise an algebra in which this holds is a division algebra.
Complex numbers[edit]
As mentioned above, the reciprocal of every nonzero complex number z = a + bi is complex. It can be found by multiplying both top and bottom of 1/z by its complex conjugate z ¯ = a − b i {\displaystyle {\bar {z}}=a-bi} and using the property that z z ¯ = ∥ z ∥ 2 {\displaystyle z{\bar {z}}=\|z\|^{2}} , the absolute value of z squared, which is the real number a2 + b2:
1 z = z ¯ z z ¯ = z ¯ ∥ z ∥ 2 = a − b i a 2 + b 2 = a a 2 + b 2 − b a 2 + b 2 i . {\displaystyle {\frac {1}{z}}={\frac {\bar {z}}{z{\bar {z}}}}={\frac {\bar {z}}{\|z\|^{2}}}={\frac {a-bi}{a^{2}+b^{2}}}={\frac {a}{a^{2}+b^{2}}}-{\frac {b}{a^{2}+b^{2}}}i.}
In particular, if ||z||=1 (z has unit magnitude), then 1 / z = z ¯ {\displaystyle 1/z={\bar {z}}} . Consequently, the imaginary units, ±i, have additive inverse equal to multiplicative inverse, and are the only complex numbers with this property. For example, additive and multiplicative inverses of i are −(i) = −i and 1/i = −i, respectively.
For a complex number in polar form z = r(cos φ + i sin φ), the reciprocal simply takes the reciprocal of the magnitude and the negative of the angle:
1 z = 1 r ( cos  ( − φ ) + i sin  ( − φ ) ) . {\displaystyle {\frac {1}{z}}={\frac {1}{r}}\left(\cos(-\varphi )+i\sin(-\varphi )\right).}
Geometric intuition for the integral of 1/x. The three integrals from 1 to 2, from 2 to 4, and from 4 to 8 are all equal. Each region is the previous region scaled vertically down by 50%, then horizontally by 200%. Extending this, the integral from 1 to 2k is k times the integral from 1 to 2, just as ln 2k = k ln 2.
Calculus[edit]
In real calculus, the derivative of 1/x = x−1 is given by the power rule with the power −1:
d d x x − 1 = ( − 1 ) x ( − 1 ) − 1 = − x − 2 = − 1 x 2 . {\displaystyle {\frac {d}{dx}}x^{-1}=(-1)x^{(-1)-1}=-x^{-2}=-{\frac {1}{x^{2}}}.}
The power rule for integrals (Cavalieri's quadrature formula) cannot be used to compute the integral of 1/x, because doing so would result in division by 0:
∫ 1 x d x = x 0 0 + C {\displaystyle \int {\frac {1}{x}}\,dx={\frac {x^{0}}{0}}\ +C}
Instead the integral is given by:
∫ 1 a 1 x d x = ln  a , {\displaystyle \int _{1}^{a}{\frac {1}{x}}\,dx=\ln a,}
∫ 1 x d x = ln  x + C . {\displaystyle \int {\frac {1}{x}}\,dx=\ln x+C.}
where ln is the natural logarithm. To show this, note that d d x e x = e x {\displaystyle {\frac {d}{dx}}e^{x}=e^{x}} , so if y = e x {\displaystyle y=e^{x}} and x = ln  y {\displaystyle x=\ln y} , we have:[2]
d y d x = y ⇒ d y y = d x ⇒ ∫ 1 y d y = ∫ 1 d x ⇒ ∫ 1 y d y = x + C = ln  y + C . {\displaystyle {\frac {dy}{dx}}=y\quad \Rightarrow \quad {\frac {dy}{y}}=dx\quad \Rightarrow \quad \int {\frac {1}{y}}\,dy=\int 1\,dx\quad \Rightarrow \quad \int {\frac {1}{y}}\,dy=x+C=\ln y+C.}
Algorithms[edit]
The reciprocal may be computed by hand with the use of long division.
Computing the reciprocal is important in many division algorithms, since the quotient a/b can be computed by first computing 1/b and then multiplying it by a. Noting that f ( x ) = 1 / x − b {\displaystyle f(x)=1/x-b} has a zero at x = 1/b, Newton's method can find that zero, starting with a guess x 0 {\displaystyle x_{0}} and iterating using the rule:
x n + 1 = x n − f ( x n ) f ′ ( x n ) = x n − 1 / x n − b − 1 / x n 2 = 2 x n − b x n 2 = x n ( 2 − b x n ) . {\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}=x_{n}-{\frac {1/x_{n}-b}{-1/x_{n}^{2}}}=2x_{n}-bx_{n}^{2}=x_{n}(2-bx_{n}).}
This continues until the desired precision is reached. For example, suppose we wish to compute 1/17 ≈ 0.0588 with 3 digits of precision. Taking x0 = 0.1, the following sequence is produced:
x1 = 0.1(2 − 17 × 0.1) = 0.03
x2 = 0.03(2 − 17 × 0.03) = 0.0447
x3 = 0.0447(2 − 17 × 0.0447) ≈ 0.0554
x4 = 0.0554(2 − 17 × 0.0554) ≈ 0.0586
x5 = 0.0586(2 − 17 × 0.0586) ≈ 0.0588
A typical initial guess can be found by rounding b to a nearby power of 2, then using bit shifts to compute its reciprocal.
In constructive mathematics, for a real number x to have a reciprocal, it is not sufficient that x ≠ 0. There must instead be given a rational number r such that 0 < r < |x|. In terms of the approximation algorithm described above, this is needed to prove that the change in y will eventually become arbitrarily small.
Graph of f(x) = xx showing the minimum at (1/e, e−1/e).
This iteration can also be generalised to a wider sort of inverses, e.g. matrix inverses.
Reciprocals of irrational numbers[edit]
Every number excluding zero has a reciprocal, and reciprocals of certain irrational numbers can have important special properties. Examples include the reciprocal of e (≈ 0.367879) and the golden ratio's reciprocal (≈ 0.618034). The first reciprocal is special because no other positive number can produce a lower number when put to the power of itself; f ( 1 / e ) {\displaystyle f(1/e)} is the global minimum of f ( x ) = x x {\displaystyle f(x)=x^{x}} . The second number is the only positive number that is equal to its reciprocal plus one: φ = 1 / φ + 1 {\displaystyle \varphi =1/\varphi +1} . Its additive inverse is the only negative number that is equal to its reciprocal minus one: − φ = − 1 / φ − 1 {\displaystyle -\varphi =-1/\varphi -1} .
The function f ( n ) = n + ( n 2 + 1 ) , n ∈ N , n > 0 {\displaystyle f(n)=n+{\sqrt {(n^{2}+1)}},n\in \mathbb {N} ,n>0} gives an infinite number of irrational numbers that differ with their reciprocal by an integer. For example, f ( 2 ) {\displaystyle f(2)} is the irrational 2 + 5 {\displaystyle 2+{\sqrt {5}}} . Its reciprocal 1 / ( 2 + 5 ) {\displaystyle 1/(2+{\sqrt {5}})} is − 2 + 5 {\displaystyle -2+{\sqrt {5}}} , exactly 4 {\displaystyle 4} less. Such irrational numbers share a curious property: they have the same fractional part as their reciprocal.
Further remarks[edit]
If the multiplication is associative, an element x with a multiplicative inverse cannot be a zero divisor (x is a zero divisor if some nonzero y, xy = 0). To see this, it is sufficient to multiply the equation xy = 0 by the inverse of x (on the left), and then simplify using associativity. In the absence of associativity, the sedenions provide a counterexample.
The converse does not hold: an element which is not a zero divisor is not guaranteed to have a multiplicative inverse. Within Z, all integers except −1, 0, 1 provide examples; they are not zero divisors nor do they have inverses in Z. If the ring or algebra is finite, however, then all elements a which are not zero divisors do have a (left and right) inverse. For, first observe that the map f(x) = ax must be injective: f(x) = f(y) implies x = y:
a x = a y ⇒ a x − a y = 0 ⇒ a ( x − y ) = 0 ⇒ x − y = 0 ⇒ x = y . {\displaystyle {\begin{aligned}ax&=ay&\quad \Rightarrow &\quad ax-ay=0\\&&\quad \Rightarrow &\quad a(x-y)=0\\&&\quad \Rightarrow &\quad x-y=0\\&&\quad \Rightarrow &\quad x=y.\end{aligned}}}
Distinct elements map to distinct elements, so the image consists of the same finite number of elements, and the map is necessarily surjective. Specifically, ƒ (namely multiplication by a) must map some element x to 1, ax = 1, so that x is an inverse for a.
Applications[edit]
The expansion of the reciprocal 1/q in any base can also act [3] as a source of pseudo-random numbers, if q is a "suitable" safe prime, a prime of the form 2p + 1 where p is also a prime. A sequence of pseudo-random numbers of length q − 1 will be produced by the expansion.
See also[edit]
Division (mathematics)
Fraction (mathematics)
Group (mathematics)
Ring (mathematics)
Division algebra
Exponential decay
Unit fractions – reciprocals of integers
Hyperbola
Repeating decimal
List of sums of reciprocals
Six-sphere coordinates
Notes[edit]
Jump up ^ " In equall Parallelipipedons the bases are reciprokall to their altitudes". OED "Reciprocal" §3a. Sir Henry Billingsley translation of Elements XI, 34.
Jump up ^ Anthony, Dr. "Proof that INT(1/x)dx = lnx". Ask Dr. Math. Drexel University. Retrieved 22 March 2013.
Jump up ^ Mitchell, Douglas W., "A nonlinear random number generator with known, long cycle length," Cryptologia 17, January 1993, 55–62.
References[edit]
Maximally Periodic Reciprocals, Matthews R.A.J. Bulletin of the Institute of Mathematics and its Applications vol 28 pp 147–148 1992
Retrieved from "https://en.wikipedia.org/w/index.php?title=Multiplicative_inverse&oldid=801480604"
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