AC (complexity) - Revision history

217.189.232.123 at 15:03, 6 March 2014

2014-03-06T15:03:01Z

← Older revision		Revision as of 17:03, 6 March 2014
Line 1:		Line 1:
	~~{{Use dmy dates\|date=April 2012}}~~		The writer is known as Araceli Gulledge. Interviewing is what she does in her day job but quickly her husband and her will begin their own business. To play badminton is something he truly enjoys performing. Kansas is where her home is but she needs to move because of her family.<br><br>Also visit my web site [http://Www.Battdouglas.com/ActivityFeed/MyProfile/tabid/61/userId/6189/Default.aspx Www.Battdouglas.com]
	~~[[Image:ArsMagna.jpg\|thumb\|right\|244px\|The title page of the '''''Ars Magna'''''.~~ The ~~full title~~ is ~~''Artis Magnæ, Sive de Regulis Algebraicis Liber Unus'' (Book number one about The Great Art, or The Rules of Algebra)~~.]]
	~~The '''''Ars Magna''''' ([[Latin language\|Latin]]: "The Great Art")~~ is ~~an important book on [[Algebra]] written by [[Girolamo Cardano]]. It was first published~~ in 1545 under the title ''Artis Magnæ, Sive de Regulis Algebraicis Liber Unus'' (''Book number one about The Great Art, or The Rules of Algebra''). There was a second edition in Cardano's lifetime, published in 1570. It is considered<ref>See, for instance, the foreword that [[Oystein Ore]] wrote for the English translation of the book, mentioned at the bibliography.</ref> one of the three greatest scientific treatises of the early [[Renaissance]], together with [[Copernicus]]' ''[[De revolutionibus orbium coelestium]]'' and ~~[[Vesalius]]' ''[[De humani corporis fabrica]]''. The first editions of these three books were published within a two year span (1543–1545)~~.

	~~==History==~~
	In 1535, [[Niccolò Fontana Tartaglia]] became famous for having solved cubics of the form ''x''<sup>3</sup> + ''ax'' = ''b'' (with ''a'',''b'' > 0). However, he chose to keep his method secret. In 1539, Cardano, then a lecturer in mathematics at the Piatti Foundation in Milan, published his first mathematical book, ''Pratica Arithmeticæ et mensurandi singularis'' (''The Practice of Arithmetic and Simple Mensuration''). That same year, he asked Tartaglia to explain to him his method for solving cubic equations. After some reluctance, Tartaglia did so, but he asked Cardano not to share the information until he published it. Cardano submerged himself in mathematics during the next several years working on how to extend Tartaglia's formula to other types of cubics. Furthermore, his student [[Lodovico Ferrari]] found a way of solving quartic equations, but Ferrari's method depended upon Tartaglia's, since it involved the use of an auxiliary cubic equation. Then Cardano become aware of the fact that [[Scipione del Ferro]] had discovered Tartaglia's formula before Tartaglia himself, a discovery that prompted him to publish these results.

	~~==Contents==~~
	~~The book, which~~ is divided into forty chapters, contains the first published solution to [[cubic equation\|cubic]] and [[quartic equation]]s. Cardano acknowledges that Tartaglia gave him the formula for solving a type of cubic equations and that the same formula had been discovered by Scipiano del Ferro. He also acknowledges that it was Ferrari who found a way of solving quartic equations.

	Since at the time negative numbers were not generally acknowledged, knowing how to solve cubics of the form ''x''<sup>3</sup> + ''ax'' = ''b'' did not mean knowing how to solve cubics of the form ''x''<sup>3</sup> = ''ax'' + ''b'' (with ''a'',''b'' > 0), for instance. Besides, Cardano, also explains how to reduce equations of the form ''x''<sup>3</sup> + ''ax''<sup>2</sup> + ''bx'' + ''c'' = 0 to cubic equations without a quadratic term, but, again, he ~~has to consider several cases. In all, Cardano was driven to the study of thirteen different types of cubic equations (chapters XI–XXIII).~~

	In ''Ars Magna'' the concept of [[Multiplicity (mathematics)#Multiplicity of a root of a polynomial\|multiple root]] appears for the first time (chapter I). The first example that Cardano provides of a polynomial equation with multiple roots is ''x''<sup>3</sup> = 12''x'' + 16, of which −2 is a double root.

	~~''Ars Magna'' also contains the first occurrence of [[complex number]]s (chapter XXXVII)~~. ~~The problem mentioned by Cardano which leads to square roots of negative numbers is: find two numbers whose sum~~ is ~~equal to 10 and whose product~~ is ~~equal~~ to 40. The answer is 5 + √<span style="text-decoration: overline">−15</span> and 5 − √<span style="text-decoration: overline">−15</span>. Cardano called this "sophistic," because he saw no physical meaning to it, but boldly wrote "nevertheless we will operate" and formally calculated that their product does indeed equal 40. Cardano then says that this answer is “as subtle as it is useless”.

	~~It is a common misconception that Cardano introduced complex numbers in solving cubic equations. Since (in modern notation) Cardano's formula for a root~~ of ~~the polynomial ''x''<sup>3</sup> + ''px'' + ''q''  is~~
	~~:<math>\sqrt[3]{-\frac q2+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}+\sqrt[3]{-\frac q2-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}},</math>~~
	~~square roots of negative numbers appear naturally in this context~~. ~~However, ''q''~~<~~sup~~>2<~~/sup~~>/4 + ''p''<sup>3</sup>/27 never happens to be negative in the specific cases in which Cardano applies the formula.<ref>This does not mean that no cubic equation occurs in ''Ars Magna'' for which ''q''<sup>2</sup>/4 + ''p''<sup>3</sup>/27 < 0. For instance, chapter I contains the equation ''x''<sup>3</sup> + 9 = 12''x'', for which ''q''<sup>2</sup>/4 + ''p''<sup>3</sup>/27 = −175/4. However, Cardano never applies his formula in those cases.</ref>

	~~==Notes==~~
	~~<references/>~~

	~~==Bibliography==~~
	*{{Citation \| last = Calinger \| first = Ronald \| year = 1999 \| title = A contextual history of Mathematics \| publisher=Prentice-Hall \| isbn = 0-02-318285-7}}
	*{{Citation \| last = Cardano \| first = Gerolamo \| author-link = Gerolamo Cardano \| publication-date = 1993 \| year = 1545 \| title = Ars magna or The Rules of Algebra \| publisher=Dover \| isbn = 0-486-67811-3}}
	*{{Citation \| last = Gindikin \| first = Simon \| publication-date = 1988 \| title = Tales of physicists and mathematicians \| publisher=Birkhäuser \| isbn = 3-7643-3317-0 \| year = 1988}}

	~~==External links==~~
	*[http://~~www~~.~~filosofia~~.~~unimi.it~~/~~cardano~~/~~testi~~/~~operaomnia~~/~~vol_4_s_4.pdf .pdf of ''Ars Magna''] (in [[Latin]])~~
	*[http://~~www-gap.dcs.st-and~~.ac.~~uk/~history/Biographies/Cardan~~.~~html Cardano's biography]~~

	~~[[Category:Mathematics books]]~~
	~~[[Category:1545 books]]~~
	~~[[Category:1545 in science]~~]

en>Yobot: /* References */WP:CHECKWIKI error fixes using AWB (9427)

2013-08-16T06:13:12Z

References: WP:CHECKWIKI error fixes using AWB (9427)

← Older revision		Revision as of 08:13, 16 August 2013
Line 1:		Line 1:
	~~Friends contact him Royal~~. The ~~preferred pastime~~ for ~~my kids~~ and ~~me is dancing~~ and ~~now I~~'~~m attempting~~ to ~~earn money with~~ it. ~~Managing people~~ is ~~how she tends~~ to ~~make cash~~ and ~~she will not change~~ it ~~anytime soon~~. ~~Some~~ time in the ~~past~~ he ~~selected~~ to ~~live in Kansas~~.<br><br>~~Here~~ is ~~my web blog~~; ~~extended auto warranty~~; ~~[http~~://~~www~~.~~thelamda~~.~~com~~/~~index~~.~~php?mod~~=~~users~~&~~action~~=~~view~~&id=~~3598 i thought about this~~],		{{Use dmy dates\|date=April 2012}}
			[[Image:ArsMagna.jpg\|thumb\|right\|244px\|The title page of the '''''Ars Magna'''''. The full title is ''Artis Magnæ, Sive de Regulis Algebraicis Liber Unus'' (Book number one about The Great Art, or The Rules of Algebra).]]
			The '''''Ars Magna''''' ([[Latin language\|Latin]]: "The Great Art") is an important book on [[Algebra]] written by [[Girolamo Cardano]]. It was first published in 1545 under the title ''Artis Magnæ, Sive de Regulis Algebraicis Liber Unus'' (''Book number one about The Great Art, or The Rules of Algebra''). There was a second edition in Cardano's lifetime, published in 1570. It is considered<ref>See, for instance, the foreword that [[Oystein Ore]] wrote for the English translation of the book, mentioned at the bibliography.</ref> one of the three greatest scientific treatises of the early [[Renaissance]], together with [[Copernicus]]' ''[[De revolutionibus orbium coelestium]]'' and [[Vesalius]]' ''[[De humani corporis fabrica]]''. The first editions of these three books were published within a two year span (1543–1545).

			==History==
			In 1535, [[Niccolò Fontana Tartaglia]] became famous for having solved cubics of the form ''x''<sup>3</sup> + ''ax'' = ''b'' (with ''a'',''b'' > 0). However, he chose to keep his method secret. In 1539, Cardano, then a lecturer in mathematics at the Piatti Foundation in Milan, published his first mathematical book, ''Pratica Arithmeticæ et mensurandi singularis'' (''The Practice of Arithmetic and Simple Mensuration''). That same year, he asked Tartaglia to explain to him his method for solving cubic equations. After some reluctance, Tartaglia did so, but he asked Cardano not to share the information until he published it. Cardano submerged himself in mathematics during the next several years working on how to extend Tartaglia's formula to other types of cubics. Furthermore, his student [[Lodovico Ferrari]] found a way of solving quartic equations, but Ferrari's method depended upon Tartaglia's, since it involved the use of an auxiliary cubic equation. Then Cardano become aware of the fact that [[Scipione del Ferro]] had discovered Tartaglia's formula before Tartaglia himself, a discovery that prompted him to publish these results.

			==Contents==
			The book, which is divided into forty chapters, contains the first published solution to [[cubic equation\|cubic]] and [[quartic equation]]s. Cardano acknowledges that Tartaglia gave him the formula for solving a type of cubic equations and that the same formula had been discovered by Scipiano del Ferro. He also acknowledges that it was Ferrari who found a way of solving quartic equations.

			Since at the time negative numbers were not generally acknowledged, knowing how to solve cubics of the form ''x''<sup>3</sup> + ''ax'' = ''b'' did not mean knowing how to solve cubics of the form ''x''<sup>3</sup> = ''ax'' + ''b'' (with ''a'',''b'' > 0), for instance. Besides, Cardano, also explains how to reduce equations of the form ''x''<sup>3</sup> + ''ax''<sup>2</sup> + ''bx'' + ''c'' = 0 to cubic equations without a quadratic term, but, again, he has to consider several cases. In all, Cardano was driven to the study of thirteen different types of cubic equations (chapters XI–XXIII).

			In ''Ars Magna'' the concept of [[Multiplicity (mathematics)#Multiplicity of a root of a polynomial\|multiple root]] appears for the first time (chapter I). The first example that Cardano provides of a polynomial equation with multiple roots is ''x''<sup>3</sup> = 12''x'' + 16, of which −2 is a double root.

			''Ars Magna'' also contains the first occurrence of [[complex number]]s (chapter XXXVII). The problem mentioned by Cardano which leads to square roots of negative numbers is: find two numbers whose sum is equal to 10 and whose product is equal to 40. The answer is 5 + √<span style="text-decoration: overline">−15</span> and 5 − √<span style="text-decoration: overline">−15</span>. Cardano called this "sophistic," because he saw no physical meaning to it, but boldly wrote "nevertheless we will operate" and formally calculated that their product does indeed equal 40. Cardano then says that this answer is “as subtle as it is useless”.

			It is a common misconception that Cardano introduced complex numbers in solving cubic equations. Since (in modern notation) Cardano's formula for a root of the polynomial ''x''<sup>3</sup> + ''px'' + ''q''  is
			:<math>\sqrt[3]{-\frac q2+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}+\sqrt[3]{-\frac q2-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}},</math>
			square roots of negative numbers appear naturally in this context. However, ''q''<sup>2</sup>/4 + ''p''<sup>3</sup>/27 never happens to be negative in the specific cases in which Cardano applies the formula.<ref>This does not mean that no cubic equation occurs in ''Ars Magna'' for which ''q''<sup>2</sup>/4 + ''p''<sup>3</sup>/27 < 0. For instance, chapter I contains the equation ''x''<sup>3</sup> + 9 = 12''x'', for which ''q''<sup>2</sup>/4 + ''p''<sup>3</sup>/27 = −175/4. However, Cardano never applies his formula in those cases.</ref>

			==Notes==
			<references/>

			==Bibliography==
			*{{Citation \| last = Calinger \| first = Ronald \| year = 1999 \| title = A contextual history of Mathematics \| publisher=Prentice-Hall \| isbn = 0-02-318285-7}}
			*{{Citation \| last = Cardano \| first = Gerolamo \| author-link = Gerolamo Cardano \| publication-date = 1993 \| year = 1545 \| title = Ars magna or The Rules of Algebra \| publisher=Dover \| isbn = 0-486-67811-3}}
			*{{Citation \| last = Gindikin \| first = Simon \| publication-date = 1988 \| title = Tales of physicists and mathematicians \| publisher=Birkhäuser \| isbn = 3-7643-3317-0 \| year = 1988}}

			==External links==
			*[http://www.filosofia.unimi.it/cardano/testi/operaomnia/vol_4_s_4.pdf .pdf of ''Ars Magna''] (in [[Latin]])
			*[http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Cardan.html Cardano's biography]

			[[Category:Mathematics books]]
			[[Category:1545 books]]
			[[Category:1545 in science]]

en>JNW: rv to previous; please start a differnet page, don't delete a current one

2010-10-26T08:51:52Z

rv to previous; please start a differnet page, don't delete a current one

New page

Friends contact him Royal. The preferred pastime for my kids and me is dancing and now I'm attempting to earn money with it. Managing people is how she tends to make cash and she will not change it anytime soon. Some time in the past he selected to live in Kansas.<br><br>Here is my web blog; extended auto warranty; [http://www.thelamda.com/index.php?mod=users&action=view&id=3598 i thought about this],