Introduction

Throughout history, mathematicians from various regions have made significant contributions to the field, bringing new innovative ideas, challenging Eurocentric notions and enriching the global mathematical landscape. One great example of such is China, which has produced ground-breaking work, often surpassing European mathematicians.

Ancient Chinese Mathematics

Chinese civilisation boasts a rich history in mathematical achievements that date back thousands of years. It is important to recognise the advancements in Chinese mathematics, particularly ones which were overshadowed by their European counterparts.

Oracle Bones

Oracle bones offer a glimpse into the ancient civilization that flourished over 3,000 years ago. They were usually made out of ox bone or turtle shell^[1] and characters were directly carved onto them, mainly for religious divination.

Beyond their role as divinatory tools and insights into ancient Chinese society, oracle bones hold intriguing connections to the world of mathematics. These artifacts not only reveal the mathematical knowledge possessed by the Chinese during the Shang Dynasty but also provide evidence of the earliest uses of numerical systems and calculations in Chinese history^[2].

The oracle bones' base 10 system was relatively simple. The first ten digits follow patterns which have mild resemblances to the current Chinese numerical characters.

Yi Jing 易經 (I Ching)

The Yi Jing is translated as the "Book of Changes" is an ancient Chinese divination text that dates back to at least the Western Zhou period (around 1046 BC - 771 BC). It was often used to provide guidance for moral decision-making..

While the Yi Jing is not inherently a mathematical text, it is one of the first with mathematical elements in its structure and interpretation and greatly influenced literature during the Zhou Dynasty.

The Yi Jing is based on a binary system, which uses two fundamental elements or symbols: the solid line (Yang) and the broken line (Yin). These two lines form the basis of the trigrams and hexagrams used in the I Ching. Yin is often discussed as the female cosmic principle and Yang as the male cosmic principle^[3].

Consulting the Yi Jing

These eight trigrams, known as bagua (八卦), represent a basic force of nature. Each Yi Jing hexagram consists of a combination of two trigrams. There are 64 hexagrams in total, each with its unique name, meaning, and divinatory text.

To consult the Yi Jing, the Chinese used yarrow stalks to create a hexagram. The interpretation of the readings found in the I Ching has been endlessly discussed and debated over the centuries. Qin dynasty records found that the hexagrams were privately consulted to answer questions about business, health, children and determining lucky days ^[4].

Counting rods

The counting board and counting rods, which emerged during the 2nd century BCE during the Warring States period, were one of the most significant mathematical innovations of that era.

The counting board was the chief computing tool, spanning from the Warring States period to 15th century AD.

Different digits are represented by the number of rods. Counting rods were used in both vertical and horizontal formats to prevent ambiguity. There were two sets of rod notations, one for digits in odd (units, hundreds, ten thousands, etc.) positions, the other for digits in even (tens, thousands, hundred thousands, etc.) positions. These two sets were represented by vertical rods and the other represented by horizontal rods and were referred as "zongs" and "hengs" ^[5], respectively.

Counting rods were primarily used to calculate numbers, not record numbers like the Oracle bone script.

Negative numbers

Counting rods were often used for commercial and tax purposes. Red painted rods represented positive numbers and black painted rods represented negative numbers i.e the amount sold was positive and the amount spent purchasing something was negative.

Interestingly, the counting rods system spread to other parts of Asia, including Japan and Korea^[6], where variations of the method were adopted and integrated into their own mathematical traditions. The lasting impact of Chinese counting rods is proof of the ingenuity of ancient mathematical systems and their ability to transcend cultural boundaries.

Suan Shu Shu 筭數書 (Writings on Reckoning)

This text was written during the early Western Han Dynasty on some 200 bamboo strips consisting of 69 mathematical problems ^[8] from a variety of sources. It is one of the earliest known Chinese mathematical treatises.

The Suan Shu Shu covers a wide range of mathematical techniques from elementary fractions through to methods for solving problems that would nowadays be formulated as linear algebraic equations.

Jiu Zhang Suanshu 九章算衔 (The Nine Chapters on the Mathematical Art)

The Jiu Zhang Suanshu or The Nine Chapters on the Mathematical Art is one of the most significant mathematical works in Chinese history. The text focuses more on problem-solving than theory, containing 246 mathematical problems with solutions covering various practical topics such as arithmetic, geometry, algebra, and measurements.

The Nine Chapters provided methods for solving everyday problems related to engineering, surveying, trade, and taxation, making it an essential resource for practical calculations and applications at the time. It was used for centuries to train civil servants in the prestigious imperial bureaucracy. ^[10]

Afterward, numerous mathematicians commented on the Jiu Zhang Suanshu, with famous figures like Liu Hui (263 AD) and Li Chunfeng (656 AD) among them. Liu Hui undoubtedly was the most notable as he provided justification of the rules and explained the methods used.

All the problems take 3 as pi for calculating circles and spheres. Some mathematicians tried to approximate a more accurate value of pi for example, Liu Xin estimated pi as 3.154 but there was not even any record of how he arrived at it.

Liu Hui's Algorithm

Liu Hui was unsatisfied with the approximations that other mathematicians had came up with at the time, commenting that they were too large. He took an approach where he calculated the area of a polygon inscribed in a circle and as he increased the number of sides of the polygon, the area became closer to the area of the circle and using that he calculated an approximate value of π.

Liu Hui argued:

"Multiply one side of a hexagon by the radius (of its circumcircle), then multiply this by three, to yield the area of a dodecagon; if we cut a hexagon into a dodecagon, multiply its side by its radius, then again multiply by six, we get the area of a 24-gon; the finer we cut, the smaller the loss with respect to the area of circle, thus with further cut after cut, the area of the resulting polygon will coincide and become one with the circle; there will be no loss". ^[13]

His own calculation with a polygon with 96 sides yielded an accuracy of four decimal places: π ≈ 3.1416 ^[14].

Yang Hui's triangle

This is the earliest depiction of Pascal’s triangle from Yang’s book Xiangjie Jiuzhang Suanfa 詳解九章算法 (A Detailed Analysis of the Nine Chapters on the Mathematical Art) in 1261 AD, in which he acknowledged that it was invented by mathematician Jia Xian. While the concept of this triangle had been known to different cultures before Yang Hui's time, such as Persian Mathematician Al-Karaji, he is often credited for systemising its properties and exploring its applications.

Each number in the triangle is the sum of the two directly above it, with applications in algebra, combinatorics, and probability theory. One example of its hidden properties is that each row forms the coefficients to the expansion of the binomial expression (a + b)ⁿ , where n is the row number.

Yang Hui 杨辉

Yang Hui was a Chinese mathematician and writer during the Song dynasty.

His work Xiangjie Jiuzhang Suanfa focused on the mathematical problem-solving techniques and further expanded upon the content of the "Nine Chapters on the Mathematical Art."

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).

Very little is known about Yang Hui but his works are of the few that have lasted in Chinese mathematics.