• Journal for History of Mathematics
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• v.18 no.3
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• pp.1-24
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• 2005

Although indeterminate equations were dealt in Jiu zhang suan shu and then in Sun zi suan fing and Zhang Giu Jian suan Jing in China, they did not get any substantial development until Qin Jiu Shao introduced da yan shu in his great book Shu shu jiu zhang which solves goneral systems of linear congruences. We first investigate his da yan shu and then study the history of indeterminate equations in Chosun Dynasty. Further, we compare Qin's da yan shu with that in San Hak Jung Eui written by Chosun mathematician Nam Byung Gil.

• Korean Journal of Mathematics
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• v.20 no.1
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• pp.137-159
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• 2012

In this paper, we deal with recent researches on Ying N$\ddot{u}$ and Ying Buzu(盈不足) which were addressed in the book Jiu Zhang Suan Shu(九章算術, The Nine Chapters on the Mathematical Art). Ying N$\ddot{u}$(Ying Buzu) is a concept on profit and loss problems. Ying Buzu Shu(盈不足術, the method of excess and deficit) represents an algorithm which has been used for solving many mathematical problems. It is known as a rule of double false position in the West. We show the importance of Ying Buzu Shu via an analysis of some problems in 'Ying Buzu' chapter. In 1202, Fibonacci(c.1170-c.1250) used Ying Buzu Shu in his book. This shows some of Asian mathematics were introduced to the West even before the year 1200. We present the origin of Ying Buzu Shu, and its relationship with Cramer's Rule. We have discovered how Asia's Ying Buzu Shu spread to Europe via Arab countries. In addition, we analyze some characters of Ying N$\ddot{u}$(Ying Buzu) in the book Suan Xue Bao Jian(算學寶鑑).

• Journal for History of Mathematics
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• v.26 no.5_6
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• pp.323-328
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• 2013

The Division Algorithm is known to be the fundamental foundation for Number Theory and it leads to the Euclidean Algorithm and hence the whole theory of divisibility properties. In JiuZhang SuanShu(九章算術), greatest common divisiors are obtained by the exactly same method as the Euclidean Algorithm in Elements but the other theory on divisibility was not pursued any more in Chinese mathematics. Unlike the other authors of the traditional Chinese mathematics, Zhu ShiJie(朱世傑) noticed in his SuanXue QiMeng(算學啓蒙, 1299) that the Division Algorithm is a really important concept. In [4], we claimed that Zhu wrote the book with a far more deeper insight on mathematical structures. Investigating the Division Algorithm in SuanXue QiMeng in more detail, we show that his theory of Division Algorithm substantiates his structural apporaches to mathematics.

• Journal for History of Mathematics
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• v.26 no.4
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• pp.213-218
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• 2013

This paper is a sequel to the paper [8], where we discussed the connection between ShiShou KaiFangFa originated from JiuZhang SuanShu and ZengCheng KaiFangFa. Investigating KaiFangShu in a Chosun mathemtics book, SanHak JeongEui and ShuLi JingYun, we show that its authors, Nam ByungGil and Lee SangHyuk clearly understood the connection and gave examples to show that the KaiFangShu in the latter is not exact. We also show that Chosun mathematicians were very much selective when they brought in Chinese mathematics.

• Journal for History of Mathematics
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• v.22 no.4
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• pp.29-40
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• 2009

we know that Zeng Cheng Kai Fang Fa is the generalization of the method of square roots and cube roots of ancient through the investigation of China mathematics. In this paper, we have research on traditional solutions equations of China mathematics and the development solutions of equations used by Chosun mathematicians.

• Journal for History of Mathematics
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• v.20 no.2
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• pp.1-18
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• 2007

In the middle of 19th century, Chosun mathematicians Nam Byung Gil(南秉吉) and Lee Sang Hyuk(李尙爀) studied mathematical structures developed in Song(宋) and Yuan(元) eras on top of their early studies on Jiu zhang suan shu(九章算術) and Shu li jing yun(數理精蘊). Their studies gave rise to a momentum for a prominent development of Chosun mathematics in the century. In this paper, we investigate Nam Byung Gil's JipGoYunDan(輯古演段) and MuIHae(無異解) and then study his theory of equations. Through a collaboration with Lee, Sang Hyuk, he consolidated the eastern and western structure of theory of equations.

• Journal for History of Mathematics
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• v.31 no.6
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• pp.273-289
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• 2018

The KaiFangFa開方法 of traditional mathematics was completed in ${\ll}$JiuZhang SuanShu九章算術${\gg}$ originally, and further organized in Song宋 $Yu{\acute{a}}n$元 dinasities. The former is the ShiSuoKaiFangFa釋鎖開方法 using the coefficients of the polynomial expansion, and the latter is the ZengChengKaiFangFa增乘開方法 obtaining the solution only by some mechanical numerical manipulations. ${\ll}$SanHak JeongEui算學正義${\gg}$ basically used the latter and improved the estimate-value array by referring to the written-calculation in ${\ll}$ShuLi JingYun數理精蘊${\gg}$. As a result, ZengChengKaiFangFa was more refined so that the KaiFangFa algorithm is more consistent.

• Journal for History of Mathematics
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• v.24 no.1
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• pp.1-13
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• 2011

In Tian mu bi lei cheng chu jie fa(田畝比類乘除捷法) of Yang Hui suan fa(楊輝算法)), Yang Hui annotated detailed comments on the method to find roots of quadratic equations given by Liu Yi in his Yi gu gen yuan(議古根源) which gave a great influence on Chosun Mathematics. In this paper, we show that 'Zeng cheng kai fang fa'(增乘開方法) evolved from a process of binomial expansions of $(y+{\alpha})^n$ which is independent from the synthetic divisions. We also show that extending the results given by Liu Yi-Yang Hui and those in Suan xue qi meng(算學啓蒙), Chosun mathematican Hong Jung Ha(洪正夏) elucidated perfectly the 'Zeng cheng kai fang fa' as the present synthetic divisions in his Gu il jib(九一集).

• Journal for History of Mathematics
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• v.25 no.3
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• pp.1-14
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• 2012

Since JiuZhang SuanShu(九章算術), the basic field of the traditional mathemtics in Eastern Asia is the field of rational numbers and hence irrational solutions of equations should be replaced by rational approximations. Thus approximate solutions of equations became a very important subject in theory of equations. We first investigate the history of approximate solutions in Chinese sources and then compare them with those in Chosun mathematics. The theory of approximate solutions in Chosun has been established in SanHakWonBon(算學原本) written by Park Yul(1621 - 1668) and JuSeoGwanGyun(籌書管見, 1718) by Cho Tae Gu(趙泰耉, 1660-1723). We show that unlike the Chinese counterpart, Park and Cho were concerned with errors of approximate solutions and tried to find better approximate solutions.

• Journal for History of Mathematics
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• v.18 no.3
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• pp.39-54
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• 2005

In Chinese Mathematics, Jia Xian(要憲) introduced Zeng cheng kai fang fa(增乘開方法) to get approximations of solutions of Polynomial equations which is a generalization of square roots and cube roots in Jiu zhang suan shu. The synthetic divisions in Zeng cheng kai fang fa give ise to two concepts of Fan il(飜積) and Yi il(益積) which were extensively used in Chosun Dynasty Mathematics. We first study their history in China and Chosun Dynasty and then investigate the historical fact that Chosun mathematicians Nam Byung Gil(南秉吉) and Lee Sang Hyuk(李尙爀) obtained the sufficient conditions for Fan il and Yi il for quadratic equations and proved them in the middle of 19th century.