Editing Chinese mathematics (section)

==Han dynasty==
{{Further|Science and technology of the Han dynasty#Mathematics and astronomy}}
[[File:九章算術.gif|thumb|''[[The Nine Chapters on the Mathematical Art]]'']]
In the Han dynasty, numbers were developed into a place value decimal system and used on a counting board with a set of [[counting rods]] called [[rod calculus]], consisting of only nine symbols with a blank space on the counting board representing zero.<ref name=":03" /> Negative numbers and fractions were also incorporated into solutions of the great mathematical texts of the period. The mathematical texts of the time, the ''[[Book on Numbers and Computation]]'' and ''[[Jiuzhang suanshu]]'' solved basic arithmetic problems such as addition, subtraction, multiplication and division.{{sfn|Needham|1959}} Furthermore, they gave the processes for square and cubed root extraction, which eventually was applied to solving quadratic equations up to the third order.{{sfn|Needham|1955}} Both texts also made substantial progress in Linear Algebra, namely solving systems of equations with multiple unknowns.{{sfn|Hart|2011}} The value of pi is taken to be equal to three in both texts.<ref name=":82">{{Cite book |last=Lennart |first=Bergren |title=Pi: A Source Book |year=1997 |isbn=978-1-4757-2738-8 |location=New York}}</ref> However, the mathematicians [[Liu Xin (scholar)|Liu Xin]] (d. 23) and [[Zhang Heng]] (78–139) gave more accurate approximations for [[pi]] than Chinese of previous centuries had used.{{sfn|Needham|1959}} Mathematics was developed to solve practical problems in the time such as division of land or problems related to division of payment.<ref name=":42">{{Cite journal |last=Yong |first=Lam Lay |date=1994 |title=Jiu zhang suanshu (nine chapters on the mathematical art): An overview |url=http://link.springer.com/10.1007/BF01881700 |journal=Archive for History of Exact Sciences |language=en |volume=47 |issue=1 |pages=1–51 |doi=10.1007/BF01881700 |issn=0003-9519 |jstor=41133972 |s2cid=123502226}}</ref> The Chinese did not focus on theoretical proofs based on geometry or algebra in the modern sense of proving equations to find area or volume. The Book of Computations and The Nine Chapters on the Mathematical Art provide numerous practical examples that would be used in daily life.{{sfn|Siu|1993}}

=== ''Book on Numbers and Computation'' ===
The ''[[Book on Numbers and Computation]]'' is approximately seven thousand characters in length, written on 190 bamboo strips.{{sfn|Dauben|2008}} It was discovered together with other writings in 1984 when [[archaeologist]]s opened a tomb at [[Zhangjiashan]] in [[Hubei]] province. From documentary evidence this tomb is known to have been closed in 186 BC, early in the Western [[Han dynasty]].{{sfn|Needham|1959}} While its relationship to the Nine Chapters is still under discussion by scholars, some of its contents are clearly paralleled there. The text of the ''Suan shu shu'' is however much less systematic than the Nine Chapters, and appears to consist of a number of more or less independent short sections of text drawn from a number of sources.{{sfn|Dauben|2008}}

The Book of Computations contains many perquisites to problems that would be expanded upon in The Nine Chapters on the Mathematical Art.{{sfn|Dauben|2008}} An example of the elementary mathematics in the ''Suàn shù shū'', the [[square root]] is approximated by using [[false position method]] which says to "combine the excess and deficiency as the divisor; (taking) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend."{{sfn|Dauben|2008}} Furthermore, The Book of Computations solves systems of two equations and two unknowns using the same false position method.{{sfn|Hart|2011|pages=11–85}}

=== ''The Nine Chapters on the Mathematical Art'' ===
''[[The Nine Chapters on the Mathematical Art]]'' dates archeologically to 179 CE, though it is traditionally dated to 1000 BCE, but it was written perhaps as early as 300–200 BCE.{{sfn|Dauben|2013}} Although the author(s) are unknown, they made a major contribution in the eastern world. Problems are set up with questions immediately followed by answers and procedure.<ref name=":42" /> There are no formal mathematical proofs within the text, just a step-by-step procedure.<ref>{{Cite journal |last=Straffin |first=Philip D. |date=1998-06-01 |title=Liu Hui and the First Golden Age of Chinese Mathematics |url=http://www.jstor.org/stable/2691200 |journal=Mathematics Magazine |language=en |volume=71 |issue=3 |pages=163–181 |doi=10.2307/2691200 |jstor=2691200}}</ref> The commentary of Liu Hui provided geometrical and algebraic proofs to the problems given within the text.<ref name=":03" />

''The Nine Chapters on the Mathematical Art'' was one of the most influential of all Chinese mathematical books and it is composed of 246 problems.{{sfn|Dauben|2013}} It was later incorporated into ''The [[Ten Computational Canons]]'', which became the core of mathematical education in later centuries.<ref name=":42" /> This book includes 246 problems on surveying, agriculture, partnerships, engineering, taxation, calculation, the solution of equations, and the properties of right triangles.<ref name=":42" /> ''The Nine Chapters'' made significant additions to solving quadratic equations in a way similar to [[Horner's method]].{{sfn|Needham|1955}} It also made advanced contributions to ''fangcheng'', or what is now known as linear algebra.{{sfn|Hart|2011|pages=11–85}} Chapter seven solves [[Rod calculus#System of linear equations|system of linear equations]] with two unknowns using the false position method, similar to The Book of Computations.{{sfn|Hart|2011|pages=11–85}} Chapter eight deals with solving determinate and indeterminate simultaneous linear equations using positive and negative numbers, with one problem dealing with solving four equations in five unknowns.{{sfn|Hart|2011|pages=11–85}} The Nine Chapters solves systems of equations using methods similar to the modern [[Gaussian elimination]] and [[Triangular matrix|back substitution]].{{sfn|Hart|2011|pp=11–85}}

The version of ''The Nine Chapters'' that has served as the foundation for modern renditions was a result of the efforts of the scholar Dai Zhen. Transcribing the problems directly from ''Yongle Encyclopedia'', he then proceeded to make revisions to the original text, along with the inclusion his own notes explaining his reasoning behind the alterations.{{sfn|Hart|2011|pp=32–33}} His finished work would be first published in 1774, but a new revision would be published in 1776 to correct various errors as well as include a version of ''The Nine Chapters'' from the Southern Song that contained the commentaries of Lui Hui and Li Chunfeng. The final version of Dai Zhen's work would come in 1777, titled ''Ripple Pavilion'', with this final rendition being widely distributed and coming to serve as the standard for modern versions of ''The Nine Chapters''.{{sfn|Dauben|2013|pp=211–216}} However, this version has come under scrutiny from Guo Shuchen, alleging that the edited version still contains numerous errors and that not all of the original amendments were done by Dai Zhen himself.{{sfn|Hart|2011|pp=32–33}}

=== Calculation of pi ===
Problems in The Nine Chapters on the Mathematical Art take pi to be equal to three in calculating problems related to circles and spheres, such as spherical surface area.{{sfn|Dauben|2013}} There is no explicit formula given within the text for the calculation of pi to be three, but it is used throughout the problems of both The Nine Chapters on the Mathematical Art and the Artificer's Record, which was produced in the same time period.<ref name=":82" /> Historians believe that this figure of pi was calculated using the 3:1 relationship between the circumference and diameter of a circle.{{sfn|Dauben|2013}} Some Han mathematicians attempted to improve this number, such as Liu Xin, who is believed to have estimated pi to be 3.154.{{sfn|Needham|1959}} Later, Liu Hui attempted to improve the calculation by calculating pi to be 3.141024. Liu calculated this number by using polygons inside a hexagon as a lower limit compared to a circle.{{sfn|Hart|2011|p=39}} Zu Chongzhi later discovered the calculation of pi to be 3.1415926 < π < 3.1415927 by using polygons with 24,576 sides. This calculation would be discovered in Europe during the 16th century.<ref>{{Cite journal |last=Wilson |first=Robin |date=2013 |title=Early Chinese Mathematics |journal=The Mathematical Intelligencer |language=en |volume=35 |issue=2 |page=80 |doi=10.1007/s00283-013-9364-x |issn=0343-6993 |s2cid=122920358 |doi-access=free}}</ref>

There is no explicit method or record of how he calculated this estimate.{{sfn|Needham|1959}}

=== Division and root extraction ===
Basic arithmetic processes such as addition, subtraction, multiplication and division were present before the Han dynasty.{{sfn|Needham|1959}} ''The Nine Chapters on the Mathematical Art'' take these basic operations for granted and simply instruct the reader to perform them.{{sfn|Hart|2011|pages=11–85}} Han mathematicians calculated square and cube roots in a similar manner as division, and problems on division and root extraction both occur in Chapter Four of ''The Nine Chapters on the Mathematical Art''.<ref name=":92">{{Cite journal |last=Yong |first=Lam Lay |date=1970 |title=The Geometrical Basis of the Ancient Chinese Square-Root Method |url=https://www.journals.uchicago.edu/doi/10.1086/350581 |journal=Isis |language=en |volume=61 |issue=1 |pages=92–102 |doi=10.1086/350581 |issn=0021-1753 |jstor=229151 |s2cid=145059170}}</ref> Calculating the square and cube roots of numbers is done through successive approximation, the same as division, and often uses similar terms such as dividend (''shi'') and divisor (''fa'') throughout the process.{{sfn|Needham|1955}} This process of successive approximation was then extended to solving quadratics of the second and third order, such as <math>x^2+a=b</math>, using a method similar to Horner's method.{{sfn|Needham|1955}} The method was not extended to solve quadratics of the nth order during the Han dynasty; however, this method was eventually used to solve these equations.{{sfn|Needham|1955}}
[[File:Fangcheng.GIF|thumb|Fangcheng on a counting board]]

=== Linear algebra ===
[[Writings on reckoning|''The Book of Computations'']] is the first known text to solve systems of equations with two unknowns.{{sfn|Hart|2011|pages=11–85}} There are a total of three sets of problems within ''The Book of Computations'' involving solving systems of equations with the false position method, which again are put into practical terms.{{sfn|Hart|2011|pages=11–85}} Chapter Seven of ''The Nine Chapters on the Mathematical Art'' also deals with solving a system of two equations with two unknowns with the false position method.{{sfn|Hart|2011|pages=11–85}} To solve for the greater of the two unknowns, the false position method instructs the reader to cross-multiply the minor terms or ''zi'' (which are the values given for the excess and deficit) with the major terms ''mu''.{{sfn|Hart|2011|pages=11–85}} To solve for the lesser of the two unknowns, simply add the minor terms together.{{sfn|Hart|2011|pages=11–85}}

Chapter Eight of ''The Nine Chapters on the Mathematical Art'' deals with solving infinite equations with infinite unknowns.{{sfn|Hart|2011|pages=11–85}} This process is referred to as the "fangcheng procedure" throughout the chapter.{{sfn|Hart|2011|pages=11–85}} Many historians chose to leave the term ''fangcheng'' untranslated due to conflicting evidence of what the term means. Many historians translate the word to [[linear algebra]] today. In this chapter, the process of Gaussian elimination and back-substitution are used to solve systems of equations with many unknowns.{{sfn|Hart|2011|pages=11–85}} Problems were done on a counting board and included the use of negative numbers as well as fractions.{{sfn|Hart|2011|pages=11–85}} The counting board was effectively a [[Matrix (mathematics)|matrix]], where the top line is the first variable of one equation and the bottom was the last.{{sfn|Hart|2011|pages=11–85}}

=== Liu Hui's commentary on ''The Nine Chapters on the Mathematical Art'' ===
[[File:Liuhui geyuanshu.svg|thumb|Liu Hui's exhaustion method]]
[[Liu Hui]]'s commentary on ''The Nine Chapters on the Mathematical Art'' is the earliest edition of the original text available.{{sfn|Dauben|2013}} Hui is believed by most to be a mathematician shortly after the Han dynasty. Within his commentary, Hui qualified and proved some of the problems from either an algebraic or geometrical standpoint.{{sfn|Siu|1993}} For instance, throughout ''The Nine Chapters on the Mathematical Art'', the value of pi is taken to be equal to three in problems regarding circles or spheres.<ref name=":82" /> In his commentary, Liu Hui [[Liu Hui's π algorithm|finds a more accurate estimation of pi]] using the [[method of exhaustion]].<ref name=":82" /> The method involves creating successive polygons within a circle so that eventually the area of a higher-order polygon will be identical to that of the circle.<ref name=":82" /> From this method, Liu Hui asserted that the value of pi is about 3.14.{{sfn|Needham|1959}} Liu Hui also presented a geometric proof of square and cubed root extraction similar to the Greek method, which involved cutting a square or cube in any line or section and determining the square root through symmetry of the remaining rectangles.<ref name=":92" />