Content deleted Content added
added links into the article |
|||
(25 intermediate revisions by 21 users not shown) | |||
Line 1:
{{Short description|Algorithm similar to Gaussian elimination}}
'''Fangcheng''' (sometimes written as '''fang-cheng''' or '''fang cheng''') ({{zh|c=方程|p=fāng chéng}}) is the title of the eighth
The procedure referred to by the term ''fangcheng'' and explained in the eighth
The ''fangcheng'' procedure was popular in ancient [[China]] and
== On the meaning of ''fangcheng'' ==
There is no ambiguity in the meaning of the first character ''fang''. It means
#The earliest extant commentary, by [[Liu Hui]], dated 263 CE defines ''cheng'' as "measures,
#Li
#[[Yang Hui]]
== Contents of the chapter titled ''Fangcheng'' ==
The eighth chapter titled ''Fangcheng'' of the ''Nine Chapters'' book contains 18 problems. (There are a total of 288 problems in the whole book.) Each of these 18 problems reduces to a problem of solving a system of simultaneous linear equations. Except for one problem, namely Problem 13, all the problems are determinate in the sense that the number of unknowns is same as the number of equations. There are problems involving 2, 3, 4 and 5 unknowns. The
▲'''Table showing the number of unknowns and number of equations <br> in the various problems in Chapter 8 of ''Nine Chapters'' '''
▲{| class="wikitable" style="Text-align: center"
|-
! Number of unknowns<br /> in the problem !! Number of equations<br /> in the problem !! Serial numbers of problems !! Number of problems || Determinacy
|-
| 2 || 2 ||2, 4, 5, 6, 7, 9, 10, 11 || 8 ||Determinate
Line 35 ⟶ 34:
| 6 || 5 || 13 || 1 ||[[Indeterminate system|Indeterminate]]
|-
|
|}
The presentations of all the 18 problems (except Problem 1 and Problem 3) follow a common pattern:
*First the problem is stated.
Line 45 ⟶ 43:
*Finally the method of obtaining the answer is indicated.
=== On Problem 1 ===
* ''Problem:''
** 3 bundles of high-quality rice straws, 2 bundles of mid-quality rice straws and 1 bundle of low-quality rice straw produce 39 units of rice
** 2 bundles of high-quality rice straws, 3 bundles of mid-quality rice straws and 1 bundle of low-quality rice straw produce 34 units of rice
** 1 bundles of high-quality rice straw, 2 bundles of mid-quality rice straws and 3 bundle of low-quality rice straws produce 26 units of rice
** Question: how many units of rice can high, mid and low quality rice straw produce respectively?
* ''Solution:''
** High-quality rice straw each produces {{sfrac|9|1|4}} units of rice
** Mid-quality rice straw each produces {{sfrac|4|1|4}} units of rice
** Low-quality rice straw each produces {{sfrac|2|3|4}} units of rice
The presentation of Problem 1 contains a description (not a crisp indication) of the procedure for obtaining the solution. The procedure has been referred to as ''fangcheng shu'', which means "''fangcheng'' procedure." The remaining problems all give the instruction "follow the ''fangcheng''" procedure sometimes followed by the instruction to use the "procedure for positive and negative numbers".
=== On Problem 3 ===
There is
=== On Problem 13 ===
In the collection of these 18 problems Problem 13 is very special. In it there are 6 unknowns but only 5 equations and so Problem 13 is indeterminate and does not have a unique solution. This is the earliest known reference to a system of linear equations in which the number of unknowns exceeds the number of equations. As per a suggestion of Jean-Claude Martzloff, a historian of Chinese mathematics, Roger Hart has named this problem "the well problem."
== References ==
{{reflist}}
== Further reading ==
*{{cite journal|
*{{cite book|
[[Category:Chinese mathematics]]
|