The Chinese needed controlling the flood prone Yangtze and Yellow Rivers. ... Hindus at the meeting place of the Hindu and Chinese cultures in south-east Asia. ...
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Chinese Numerals one number above the other. The digits added (merged) left to right with carries where needed. Subtraction similar.
In 1899 a major discovery was made at the archaeological site at the village of Xiao dun in the An-yang district of Henan province. Thousands of bones and tortoise shells were discovered there which had been inscribed with ancient Chinese characters. The site had been the capital of the kings of the Late Shang dynasty (this Late Shang is also called the Yin) from the 14th century BC. The last twelve of the Shang kings ruled here until about 1045 BC and the bones and tortoise shells discovered there had been used as part of religious ceremonies. Questions were inscribed on one side of a tortoise shell, the other side of the shell was then subjected to the heat of a fire, and the cracks which appeared were interpreted as the answers to the questions coming from ancient ancestors.
The importance of these finds, as far as learning about the ancient Chinese number system, was that many of the inscriptions contained numerical information about men lost in battle, prisoners taken in battle, the number of sacrifices made, the number of animals killed on hunts, the number of days or months, etc. The number system which was used to express this numerical information was based on the decimal system and was both additive and multiplicative in nature.
The four "corner" triangles each have area ab/2
giving a total area of 2ab for the four added
together. Hence the inside square (whose
vertices are on the outside square) has area
(a^2 +2ab + b^2) - 2ab = a^2 + b^2.
Its side therefore has length ( a^2 + b^2). Therefore the hypotenuse of the right angled triangle with sides of length a and b has length ( a^2 + b^2).
Eastern Han Dynasty (about 1st century), indicating the formation of ancient Chinese mathematical system. It became the criterion of mathematical learning and research for mathematicians of later generations ever since then.
A = 1/2 (c + s) s
, where A is the area, c the chord & s the sagitta of the segment.
The same expression is found in the works of the Indian mathematician Mahavira about 850 AD.
The book's major achievements: in connection with the time required for people to carry their grain contributions from their native towns to the capital. There are also problems of ratios in connection with the allocation of tax burdens according to population.
1. Devising a systematic treatment of arithmetic operations with fractions, 1,400 years earlier than the Europeans.
2. Dealing with various types of problems on proportions, 1,400 years earlier than the Europeans.
3. Devising methods for extracting square root and cubic root, which is quite similar to today's method, several hundred years earlier than the Western mathematicians.
4. Developing solutions for a system of linear equations, about 1,600 years earlier than the Western mathematicians.
5. Introducing the concepts of positive and negative numbers, more than 600 years earlier than the West.
6. Developing a general solution formula for the Pythagorean problems (problems of Gou gu), 300 years earlier than the West.
7. Putting forward theories of calculating areas and volumes of different shapes and figures.
The Abacus numbers, more than 600 years earlier than the West.
For numbers up to 4 slide the required number of beads in the lower part up to the middle bar.
For example on the right most wire two is represented. For five or above, slide one bead above the middle bar down (representing 5), and 1, 2, 3 or 4 beads up to the middle bar for the numbers 6, 7, 8, or 9 respectively. For example on the wire three from the right hand side the number 8 is represented (5 for the bead above, three beads below).