History of Chinese Linear Algebra Written by Charles Luettgen Citadel Linear Algebra 530 Professor Mei Chen, Instructor Abbreviated history of Chinese Linear Algebra with references to Modern Western Linear Algebra Techniques of Gaussian Elimination and Solution by Determinants History of Chinese Linear Algebra • Chinese Character Development and Problems with Translating Them • General History of Chinese Mathematics and Linear Algebra • Chinese Multple Equation Solving compared to Gaussian Elimination • Chinese Determinants compared to Cramers’ Rule • Why you didn’t know this History of Chinese Linear Algebra • Writing developed 6600-6200 BC • >100,000 Chinese Characters (not alphabets) • Need for Standardization – Qin Dynasty (circa 220 BC) • Modern Simplification – Multiple Scripts added over time – 19 Century (some before but most after the reforms of the Peoples Republic of China) History of Chinese Linear Algebra History of Chinese Linear Algebra • Multiple Uses and Variants of Numbers • Development of the Zero, Negative Numbers, Powers and Fractions • Need to Solve multiple Equations for Taxes, Astronomy, Engineering and Surveying History of Chinese Linear Algebra Typical Counting Rods Numerals and counting board History of Chinese Linear Algebra History of Chinese Linear Algebra History of Chinese Linear Algebra • Chinese Multple Equation Solving compared to Gaussian Elimination C B A 0 0 3 0 4 0 4 0 0 11 17 38 (Transposed) A 3 0 0 38 B 0 4 0 24 C 4 0 0 11 History of Chinese Linear Algebra • Chinese Determinants compared to Cramers’ Rule – Chinese Method Used Guesses to Formulate Matrix Coefficients in Example Format – Chinese Method intended for Practical Daily Application • Cramer’s Rule uses General Proof Theorem to State Rule for all Examples – Cramer’s Rule for Mathematicians History of Chinese Linear Algebra • Chinese approach: • The 20 problems give a rule of double false position. Essentially linear equations are solved by making two guesses at the solution, then computing the correct answer from the two errors. For example to solve • ax + b = c • we try x = i, and instead of c we get c + d. Then we try x = j, and instead of c we obtain c + e. Then the correct solution is • x = (jd - ie)/(d - e). • The first problem essentially contains the "guesses" in its formulation:- • Certain items are purchased jointly. If each person pays 8 coins, the surplus is 3 coins, and if each person gives 7 coins, the deficiency is 4 coins. Find the number of people and the total cost of the items. [Answer: There are 7 people and the total cost of the items is 53 coins.] History of Chinese Linear Algebra • General case • Consider a system of n linear equations for n unknowns, represented in matrix multiplication form as follows: • where the n by n matrix has a nonzero determinant, and the vector is the column vector of the variables. • Then the theorem states that in this case the system has a unique solution, whose individual values for the unknowns are given by: • where is the matrix formed by replacing the ith column of by the column vector . • The rule holds for systems of equations with coefficients and unknowns in any field, not just in the real numbers. It has recently been shown that Cramer's rule can be implemented in O(n3) time,[6] which is comparable to more common methods of solving systems of linear equations, such as Gaussian elimination. History of Chinese Linear Algebra • Why You Didn’t Know This? – Complicated Character/ Number System – Great Wall of China/ Isolationism (up to 1500 AD) – Deterioration of Written Materials (2000 yrs old) – Burning of Books/ Erasure of Previous Regime – Lack of Math Skills amongst Translators – Vatican Conspiracy to Plagiarize Chinese Texts without Acknowledgement of Chinese Authors Starting as Early as 1607 AD to 20th century References • The Chinese Roots of Linear Algebra, (book review)Roger Hart, John Hopkins University Press, 2010, http://books.google.com/books?id=zLPm3xE2qWgC&dq=chinese+linear+algebra&source=gbs_navlinks_s • Chinese Roots of Linear Algebra, Roger Hart, 2009, http://uts.cc.utexas.edu/~rhart/algebra/main.html • • History of Math, Historical Modules for the Mathematics Classroom; Solving Systems of Linear Equations: Ancient Methods, , Creative Commons Attribution-ShareAlike 3.0 License, 4 Aug 2009, http://hom.wikidot.com/cramer-s-method-and-cramer- s-paradox • Chinese Mathematics, Wikipedia®, last modified on 6 November 2012,(two linked pages) http://www.crystalinks.com/chinamath.html ,http://en.wikipedia.org/wiki/Chinese_mathematics • Counting rods, Wikipedia®, last modified on 9 November 2012, http://en.wikipedia.org/wiki/Counting_rods • Rod calculus, Wikipedia®, last modified on 20 November 2012, http://en.wikipedia.org/wiki/Rod_calculus#System_of_linear_equations • Chinese Characters, Wikipedia®, last modified on 20 November 2012, http://en.wikipedia.org/wiki/Chinese_characters • Chinese Characters (Google Images), https://www.google.com/search?q=chinese+characters&hl=en&client=firefox- a&hs=uER&tbo=u&rls=org.mozilla:en- US:official&tbm=isch&source=univ&sa=X&ei=KVSrUJqaPIPC9QTKroGwCw&ved=0CEMQsAQ&biw=1440&bih=796&sei=a2Or UNLWM43e8AS5-4C4DA • Chinese Numbers (Google Images) https://www.google.com/search?q=chinese+numbers&hl=en&client=firefox- a&hs=NvT&tbo=u&rls=org.mozilla:en- US:official&tbm=isch&source=univ&sa=X&ei=enyrUPbrIIn48gTR0YGoAw&ved=0CEcQsAQ&biw=1440&bih=796&sei=fXyrULX 4LI3W8gSpm4HQAQ • Ancient Chinese Counting Rods, (Google Images), https://www.google.com/search?q=ancient+chinese+counting+rods&hl=en&client=firefox- a&hs=Tww&tbo=u&rls=org.mozilla:en- US:official&tbm=isch&source=univ&sa=X&ei=4PmrUKqeIISs9ATjs4GgAQ&ved=0CDkQsAQ&biw=1440&bih=796&sei=5PmrU PCzEZKE9QTh6YCYDQ • Cramer’s Rule, Wikipedia®, last modified on 10 November 2012, http://en.wikipedia.org/wiki/Cramer%27s_rule • Renowned Mathematicians of Ancient China, Kaleidoscope-> Science and Invention Cultural China, 2007-2010, http://kaleidoscope.cultural-china.com/en/10Kaleidoscope2079.html • Matrices and Determinants, JJ O’Connor and EF Robertson, MCS St Andrews (UK), February 1996, http://www- history.mcs.st-and.ac.uk/HistTopics/Matrices_and_determinants.html .