Willy Ngin
AMAT 452: History of Mathematics

Mathematical History of China and India

Since the beginning of time mathematics has been a part of history. Throughout time without mathematics we wouldn’t have been able to make fundamental advances in science, engineering, technology and much more. Although every country has different histories, cultures and lifestyles; one thing that remains the same is the universal language of Mathematics. If you go to any country in the world, mathematics will always be the same. Addition will always be addition and subtraction will always be subtraction anywhere. Some of the countries who have been able to help further our discoveries and advances in mathematics were China and India. China’s history included many different wars which led to a lot of different dynasties taking over the country. Still, ”the demands of the empire for administrative services, including surveying, taxation, and calendar making, required that many civil servants be competent in certain areas of mathematics” (Katz, 2009, p. 197). It wasn’t until 1984 when they opened the tombs that they found some of the mathematic history. “Among the books was discovered a mathematics text written on 200 bamboo strips. This work, called the Suan shu shu (Book of Numbers and Computation), is the earliest extant text of Chinese mathematics.” (Katz, 2009, p. 196). This work was created during the Han Dynasty. It consisted of different problems and their solution. Alongside Suan Shu Shu was the Sea Island Mathematical Manual, which probably was the most practiced and important to ancient Chinese mathematics history. During the Tang Dynasty, a man named Li Chunfeng decided to collect all the history of mathematics in china which led to what became known as the Ten Mathematical Classics. These included methods of solutions to various problems. The four major areas included numerical calculations, geometry, equation solving and the solutions of linear congruencies. Through their numerical calculations the Chinese use their own symbols, based on the powers of 10. It was about the fourth century BCE where they used counting rods (Bamboo rods) about 10 CM long. Here’s an example of their system:

(Katz, 2009, p. 198)
It was through these symbols were they able to make their computations based on their numbers. But it wasn’t until the later that there was concrete evidence of the Chinese using fractions. Fen Zhi was used in fractions, “For example, 2/3 would be written 3 fen zhi 2 and could be translated as ‘2 parts from a whole broken into 3 equal parts.’” (Katz, 2009, p. 198). In geometry Chinese mathematicians developed important theoretical principles to solve harder problems. They created formulas for solving both volume and area. It is amazing that in their time they were able to create formulas for rectangles, triangles, parallelepipeds and pyramids. Here’s an example where the Chinese gave four separate formulas to calculate the area of a circle:

(Katz, 2009, p. 201)
The only difference is that they didn’t use Pi but instead 3. One thing they did fail to do was to tell us why these formulas are correct. The Chinese used two methods to solve linear equations, the first method was known as the “Surplus and deficiency” method and the second one was identical to that of Gaussian Elimination Method. The first algorithm was not told about how he arrived at this method, but today we know the formula as: x= (b1x2 +b2x1) / (b1+b2). As for the second method, the Chinese presented the system of equations in matrix form. Just like the first method, the author once again did not prove why this worked but later Lui Hui was able to “justifying the procedure by quoting the “axiom” that when one subtracts equals from equals, the remainders are equal.” (Katz, 2009, p. 212). Lastly, the most famous mathematical technique which originated from china was the Chinese Remainder Theorem. The earliest use of this theorem was by a man named, Sun Zi. He was given the problem: N = 3x + 2, N = 5y +3 and N = 7z + 2. He was able to give his answer and method to the solution: “If you count by threes and have the remainder 2, put 140. If you count by fives and have the remainder 3, put 63. If you count by sevens and have the remainder 2, put 30. Add these numbers and you get 233. From this subtract 210 and you get 23.” (Katz, 2009, p. 222). India was just like China in many ways. They dealt with invasions from people like Alexander the Great. They also dealt with many different falls and rises of new kingdoms. It was evident that most Indians were told to study astronomy for calendrical and astronomical advice. Just like many Chinese mathematicians, they went beyond their job task and fed their intellectual minds. Some of the things they were able to accomplish through their many hardships were, “number systems and methods of calculation, then the geometry of the Sulbasutras and later, next the algebraic methods developed in the medieval period to solve equations (including the so-called Pell equation), next the beginning of combinatorics, and then the development of trigonometry and associated techniques” (Katz, 2009, p. 232). Their number writing systems dates back to the third century BCE it originates from the Brahmi system of writing in India during King Ashoka reign. It was also noted that around the 9th century is when Indians started using decimal, although it may have been earlier this was the only time with evidence. Aryabhata was a indian mathematician who was able to present the methods of calculating square and cube roots. His method was: “One should divide the second noncube [place] by three times the square of the root of the cube. The square [of the quotient] multiplied by three and the former [quantity] should be subtracted from the first [noncube place] and the cube from the cube [place]” (Katz, 2009, p. 235). Another Indian mathematician Brahmagupta had details of arithmetic calculations rules for fractions, positive and negative numbers as well as 0. During the mathematical history of India, they were able to somewhat prove the Pythagorean Theorem by mathematician Jyesthadeva in the book, Yuktibhasa. He used two right triangles and squares. During the fifth century it was known that Indian mathematicians were able to find a rule for solving quadratic equations. Aryabhata was able to somewhat figure the formula:

(Katz, 2009, p. 242)
This translated into the formula we know today as: Sn = n[((((n-1)/2)*d) + a)] = (n/2)[ a + (a + (n-1)/d)]. It is still unknown today if Indian mathematician’s learned this from the Babylonians or Diophantus. They did however originated the method for solving linear congruencies. Although it was Aryabhata’s work that used the method for solving the problem, it wasn’t until Brahmagupta was able to clear up some of the translating problems. This showed that the Indians and Chinese both had the same use for solving congruence problems which was for astronomy. There were many other important mathematical advances done by the Indians. One of them includes the constructions of sines tables. “The earliest known Indian work containing trigonometry is the Pait¯amahasiddh¯anta, written in the early fifth century.” (Katz, 2009, p. 252). Combinatorical rules were also founded in India but did not contain any justifications or proof. The Pell Equation was the first developed case and the high point in Indian algebra. Lastly, the Indians used power series to help with their navigation. It was all of these advances which help lay foundation to future mathematicians. Chinese and Indian mathematical history were closely related. A lot of their discoveries came from the demands of their various rulers. It was the mathematicians who took it upon themselves to continue their desire to learn more which led them to solving much more.