Editing Negative number (section)

==History==
<!-- This section is linked from [[History of negative numbers]] (R to section) -->
{{anchor|First usage of negative numbers}}<!-- linked from a 2007 article on the Web -->
{{See also|Complex number#History}}
For a long time, understanding of negative numbers was delayed by the impossibility of having a negative-number amount of a physical object, for example "minus-three apples", and negative solutions to problems were considered "false".

In [[Hellenistic Egypt]], the [[Greek mathematics|Greek]] mathematician [[Diophantus]] in the 3rd century AD referred to an equation that was equivalent to <math>4x + 20 = 4</math> (which has a negative solution) in ''[[Arithmetica]]'', saying that the equation was absurd.<ref name="Needham volume 3 p90">{{cite book|last1=Needham|first1=Joseph|last2=Wang|first2=Ling|title=Science and Civilisation in China: Volume 3; Mathematics and the Sciences of the Heavens and the Earth|url=https://books.google.com/books?id=jfQ9E0u4pLAC|year=1995|orig-year=1959|edition=reprint|location=Cambridge|publisher=Cambridge University Press| isbn=0-521-05801-5|page=90}}</ref> For this reason Greek geometers were able to solve geometrically all forms of the quadratic equation which give positive roots, while they could take no account of others.<ref>{{cite book|last1=Heath|first1=Thomas L.| title=The works of Archimedes|date=1897|publisher=Cambridge University Press|pages=cxxiii|url=https://archive.org/details/worksofarchimede029517mbp/page/n123/mode/2up}}</ref>

Negative numbers appear for the first time in history in the ''[[Nine Chapters on the Mathematical Art]]'' (九章算術, ''Jiǔ zhāng suàn-shù''), which in its present form dates from the [[Han dynasty|Han period]], but may well contain much older material.<ref name="struik33"/> The mathematician [[Liu Hui]] (c. 3rd century) established rules for the addition and subtraction of negative numbers. The historian Jean-Claude Martzloff theorized that the importance of duality in Chinese [[natural philosophy]] made it easier for the Chinese to accept the idea of negative numbers.<ref name="Hodgkin">{{cite book|last=Hodgkin|first=Luke|title=A History of Mathematics: From Mesopotamia to Modernity|url=https://archive.org/details/historyofmathema0000hodg|url-access=registration|year=2005|publisher=Oxford University Press|isbn=978-0-19-152383-0|page=[https://archive.org/details/historyofmathema0000hodg/page/88 88]|quote=Liu is explicit on this; at the point where the ''Nine Chapters'' give a detailed and helpful 'Sign Rule'}}</ref> The Chinese were able to solve simultaneous equations involving negative numbers. The ''Nine Chapters'' used red [[counting rods]] to denote positive [[coefficient]]s and black rods for negative.<ref name="Hodgkin"/><ref name="Needham volume 3 pp90-91">{{cite book|last1=Needham|first1=Joseph| last2=Wang| first2=Ling| title=Science and Civilisation in China: Volume 3; Mathematics and the Sciences of the Heavens and the Earth| url=https://books.google.com/books?id=jfQ9E0u4pLAC|year=1995|orig-year=1959|edition=reprint|location=Cambridge| publisher=Cambridge University Press|isbn=0-521-05801-5|pages=90–91}}</ref> This system is the exact opposite of contemporary printing of positive and negative numbers in the fields of banking, accounting, and commerce, wherein red numbers denote negative values and black numbers signify positive values. Liu Hui writes:
{{blockquote|Now there are two opposite kinds of counting rods for gains and losses, let them be called positive and negative. Red counting rods are positive, black counting rods are negative.<ref name="Hodgkin"/>}}

The ancient Indian ''[[Bakhshali Manuscript]]'' carried out calculations with negative numbers, using "+" as a negative sign.<ref>Teresi, Dick. (2002). ''Lost Discoveries: The Ancient Roots of Modern Science–from the Babylonians to the Mayas''. New York: Simon & Schuster. {{isbn|0-684-83718-8}}. Page 65.</ref> The date of the manuscript is uncertain. LV Gurjar dates it no later than the 4th century,<ref>{{cite web|last=Pearce|first=Ian|title=The Bakhshali manuscript|publisher=The MacTutor History of Mathematics archive|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Bakhshali_manuscript.html|date=May 2002|access-date=2007-07-24}}</ref> Hoernle dates it between the third and fourth centuries, Ayyangar and Pingree dates it to the 8th or 9th centuries,<ref name=HayashiEncy>{{citation|last=Hayashi|first=Takao|title=Bakhshālī Manuscript|encyclopedia=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures|volume=1|year=2008|publisher=Springer| isbn=9781402045592| editor=Helaine Selin|editor-link=Helaine Selin|page=B2|url=https://books.google.com/books?id=kt9DIY1g9HYC&pg=RA1-PA1}}</ref> and George Gheverghese Joseph dates it to about AD 400 and no later than the early 7th century.<ref>Teresi, Dick. (2002). ''Lost Discoveries: The Ancient Roots of Modern Science–from the Babylonians to the Mayas''. New York: Simon & Schuster. {{isbn|0-684-83718-8}}. Page 65–66.</ref>

During the 7th century AD, negative numbers were used in India to represent debts. The [[Indian mathematics|Indian mathematician]] [[Brahmagupta]], in ''[[Brahmasphutasiddhanta|Brahma-Sphuta-Siddhanta]]'' (written c. AD 630), discussed the use of negative numbers to produce a general form [[quadratic formula]] similar to the one in use today.<ref name="Needham volume 3 p90"/>

In the 9th century, [[Islamic mathematicians]] were familiar with negative numbers from the works of Indian mathematicians, but the recognition and use of negative numbers during this period remained timid.<ref name=Rashed>{{Cite book|last=Rashed|first=R.| publisher=Springer|isbn=9780792325659|title=The Development of Arabic Mathematics: Between Arithmetic and Algebra| date=1994-06-30| pages=36–37}}</ref> [[Al-Khwarizmi]] in his ''[[The Compendious Book on Calculation by Completion and Balancing|Al-jabr wa'l-muqabala]]'' (from which the word "algebra" derives) did not use negative numbers or negative coefficients.<ref name=Rashed /> But within fifty years, [[Abu Kamil]] illustrated the rules of signs for expanding the multiplication <math>(a \pm b)(c \pm d)</math>,<ref name=Ismail>{{citation|last=Bin Ismail|first=Mat Rofa|author-link=Mat Rofa bin Ismail|title=Algebra in Islamic Mathematics|encyclopedia=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures|volume=1|year=2008|publisher=Springer|isbn=9781402045592|editor=Helaine Selin|editor-link=Helaine Selin|page=115|edition=2nd}}</ref> and [[al-Karaji]] wrote in his ''al-Fakhrī'' that "negative quantities must be counted as terms".<ref name=Rashed /> In the 10th century, [[Abū al-Wafā' al-Būzjānī]] considered debts as negative numbers in ''[[A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen]]''.<ref name=Ismail />

By the 12th century, al-Karaji's successors were to state the general rules of signs and use them to solve [[polynomial division]]s.<ref name=Rashed /> As [[al-Samaw'al]] writes:
<blockquote>the product of a negative number—''al-nāqiṣ'' (loss)—by a positive number—''al-zāʾid'' (gain)—is negative, and by a negative number is positive. If we subtract a negative number from a higher negative number, the remainder is their negative difference. The difference remains positive if we subtract a negative number from a lower negative number. If we subtract a negative number from a positive number, the remainder is their positive sum. If we subtract a positive number from an empty power (''martaba khāliyya''), the remainder is the same negative, and if we subtract a negative number from an empty power, the remainder is the same positive number.<ref name=Rashed /></blockquote>

In the 12th century in India, [[Bhāskara II]] gave negative roots for quadratic equations but rejected them because they were inappropriate in the context of the problem. He stated that a negative value is "in this case not to be taken, for it is inadequate; people do not approve of negative roots."

[[Leonardo of Pisa#Important publications|Fibonacci]] allowed negative solutions in financial problems where they could be interpreted as debits (chapter 13 of ''[[Liber Abaci]]'', 1202) and later as losses (in [[Leonardo of Pisa#Works|''Flos'']], 1225).

In the 15th century, [[Nicolas Chuquet]], a Frenchman, used negative numbers as [[Exponentiation|exponents]]<ref>{{citation| last1=Flegg|first1=Graham|last2=Hay|first2=C.|last3=Moss|first3=B.|title=Nicolas Chuquet, Renaissance Mathematician: a study with extensive translations of Chuquet's mathematical manuscript completed in 1484|publisher=D. Reidel Publishing Co.|year=1985| isbn=9789027718723|page=354|url=https://books.google.com/books?id=_rO6lVwdbjcC&pg=PA354}}.</ref> but referred to them as "absurd numbers".<ref>{{citation|last1=Johnson|first1=Art|title=Famous Problems and Their Mathematicians|publisher=Greenwood Publishing Group|year=1999|isbn=9781563084461|page=56|url=https://books.google.com/books?id=STKX4qadFTkC&pg=PA56}}.</ref>

[[Michael Stifel]] dealt with negative numbers in his [[1544]] AD ''[[Arithmetica Integra]]'', where he also called them ''numeri absurdi'' (absurd numbers).

In 1545, [[Gerolamo Cardano]], in his [[Ars Magna (Gerolamo Cardano)|''Ars Magna'']], provided the first satisfactory treatment of negative numbers in Europe.<ref name="Needham volume 3 p90"/> He did not allow negative numbers in his consideration of [[cubic equation]]s, so he had to treat, for example, <math>x^3 + a x = b</math> separately from <math>x^3 = a x + b</math> (with <math>a, b > 0</math> in both cases). In all, Cardano was driven to the study of thirteen types of cubic equations, each with all negative terms moved to the other side of the = sign to make them positive. (Cardano also dealt with [[complex numbers]], but understandably liked them even less.)