Yang Hui (simplified Chinese :杨辉; traditional Chinese :楊輝; pinyin :Yáng Huī, ca. 1238–1298), courtesy name Qianguang (謙光), was a Chinese mathematician and writer during the Song dynasty. Originally, from Qiantang (modern Hangzhou, Zhejiang), Yang worked on magic squares, magic circles and the binomial theorem, and is best known for his contribution of presenting Yang Hui's Triangle. This triangle was the same as Pascal's Triangle, discovered by Yang's predecessor Jia Xian. Yang was also a contemporary to the other famous mathematician Qin Jiushao.
The earliest extant Chinese illustration of 'Pascal's Triangle' is from Yang's book Xiangjie Jiuzhang Suanfa (詳解九章算法) of 1261 AD, in which Yang acknowledged that his method of finding square roots and cubic roots using "Yang Hui's Triangle" was invented by mathematician Jia Xian who expounded it around 1100 AD, about 500 years before Pascal. In his book (now lost) known as Rújī Shìsuǒ (如積釋鎖) or Piling-up Powers and Unlocking Coefficients, which is known through his contemporary mathematician Liu Ruxie (劉汝諧). Jia described the method used as 'li cheng shi suo' (the tabulation system for unlocking binomial coefficients). It appeared again in a publication of Zhu Shijie's book Jade Mirror of the Four Unknowns (四元玉鑒) of 1303 AD.
Around 1275 AD, Yang finally had two published mathematical books, which were known as the Xugu Zhaiqi Suanfa (續古摘奇算法) and the Suanfa Tongbian Benmo (算法通變本末, summarily called Yang Hui suanfa 楊輝算法). In the former book, Yang wrote of arrangement of natural numbers around concentric and non concentric circles, known as magic circles and vertical-horizontal diagrams of complex combinatorial arrangements known as magic squares, providing rules for their construction. In his writing, he harshly criticized the earlier works of Li Chunfeng and Liu Yi (劉益), the latter of whom were both content with using methods without working out their theoretical origins or principle. Displaying a somewhat modern attitude and approach to mathematics, Yang once said:
In his written work, Yang provided theoretical proof for the proposition that the complements of the parallelograms which are about the diameter of any given parallelogram are equal to one another.This was the same idea expressed in the Greek mathematician Euclid's (fl. 300 BC) forty-third proposition of his first book, only Yang used the case of a rectangle and gnomon. There were also a number of other geometrical problems and theoretical mathematical propositions posed by Yang that were strikingly similar to the Euclidean system. However, the first books of Euclid to be translated into Chinese was by the cooperative effort of the Italian Jesuit Matteo Ricci and the Ming official Xu Guangqi in the early 17th century.
Yang's writing represents the first in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi.Yang was also well known for his ability to manipulate decimal fractions. When he wished to multiply the figures in a rectangular field with a breadth of 24 paces 3 4⁄10 ft. and length of 36 paces 2 8⁄10, Yang expressed them in decimal parts of the pace, as 24.68 X 36.56 = 902.3008.
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers (arithmetic).
Zhu Shijie, courtesy name Hanqing (漢卿), pseudonym Songting (松庭), was a Chinese mathematician and writer. He was a Chinese mathematician during the Yuan Dynasty. Zhu was born close to today's Beijing. Two of his mathematical works have survived. Introduction to Computational Studies, and Jade Mirror of the Four Unknowns.
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy.
Zu Chongzhi, courtesy name Wenyuan, was a Chinese astronomer, mathematician, politician, inventor, and writer during the Liu Song and Southern Qi dynasties. He was most notable for calculating pi as between 3.1415926 and 3.1415927, a record which would not be surpassed for 800 years.
Liu Hui was a Chinese mathematician and writer who lived in the state of Cao Wei during the Three Kingdoms period (220–280) of China. In 263, he edited and published a book with solutions to mathematical problems presented in the famous Chinese book of mathematics known as The Nine Chapters on the Mathematical Art, in which he was possibly the first mathematician to discover, understand and use negative numbers. He was a descendant of the Marquis of Zi District (菑鄉侯) of the Eastern Han dynasty, whose marquisate is in present-day Zichuan District, Zibo, Shandong. He completed his commentary to the Nine Chapters in the year 263. He probably visited Luoyang, where he measured the sun's shadow.
The Nine Chapters on the Mathematical Art is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest surviving mathematical texts from China, the first being Suan shu shu and Zhoubi Suanjing. It lays out an approach to mathematics that centres on finding the most general methods of solving problems, which may be contrasted with the approach common to ancient Greek mathematicians, who tended to deduce propositions from an initial set of axioms.
Seki Takakazu, also known as Seki Kōwa, was a Japanese mathematician and author of the Edo period.
Liu Xin, courtesy name Zijun, was a Chinese astronomer, mathematician, historian, librarian and politician during the Western Han Dynasty and Xin Dynasty. He later changed his name to Liu Xiu due to the naming taboo of Emperor Ai of Han. He was the son of Confucian scholar Liu Xiang and an associate of other prominent thinkers such as the philosopher Huan Tan. Liu founded the Old Text school of Confucianism.
Mathematics in China emerged independently by the 11th century BC. The Chinese independently developed a real number system that includes significantly large and negative numbers, more than one numeral system, algebra, geometry, number theory and trigonometry.
The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the nth polygon and the containing shape will become arbitrarily small as n becomes large. As this difference becomes arbitrarily small, the possible values for the area of the shape are systematically "exhausted" by the lower bound areas successively established by the sequence members.
The Zhoubi Suanjing is one of the oldest Chinese mathematical texts. "Zhou" refers to the ancient Zhou dynasty ; "Bi" means thigh and according to the book, it refers to the gnomon of the sundial. The book is dedicated to astronomical observation and calculation. "Suan Jing" or "classic of arithmetics" were appended in later time to honor the achievement of the book in mathematics.
The Book on Numbers and Computation, or the Writings on Reckoning, is one of the earliest known Chinese mathematical treatises. It was written during the early Western Han dynasty, sometime between 202 BC and 186 BC. It was preserved among the Zhangjiashan Han bamboo texts and contains similar mathematical problems and principles found in the later Eastern Han period text of The Nine Chapters on the Mathematical Art.
Ancient Chinese scientists and engineers made significant scientific innovations, findings and technological advances across various scientific disciplines including the natural sciences, engineering, medicine, military technology, mathematics, geology and astronomy.
Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī was a Persian astronomer and mathematician during the reign of Tamerlane.
Magic circles were invented by the Song dynasty (960–1279) Chinese mathematician Yang Hui. It is the arrangement of natural numbers on circles where the sum of the numbers on each circle and the sum of numbers on diameter are identical. One of his magic circles was constructed from 33 natural numbers from 1 to 33 arranged on four concentric circles, with 9 at the center.
Jia Xian was a Chinese mathematician from Kaifeng of the Song dynasty.
Liu Hui's π algorithm was invented by Liu Hui, a mathematician of the Cao Wei Kingdom. Before his time, the ratio of the circumference of a circle to its diameter was often taken experimentally as three in China, while Zhang Heng (78–139) rendered it as 3.1724 or as . Liu Hui was not satisfied with this value. He commented that it was too large and overshot the mark. Another mathematician Wang Fan (219–257) provided π ≈ 142/45 ≈ 3.156. All these empirical π values were accurate to two digits. Liu Hui was the first Chinese mathematician to provide a rigorous algorithm for calculation of π to any accuracy. Liu Hui's own calculation with a 96-gon provided an accuracy of five digits: π ≈ 3.1416.
The mathematical field of combinatorics was studied to varying degrees in numerous ancient societies. Its study in Europe dates to the work of Leonardo Fibonacci in the 13th century AD, which introduced Arabian and Indian ideas to the continent. It has continued to be studied in the modern era.
Suanfa tongzong is a mathematical text written by sixteenth century Chinese mathematician Cheng Dawei (1533–1606) and published in the year 1592. The book contains 595 problems divided into 17 chapters. The book is essentially general arithmetic for the abacus. The book was the main source available to scholars concerning mathematics as it developed in China's tradition. Six years after the publication of Suanfa Tongzong, Cheng Dawei published another book titled Suanfa Zuanyao. About 90% of the content of the new book came from the contents of four chapters of the first book with some rearrangement. It is said that when Suanfa Tongzong was first published, it sold so many copies that the cost of paper went up and the lucrative sales resulted in unscrupulous people beginning to print pirated copies of the book with many errors. It was this that forced the author to print an abridged version.