The sciences of astronomy, physics, chemistry, meteorology, seismology, technology, engineering, and mathematics can trace their early origins to China. From 600 AD until 1500 AD, China was the world's most technologically advanced society. Beginning in the 14th Century BC, the Chinese developed a decimal, or base ten system of recording numbers. This is one of the earliest such systems known.

There are many notable contributors to the field of Chinese science throughout the ages. One of the best examples would be Shen Kuo (1031–1095), a polymath scientist and statesman who was the first to describe the magnetic-needle compass used for navigation, discovered the concept of true north, improved the design of the astronomical gnomon, armillary sphere, sight tube, and clepsydra, and described the use of drydocks to repair boats. After observing the natural process of the inundation of silt and the find of marine fossils in the Taihang Mountains, hundreds of miles from the Pacific Ocean, Shen Kuo devised a theory of land formation, or geomorphology. He also adopted a theory of gradual climate change in regions over time, after observing petrified bamboo found underground at Yan'an, Shaanxi province. If not for Shen Kuo's writing, the architectural works of Yu Hao would be little known, along with the inventor of movable type printing, Bi Sheng (990-1051). Shen's contemporary Su Song (1020–1101) was also a brilliant polymath, an astronomer who created a celestial atlas of star maps, wrote a pharmaceutical treatise with related subjects of botany, zoology, mineralogy, and metallurgy, and had erected a large astronomical clocktower in Kaifeng city in 1088. To operate the crowning armillary sphere, his clocktower featured an escapement mechanism and the world's oldest known use of an endless power-transmitting chain drive.

The Jesuit - China missions of the 16th and 17th centuries learned to appreciate the scientific achievements of this ancient culture and made them known in Europe. Through their correspondence European scientists first learned about the Chinese science and culture. Modern science is indebted to the ancient Chinese scientists in number of ways. Even in disciplines as diverse as physics and social science you will be able to find at least some Chinese influence. In astronomy, the first planetarium was invented by the ancient Chinese. The official astronomers were paid to keep track of the solar, lunar, and planetary motions. In chemistry ancient Chinese chemists with extra ordinary talent used to experiment with number of herbs, animals organs, minerals and many other elements and in the process they invented a range of life saving and healing drugs.

Emperor Gaozong (r. 649-683) of the Tang Dynasty (618-907) commissioned the scholarly compilation of a materia medica in 657 that documented 833 medicinal substances taken from stones, minerals, metals, plants, herbs, animals, vegetables, fruits, and cereal crops. In his Bencao Tujing (Illustrated Pharmacopoeia), the scholar-official Su Song (1020-1101) not only systematically categorized herbs and minerals according to their pharmaceutical uses, but he also took an interest in zoology. For example, Su made systematic descriptions of animal species and the environmental regions they could be found, such as the freshwater crab Eriocher sinensis found in the Huai River running through Anhui, in waterways near the capital city, as well as reservoirs and marshes of Hebei.

Experimentation with various materials and ingredients in China during the middle period led to the discovery of many ointments, creams, and other mixtures with practical uses. In a 9th century Arab work Kitab al-Khawass al Kabir, there are numerous products listed that were native to China, including waterproof and dust-repelling cream or varnish for clothes and weapons, a Chinese lacquer, varnish, or cream that protected leather items, a completely fire-proof cement for glass and porcelain, recipes for Chinese and Indian ink, a waterproof cream for the silk garments of underwater divers, and a cream specifically used for polishing mirrors.

The expertise of the ancient Chinese in the area of physics led the first among other civilizations to invent complex machines. The wheel burrow, the blast furnace, the grooves, all are the evidence of Chinese scientists’ familiarity with the basics laws of physics. Their invention and uses of compass or their experimentations with the several aspects of earthquake suggests that they took great interest in the earth science also.

Ancient Chinese Mathematics

Several factors led to the development of mathematics in China. The geographical nature of the country meant that there were natural boundaries, mountains and seas, which isolated it. On the other hand, when the country was conquered by foreign invaders, they were assimilated into the Chinese culture rather than changing the culture to their own. As a consequence there was a continuous cultural development in China from around 1000 BC and it is fascinating to trace mathematical development within that culture. There are periods of rapid advance, periods when a certain level was maintained, and periods of decline.

Unlike Greek mathematics there is no axiomatic development of mathematics. The Chinese concept of mathematical proof is radically different from that of the Greeks, yet one must not in any sense think less of it because of this. Rather one must marvel at the Chinese approach to mathematics and the results to which it led.

Chinese mathematics was, like the language, very concise. It was very much problem based, motivated by problems of the calendar, trade, land measurement, architecture, government records and taxes. By the fourth century BC counting boards were used for calculating, which effectively meant that a decimal place valued number system was in use. It is worth noting that counting boards are uniquely Chinese, and do not appear to have been used by any other civilisation.

Our knowledge of Chinese mathematics before 100 BC is very sketchy although in 1984 the Suan Shu Shu (A Book on Arithmetic) dating from around 180 BC was discovered. It is a book written on bamboo strips and was found near Jiangling in Hubei province. The next important books of which we have records are a sixteen chapter work Suanshu (Computational prescriptions) written by Du Zhong and a twenty-six chapter work Xu Shang suanshu (Computational prescriptions of Xu Shang) written by Xu Shang. Neither of these texts has survived and little is known of their content. The oldest complete surviving text is the Zhoubi suanjing (Zhou Shadow Gauge Manual) which was compiled between 100 BC and 100 AD (see the article on The Ten Classics). It is an astronomy text, showing how to measure the positions of the heavenly bodies using shadow gauges which are also called gnomons, but it contains important sections on mathematics.

The method of calculation is very simple to explain but has wide application. This is because a person gains knowledge by analogy, that is, after understanding a particular line of argument they can infer various kinds of similar reasoning ... Whoever can draw inferences about other cases from one instance can generalise ... really knows how to calculate... . To be able to deduce and then generalize.. is the mark of an intelligent person.

Much of Chinese mathematics from this period was produced because of the need to make calculations for constructing the calendar and predicting positions of the heavenly bodies. The Chinese word 'chouren' refers to both mathematicians and astronomers showing the close link between the two areas. One early 'choren' was Luoxia Hong (about 130 BC - about 70 BC) who produced a calendar which was based on a cycle of 19 years.

The most famous Chinese mathematics book of all time is the Jiuzhang suanshu or, as it is more commonly called, the Nine Chapters on the Mathematical Art. The book certainly contains contributions to mathematics which had been made over quite a long period, but there is little in the original text to distinguish the precise period of each. This important work, which came to dominate mathematical development and style for 1500 years, is discussed in the article Nine Chapters on the Mathematical Art. Many later developments came through commentaries on this text, one of the first being by Xu Yue (about 160 - about 227) although this one has been lost.

A significant mathematical advance was made by Liu Hui (about 220 - about 280) who wrote his commentary on the Jiuzhang suanshu or Nine Chapters on the Mathematical Art in about 263. Liu Hui, a great mathematician in the Wei Jin Dynasty, ushered in an era of mathematical theorisation in ancient China, and made great contributions to the domain of mathematics. From the "Jiu Zhang Suan Shu Zhu" and the "Hai Dao Suan Jing" it can be seen that Liu Hui made skilful use of thinking in images as well as in logical and dialectical ways. He solved many mathematical problems, pushing his mathematical reasoning further along the dialectical way.

Liu Hui gave a more mathematical approach than earlier Chinese texts, providing principles on which his calculations are based. He found approximations to using regular polygons with 3 2n sides inscribed in a circle. His best approximation of was 3.14159 which he achieved from a regular polygon of 3072 sides. It is clear that he understood iterative processes and the notion of a limit. Liu also wrote Haidao suanjing or Sea Island Mathematical Manual (see the article on The Ten Classics) which was originally an appendix to his commentary on Chapter 9 of the Nine Chapters on the Mathematical Art. In it Liu uses Pythagoras's theorem to calculate heights of objects and distances to objects which cannot be measured directly. This was to become one of the themes of Chinese mathematics.

About fifty years after Liu's remarkable contributions, a major advance was made in astronomy when Yu Xi discovered the precession of the equinoxes. In mathematics it was some time before mathematics progressed beyond the depth achieved by Liu Hui. For example Sun Zi (about 400 - about 460) wrote his mathematical manual the Sunzi suanjing which on the whole provides little new. However, it does contains a problem solved using the Chinese remainder theorem, being the earliest known occurrence of this type of problem.

This text by Sun Zi was the first of a number of texts over the following two hundred years which made a number of important contributions. Xiahou Yang (about 400 - about 470) was the supposed author of the Xiahou Yang suanjing (Xiahou Yang's Mathematical Manual) which contains representations of numbers in the decimal notation using positive and negative powers of ten. Zhang Qiujian (about 430 - about 490) wrote his mathematical text Zhang Qiujian suanjing (Zhang Qiujian's Mathematical Manual) some time between 468 and 486. Its 92 problems illustrate the formula for summing an arithmetic progression. Perhaps it is most famous for presenting the 'Hundred fowls problem' which is an indeterminate problem with three non-trivial solutions.

One of the most significant advances was by Zu Chongzhi (429-501) and his son Zu Geng (about 450 - about 520). Zu Chongzhi was an astronomer who made accurate observations which he used to produce a new calendar, the Tam-ing Calendar (Calendar of Great Brightness), which was based on a cycle of 391 years. He wrote the Zhui shu (Method of Interpolation) in which he proved that 3.1415926 < p < 3.1415927. He recommended using 355/113 as a good approximation and 22/7 in less accurate work. With his son Zu Geng he computed the formula for the volume of a sphere using Cavalieri's principle. The beginnings of Chinese algebra is seen in the work of Wang Xiaotong (about 580 - about 640). He wrote the Jigu suanjing (Continuation of Ancient Mathematics), a text with only 20 problems which later became one of the Ten Classics. He solved cubic equations by extending an algorithm for finding cube roots. His work is seen as a first step towards the "tian yuan" or "coefficient array method" or "method of the celestial unknown" of Li Zhi for computing with polynomials.

Interpolation was an important tool in astronomy and Liu Zhuo (544-610) was an astronomer who introduced quadratic interpolation with a second order difference method. Certainly Chinese astronomy was not totally independent of developments taking place in the subject in India and similarly mathematics was influenced to some extent by Indian mathematical works, some of which were translated into Chinese. Historians argue today about the extent of the influence on the Chinese development of Indian, Arabic and Islamic mathematics. It is fair to say that their influence was less than it might have been, for the Chinese seemed to have little desire to embrace other approaches to mathematics. Early trigonometry was described in some of the Indian texts which were translated and there was also development of trigonometry in China. For example Yi Xing (683-727) produced a tangent table.

From the sixth century mathematics was taught as part of the course for the civil service examinations. Li Chunfeng (602 - 670) was appointed as the editor-in-chief for a collection of mathematical treatises to be used for such a course, many of which we have mentioned above. The collection is now called The Ten Classics, a name given to them in 1084.

The period from the tenth to the twelfth centuries is one where few advances were made and no mathematical texts from this period survive. However Jia Xian (about 1010 - about 1070) made good contributions which are only known through the texts of Yang Hui since his own writings are lost. He improved methods for finding square and cube roots, and extended the method to the numerical solution of polynomial equations computing powers of sums using binomial coefficients constructed with Pascal's triangle. Although Shen Kua (1031 - 1095) made relatively few contributions to mathematics, he did produce remarkable work in many areas and is regarded by many as the first scientist. He wrote the Meng ch'i pi t'an (Brush talks from Dream Brook) which contains many accurate scientific observations.

The next major mathematical advance was by Qin Jiushao (1202 - 1261) who wrote his famous mathematical treatise Shushu Jiuzhang (Mathematical Treatise in Nine Sections) which appeared in 1247. He was the first of the great thirteenth century Chinese mathematicians. This was a period of major progress during which mathematics reached new heights. The treatise contains remarkable work on the Chinese remainder theorem, gives an equation whose coefficients are variables and, among other results, Heron's formula for the area of a triangle. Equations up to degree ten are solved using the Ruffini-Horner method.

Li Zhi (also called Li Yeh) (1192-1279) was the next of the great thirteenth century Chinese mathematicians. His most famous work is the Ce yuan hai jing (Sea mirror of circle measurements). written in 1248. It contains the "tian yuan" or "coefficient array method" or "method of the celestial unknown" which was a method to work with polynomial equations. He also wrote Yi gu yan duan (New steps in computation) in 1259 which is a more elementary work containing geometric problems solved by algebra. The next major figure from this golden age of Chinese mathematics was Yang Hui (about 1238 - about 1298). He wrote the Xiangjie jiuzhang suanfa (Detailed analysis of the mathematical rules in the Nine Chapters and their reclassifications) in 1261, and his other works were collected into the Yang Hui suanfa (Yang Hui's methods of computation) which appeared in 1275. He described multiplication, division, root-extraction, quadratic and simultaneous equations, series, computations of areas of a rectangle, a trapezium, a circle, and other figures. He also gave a wonderful account of magic squares and magic circles.

Guo Shoujing (1231-1316), although not usually included among the major mathematicians of the thirteen century, nevertheless made important contributions. He produced the Shou shi li (Works and Days Calendar), worked on spherical trigonometry, and solved equations using the Ruffini-Horner numerical method. He also developed a cubic interpolation formula tabulating differences of the accumulated difference as in Newton's forward difference interpolation method.

The last of the mathematicians from this golden age was Zhu Shijie (about 1260 - about 1320) who wrote the Suanxue Qimeng (Introduction to Mathematical Studies) published in 1299, and the Siyuan Yujian (True Reflections of the Four Unknowns) published in 1303. He used an extension of the "coefficient array method" or "method of the celestial unknown" to handle polynomials with up to four unknowns. He also gave many results on sums of series. This represents a high point in ancient Chinese mathematics.

Other notable ancient Chinese mathematical discoveries later practiced in the West include:

A Refined Pi - The first advanced knowledge of the value of pi originated in China, but was forgotten there in the fourteenth century . When the Jesuits went to China in the seventeenth century, the Chinese were impressed by the European knowledge of pi. Here we see a diagram explaining Liu Hui's exhaustion method in 264 AD for finding the value of pi. By inscribing 3072 sides of a polygon in a circle, Liu Hui was able to overtake the Greeks and compute the value to a fifth decimal place at 3.14159. By the fifth century, the value was computed to ten decimal places. In Europe, pi was only approximately calculated to seven places by the year 1600, a full twelve hundred years later.

The irrational number pi can be computed to an infinite number of decimal places. It expresses the ratio of the circumference of a circle to its diameter, a relationship which cannot be framed in terms of whole numbers. (Pi is needed to compute the area of a circle or volume of a sphere.) The value of pi was computed by Archimedes to three decimal places, and by Ptolemy to four decimal places. But after that, for 1450 years, no greater accuracy was achieved in the Western world. The Chinese, however, made great strides forward in computing pi.

Pascal Triangle - Pascal's Triangle was not invented by Blaise Pascal in 1654: it came from China. The so called Pascal triangle was known in China as early as 1261. In 1261 the triangle appears to a depth of six in Yang Hui and to a depth of eight in Zhu Shijiei in 1303. Yang Hui attributes the triangle to Jia Xian, who lived in the eleventh century and used as a means of generating the binomial coefficients. It wasn't until the eleventh century that a method for solving quadratic and cubic equations was recorded, although they seemed to have existed since the first millennium. At this time Jia Xian 'generalized the square and cube root procedures to higher roots by using the array of numbers known today as the Pascal triangle and also extended and improved the method into one useable for solving polynomial equations of any degree' Pascal's triangle was first illustrated in China by Yang Hui in his book Xiangjie Jiuzhang Suanfa, although it was described earlier around 1100 by Jia Xian.

In the first century AD, Chinese scholars compiled a volume of mathematics, Jin Zhang Suanshu, (Arithmetic in Nine Chapters). Mathematician Zu Chongzhi (429-500) calculated the first 12 digits of the value of pi, while his son, Zu Gengzhi, updated the Jin Zhang Suanshu and determined the correct formula for the volume of a sphere, V= (pi/4)d^3, where d is the diameter. By 1100, the mathematician Chia Hsien expounded it at that time as 'the tabulation system for unlocking binomial coefficients'; but its first appearance is thought to have been in a book of that date, now lost, entitled Piling-up Powers and Unlocking Coefficients, by Liu Ju-Hsieh. Yang Hui was also the first person in history to discover and prove "Pascal's Triangle", along with its binomial proof, although the earliest mention of the Pascal's triangle in China exists before the eleventh century C.E. Li Zhi on the other hand, investigated on a form of algebraic geometry. His book; Ce Hai Yuan Jing revolutionized the idea of inscribing a circle into triangles, which could be calculated using equations with the Pythagorean theorem.

The mathematician and poet Omar Khayyam discussed the Pascal Triangle somewhat indirectly about I 100. We do not know whether he got it from China or invented the elements of the system independently. But the first appearance of the Triangle in print in Europe was on the title page of the book on arithmetic of Petrus Apianus in 1527. Several succeeding mathematicians, such as Michael Stifel, considered it. And the Italian Nicolo Tartaglia, who was something of a scoundrel, claimed it as his own invention. But as far as we know, the inventor was indeed Liu Ju-Hsich, 427 years before the appearance of the 'Pascal' Triangle in Europe. This diagram comes from Chu Shih-Chieh's Precious Mirror of the Four Elements, published in 1303. The caption refers to the triangle as the 'Old Method'; it had been expounded by the year I 100 by the mathematician Chia Hsien, who called it 'the tabulation system for unlocking binomial coefficients'. The Zhou Bi Suan Jing dates around 1200-1000BCE, yet many scholars believed it was written between 300-250BCE. The Zhou Bi Suan Jing contains an in depth proof of the Gougu Theorem (Pythagorean Theorem) but focuses more on astronomical calculations.

A Mathematical Place for Zero - It is recognized the world-over that the Chinese took the first step in developing the concept of zero, necessary for carrying out even the most simple of mathematical computations. As early as the 4th century BCE, the Chinese started leaving a blank space for the zero symbol, used in conjunction with the traditional Chinese counting board and the smaller abacus; and evidence exists attributing to the Chinese the use of the actual "0" before 686 AD. Later, Qin Jiushao (c. 1202-1261) was the first to introduce the zero symbol into Chinese mathematics. Before this innovation, blank spaces were used instead of zeros in the system of counting rods.

Geometry - The oldest existent work on geometry in China comes from the philosophical Mohist canon of c. 330 BC, compiled by the followers of Mozi (470 BC-390 BC). The Mo Jing described various aspects of many fields associated with physical science, and provided a small wealth of information on mathematics as well. It provided an 'atomic' definition of the geometric point, stating that a line is separated into parts, and the part which has no remaining parts (i.e. cannot be divided into smaller parts) and thus forms the extreme end of a line is a point. Much like Euclid's first and third definitions and Plato's 'beginning of a line', the Mo Jing stated that "a point may stand at the end (of a line) or at its beginning like a head-presentation in childbirth. (As to its invisibility) there is nothing similar to it." Similar to the atomists of Democritus, the Mo Jing stated that a point is the smallest unit, and cannot be cut in half, since 'nothing' cannot be halved. It stated that two lines of equal length will always finish at the same place, while providing definitions for the comparison of lengths and for parallels, along with principles of space and bounded space. It also described the fact that planes without the quality of thickness cannot be piled up since they cannot mutually touch. The book provided definitions for circumference, diameter, and radius, along with the definition of volume.

Chinese Remainder Theorem - The Chinese remainder theorem, including simultaneous congruences in number theory, was first created by the mathematician Sun Zi in the 3rd century AD, whose Mathematical Classic by Sun Zi (Sunzi suanjing) posed the problem: "There is an unknown number of things, when divided by 3 it leaves 2, when divided by 5 it leaves 3, and when divided by 7 it leaves a remainder of 2. Find the number." This method of calculation was used in calendrical mathematics by Tang Dynasty (618-907) mathematicians such as Li Chunfeng (602-670) and Yi Xing (683-727) in order to determine the length of the "Great Epoch", the lapse of time between the conjunctions of the moon, sun, and Five Planets (those discerned by the naked eye). Thus, it was strongly associated with the divination methods of the ancient Yijing. Its use was lost for centuries until Qin Jiushao (c. 1202-1261) revived it in his Mathematical Treatise in Nine Sections of 1247, providing constructive proof for it.

Decimal fractions - As proven by inscriptions from the 13th century BC, the decimal system existed in China since the Shang Dynasty (c. 1600-c. 1050 BC). The earliest evidence of a decimal fraction, where the fraction's denominator is a power of ten, appears on an inscription of a standard measure of volume used by the mathematician and astronomer Liu Xin (c. 46 BC-23 AD), dated precisely 5 AD. The first significant piece of Chinese literature to feature decimal fractions was the The Nine Chapters on the Mathematical Art. This text was first mentioned in 179 AD, although Liu Hui (fl. 3rd century AD) asserts that some of its material predates the infamous Qin book burning in 213 BC (i.e. older than the oldest surviving Chinese mathematical treatise, the Book on Numbers and Computation, 202-186 BC). Liu Hui used decimal fractions with measurements and as solutions to equations. At first decimal fractions were written in word form, since it was Han Yan (fl. late 8th century) of the Tang Dynasty (607-907) who first used modern decimal notation to write out decimal fractions. Decimal fractions were vital to the work of Song (960-1279) mathematicians such as Yang Hui (1238-1298) and Qin Jiushao (c. 1201-1261). Ghiyath al-Din Jamshid Mas'ud al-Kashi (1380-1429), director of the astronomical observatory at Samarkand, adopted the use of decimal fractions; they were first mentioned in Europe by Christoff Rudolff of Augsburg in his Exempel-Buechlin of 1530, yet not given serious attention until the 1585 work of the Flemish mathematician Simon Stevin (1548-1620).

Equal Temperament - During the Han Dynasty (202 BC-220 AD), the music theorist and mathematician Jing Fang (78-37 BC) extended the 12 tones found in the 2nd century BC Huainanzi to 60. While generating his 60-divisional tuning, he discovered that 53 just fifths is approximate to 31 octaves, calculating the difference at ; this was the exact same value for 53 equal temperament calculated by the German mathematician Nicholas Mercator (c. 1620-1687) as 353/284, a value known as Mercator's Comma. The Ming Dynasty (1368-1644) music theorist Zhu Zaiyu (1536-1611) elaborated in three separate works beginning in 1584 the tuning system of equal temperament; in an unusual event in music theory's history, the Flemish mathematician Simon Stevin (1548-1620) discovered the mathematical formula for equal temperament at roughly the same time (within 1 to 25 years of Zhu), yet he did not publish his work and it remained unknown until 1884; therefore, it is debatable who discovered equal temperament first, Zhu or Stevin. In order to obtain equal intervals, Zhu divided the octave (each octave with a ratio of 1:2, which can also be expressed as 1:212/12) into twelve equal semitones while each length was divided by the 12th root of 2 He did not simply divide the string into twelve equal parts (i.e. 11/12, 10/12, 9/12, etc.) since this would give unequal temperament; instead, he altered the ratio of each semitone by an equal amount (i.e. 1:2 11/12, 1:210/12, 1:29/12, etc.) and determined the exact length of the string by dividing it by 12?2 (same as 21/12). The Harmonie Universelle (1636) written by Marin Mersenne (1588-1648) was the first publication in Europe outlining equal temperament, a new system of tuning that was passionately defended by J.S. Bach (1685-1750) in his Well-Tempered Clavier of 1722.

First Law of Motion - The Mohist philosophical canon of the Mojing, compiled by the followers of Mozi (c. 470 - c. 390 BC), provides the earliest known attempt to describe inertia: "The cessation of motion is due to the opposing force...If there is no opposing force...the motion will never stop. This is as true as that an ox is not a horse." However, like many of the Hundred Schools of Thought during the Warring States Period (403-221 BC), the doctrine of the Mohist sect had little impact on the course of later Chinese thought, while this passage and others from the Mojing were only given serious attention by modern scholarship after the work of Joseph Needham in 1962.

Gaussian Elimination - First published in the West by Carl Friedrich Gauss (1777-1855) in 1826, the algorithm for solving linear equations known as Gaussian elimination is named after this Hanoverian mathematician, yet it was first expressed as the Array Rule in the Chinese Nine Chapters on the Mathematical Art, written at least by 179 AD during the Han Dynasty (202 BC-220 AD) and commented on by the 3rd century mathematician Liu Hui.

Horner Scheme - Although named after English mathematician William George Horner (1786-1837), the Horner scheme, an algorithm used to estimate the root of an equation and evaluate polynomials in monomial form, was actually first invented in China to find the cube root of the number 1,860,867 (the answer given being 123). This is found in the Han Dynasty (202 BC-220 AD) work The Nine Chapters on the Mathematical Art, commented on by Liu Hui (fl. 3rd century) in 263 AD. The original Nine Chapters found the root of equations through continued fractions, just like the later Italian mathematician Joseph Louis Lagrange (1736-1813), while Liu Hui achieved this by increasing decimals, just like William George Horner in his work of 1819.

Negative Numbers, Symbols For and Use Of - In the Nine Chapters on the Mathematical Art compiled during the Han Dynasty (202 BC-220 AD) by 179 AD and commented on by Liu Hui (fl. 3rd century) in 263,[12] negative numbers appear as black rods and positive numbers as red rods in the Chinese counting rods system. Liu Hui also used slanted counting rods to denote negative numbers. Negative numbers denoted by a "+" sign also appear in the ancient Bakhshali manuscript of India, yet scholars disagree as to when it was compiled, giving a collective range of 200 to 600 AD. Negative numbers were known in India certainly by about 630 AD, when the mathematician Brahmagupta (598-668) used them. Negative numbers were first used in Europe by the Greek mathematician Diophantus (fl. 3rd century) in about 275 AD, yet were considered absurd in the West until The Great Art written in 1545 by the Italian mathematician Girolamo Cardano (1501-1576).