Suanxue qimeng

The Suan hsüeh Ch'i-mong, written in 1299, is an elementary textbook on mathematics in three volumes, 20 chapters and 259 problems. This book also showed how to measure different two-dimensional shapes and three-dimensional solids. The Introduction had an important influence on the development of mathematics in Japan. The book was once lost in China until Qing dynasty mathematician Luo Shilin bought a Korean printed edition, and re-published in Yangzhou. Since then this book was reprinted several times.

Jade Mirror of the Four Unknowns

Zhu's second book, Jade Mirror of the Four Unknowns, written in 1303, is his most important work. With this book, Zhu advanced Chinese algebra. The first four of the 288 problems for solution illustrate his method of the four unknowns. He shows how to convert a problem stated verbally into a system of polynomial equations (up to 14th order), by using up to four unknowns: 天 Heaven, 地 Earth, 人 Man, 物 Matter, and then how to reduce the system to a single polynomial equation in one unknown by successive elimination of unknowns. He then solved the high order equation by Southern Song dynasty mathematician Qin Jiushao's "Ling long kai fang" method published in Shùshū Jiǔzhāng (「Mathematical Treatise in Nine Sections」) in 1247 (more than 570 years before English mathematician William Horner's method using synthetic division). To do this, he makes use of what is currently known as the Pascal triangle, which he labels as the diagram of an ancient method first discovered by Jia Xian before 1050. The final equation and one of its solutions is given for each of the 288 problems.

Zhu also found square and cube roots by solving quadratic and cubic equations, and added to the understanding of series and progressions, classifying them according to the coefficients of the Pascal triangle. He also showed how to solve systems of linear equations by reducing the matrix of their coefficients to diagonal form. His methods pre-date Blaise Pascal, William Horner, and modern matrix methods by many centuries. The preface of the book describes how Zhu traveled around China for 20 years as a teacher of mathematics.

The methods of Jade Mirror of the Four Unknowns form the foundation for Wu's method of characteristic set.

生平與著作

古籍中對朱世傑的生平介紹不多，只知道他曾以數學名家的身份遊歷四方，從事教學。朱世傑年輕時遍讀北方算學家著作，李冶的《測圓海鏡》一書對他影響很大。後來他還學習了李德載的二元術和劉大鑒的三元術，懂得了如何建立並解出二元、三元的高次方程組。在1270年代時，他已經是北方知名的算學家了。

公元1279年，元滅南宋，朱世傑也來到南方遊學。他接結識了不少南方的數學家，接觸到了南方的算書，尤其是秦九韶的《數書九章》和楊輝的著作。後來，朱世傑到揚州定居，慕名而來求學的人絡繹不絕。大德三年（公元1299年），他的著作《算學啟蒙》在揚州刊刻。《算學啟蒙》分三卷，二十門，259問，由淺入深，循序漸進，從一位數的乘法開始，內容包括了各類乘除法歌訣、各類面積和體積以及算術問題，還有分數運算、垛積法、盈不足術，一直講到天元術。大德七年（1303年），他的代表作《四元玉鑒》也書成付梓。《四元玉鑒》分三卷，二十四門，共收錄288個問題，都與方程或方程組的求解相關。其中關于四元方程組的問題有七個，三元的有13個，二元的有36個。書中給出了多元高次方程組的消元方法，以及用正負開方術求數值解的方法。此外，受到南方數學重實用，重口訣的風氣，朱世傑在書中吸納了一些日用算法、商用算法和通俗歌訣。

數學思想

朱世傑的數學成果代表了宋元以來的最高水平，正是因為他吸取了各種先進的思想，並加以創造性的發展。朱世傑對算理十分重視，認為數學的基礎是數學理論。朱世傑的方程理論已經超出了實際中的計算需要，而具有更加純粹的數學性質，提高了數學的抽象程度和一般化程度。《四元玉鑒》中列出過高達十次的方程。但同時朱世傑也很重視天元術等數學理論的實際應用，著作中的許多問題都有實際的背景。

輔助未知數

在解分數係數的方程組時，傳統的方法是用各項乘以分母的最小公倍數，以轉換為整係數方程組。在《算學啟蒙》一書中，朱世傑採用設輔助未知數的方法，將分數係數轉化為整數係數。李冶在其著作中曾用設輔助未知數的方法轉化方程，而朱世傑將這種方法推廣應用到方程組上。

無理方程的轉化

無理方程是指出現了關于未知數的無理表達式的方程。李冶處理過根式，但並未解過無理方程。朱世傑著作中的無理方程是中國算學史上的首創。朱世傑的處理方法是將無理式設為輔助未知數，通過變量代換將無理方程轉化為有理方程來解決。這種方法只能針對只有一個無理式的無理方程，當出現形如\sqrt{A} + \sqrt{B} = C 的方程時，朱世傑則通過兩次平方，將其轉為有理方程。

消元法

解多元高次方程組的關鍵是將其中的多個未知數消去，轉化為一元方程求解。朱世傑創造了一套完整的消元方法，稱為四元消法。他通過方程組中不同方程的配合，依次消掉各個未知數，化四元為三元、二元以至一元。

著作

• 《算學啟蒙》（1299年）：曾傳到朝鮮和日本

• 《四元玉鑒》（1303年）

重要貢獻

• 四元術（四元高次方程式）

• 垛積術（高階等差級數）

• 招差術