ROMAN MATHEMATICS

By the middle of the 1st Century BC, the Roman had tightened their grip on the old Greek and Hellenistic empires, and the mathematical revolution of the Greeks ground to halt. Despite all their advances in other respects, no mathematical innovations occurred under the Roman Empire and Republic, and there were no mathematicians of note. The Romans had no use for pure mathematics, only for its practical applications, and the Christian regime that followed it (after Christianity became the official religion of the Roman empire) even less so.

Roman arithmetic

Roman numerals are well known today, and were the dominant number system for trade and administration in most of Europe for the best part of a millennium. It was decimal (base 10) system but not directly positional, and did not include a zero, so that, for arithmetic and mathematical purposes, it was a clumsy and inefficient system. It was based on letters of the Roman alphabet - I, V, X, L, C, D and M - combines to
signify the sum of their values (e.g. VII = V + I + I = 7).
Later, a subtractive notation was also adopted, where VIIII, for example, was replaced by IX (10 - 1 = 9), which simplified the writing of numbers a little, but made calculation even more difficult, requiring conversion of the subtractive notation at the beginning of a sum and then its re-application at the end (see image at right). Due to the difficulty of written arithmetic using Roman numeral notation, calculations were usually performed with an abacus, based on earlier Babylonian and Greek abaci

MAYAN MATHEMATICS

Mayan numerals

The Mayan civilisation had settled in the region of Central America from about 2000 BC, although the so-called Classic Period stretches from about 250 AD to 900 AD. At its peak, it was one of the most densely populated and culturally dynamic societies in the world.
The importance of astronomy and calendar calculations in Mayan society required mathematics, and the Maya constructed quite early a very sophisticated number system, possibly more advanced than any other in the world at the time (although the dating of developments is quite difficult).
The Mayan and other Mesoamerican cultures used a vigesimal number system based on base 20 (and, to some extent, base 5), probably originally developed from counting on fingers and toes. The numerals consisted of only three symbols: zero, represented as a shell shape; one, a dot; and five, a bar. Thus, addition and subtraction was a relatively simple matter of adding up dots and bars. After the number 19, larger numbers were written in a kind of vertical place value format using powers of 20: 1, 20, 400, 8000, 160000, etc (see image above), although in their calendar calculations they gave the third position a value of 360 instead of 400 (higher positions revert to multiples of 20).
The pre-classic Maya and their neighbours had independently developed the concept of zero by at least as early as 36 BC, and we have evidence of their working with sums up to the hundreds of millions, and with dates so large it took several lines just to represent them. Despite not possessing the concept of a fraction, they produced extremely accurate astronomical observations using no instruments other than sticks, and were able to measure the length of the solar year to a far higher degree of accuracy than that used in Europe (their calculations produced 365.242 days, compared to the modern value of 365.242198), as well as the length of the lunar month (their estimate was 29.5308 days, compared to the modern value of 29.53059).
However, due to the geographical disconnect, Mayan and Mesoamerican mathematics had absolutely no influence on Old World (European and Asian) numbering systems and mathematics.

CHINESE MATHEMATICS

Ancient Chinese number system

Even as mathematical developments in the ancient Greek world were beginning to falter during the final centuries BC, the burgeoning trade empire of China was leading Chinese mathematics to ever greater heights.
The simple but efficient ancient Chinese numbering system, which dates back to at least the 2nd millennium BC, used small bamboo rods arranged to represent the numbers 1 to 9, which were then places in columns representing units, tens, hundreds, thousands, etc. It was therefore a decimal place value system, very similar to the one we use today - indeed it was the first such number system, adopted by the Chinese over a thousand years before it was adopted in the West - and it made even quite complex calculations very quick and easy.
Written numbers, however, employed the slightly less efficient system of using a different symbol for tens, hundreds, thousands, etc. This was largely because there was no concept or symbol of zero, and it had the effect of limiting the usefulness of the written number in Chinese.
The use of the abacus is often thought of as a Chinese idea, although some type of abacus was in use in Mesopotamia, Egypt and Greece, probably much earlier than in China (the first Chinese abacus, or “suanpan”, we know of dates to about the 2nd Century BC).

Lo Shu magic square, with its traditional graphical representation

There was a pervasive fascination with numbers and mathematical patterns in ancient China, and different numbers were believed to have cosmic significance. In particular, magic squares - squares of numbers where each row, column and diagonal added up to the same total - were regarded as having great spiritual and religious significance.
The Lo Shu Square, an order three square where each row, column and diagonal adds up to 15, is perhaps the earliest of these, dating back to around 650 BC (the legend of Emperor Yu’s discovery of the the square on the back of a turtle is set as taking place in about 2800 BC). But soon, bigger magic squares were being constructed, with even greater magical and mathematical powers, culminating in the elaborate magic squares, circles and triangles of Yang Hui in the 13th Century (Yang Hui also produced a trianglular representation of binomial coefficients identical to the later Pascals’ Triangle, and was perhaps the first to use decimal fractions in the modern form).

Early Chinese method of solving equations

But the main thrust of Chinese mathematics developed in response to the empire’s growing need for mathematically competent administrators. A textbook called “Jiuzhang Suanshu” or “Nine Chapters on the Mathematical Art” (written over a period of time from about 200 BC onwards, probably by a variety of authors) became an important tool in the education of such a civil service, covering hundreds of problems in practical areas such as trade, taxation, engineering and the payment of wages.
It was particularly important as a guide to how to solve equations - the deduction of an unknown number from other known information - using a sophisticated matrix-based method which did not appear in the West until Carl Friedrich Gauss re-discovered it at the beginning of the 19th Century (and which is now known as Gaussian elimination).
Among the greatest mathematicians of ancient China was Liu Hui, who produced a detailed commentary on the “Nine Chapters” in 263 AD, was one of the first mathematicians known to leave roots unevaluated, giving more exact results instead of approximations. By an approximation using a regular polygon with 192 sides, he also formulated an algorithm which calculated the value of π as 3.14159 (correct to five decimal places), as well as developing a very early forms of both integral and differential calculus.

The Chinese Remainder Theorem

The Chinese went on to solve far more complex equations using far larger numbers than those outlined in the “Nine Chapters”, though. They also started to pursue more abstract mathematical problems (although usually couched in rather artificial practical terms), including what has become known as the Chinese Remainder Theorem. This uses the remainders after dividing an unknown number by a succession of smaller numbers, such as 3, 5 and 7, in order to calculate the smallest value of the unknown number. A technique for solving such problems, initially posed by Sun Tzu in the 3rd Century AD and considered one of the jewels of mathematics, was being used to measure planetary movements by Chinese astronomers in the 6th Century AD, and even today it has practical uses, such as in Internet cryptography.
By the 13th Century, the Golden Age of Chinese mathematics, there were over 30 prestigious mathematics schools scattered across China. Perhaps the most brilliant Chinese mathematician of this time was Qin Jiushao, a rather violent and corrupt imperial administrator and warrior, who explored solutions to quadratic and even cubic equations using a method of repeated approximations very similar to that later devised in the West by Sir Isaac Newton in the 17th Century. Qin even extended his technique to solve (albeit approximately) equations involving numbers up to the power of ten, extraordinarily complex mathematics for its time