Essay on Origins final

Leo Huang
Origins of world mathematics
Oct. 29

Talents in the east The history of mathematics is consisted of a lot of different mathematical systems in different parts of the world. One of the systems, the ancient Chinese mathematics, is an advanced mathematics system that focused more on problem solving. The ancient Chinese people found out clever ways of solving problems different from those in modern day or in other cultures. The algebra level of ancient China is very high. Most of the notable books of Ancient China talk a lot about Algebra, especially number theory. The book, Sun Zi Suan Jing, contains the earliest written method of doing square root,1 and the method is still used by the calculators nowadays.
Chapter 3 of the book, the most well-known one, the Chinese remainder theorem, is known as the four basic theorems of number theory in modern days. The problem was, “Now we have an unknown number of peaches. If you count them by three, you get two left. If you count them by five, you get three left. If you count them by seven, you get two left. Then what is the number of peaches?” To solve this problem, the Chinese found out that 70 is a multiple of five and seven and the remainder is one when it is divided by three. Because of this property, if you multiply the remainder of three by seventy, the result is a number that is a multiple of five and seven, and when it is divided by three, the remainder is the same as the remainder of the original number. Similarly, if you multiply the remainder of five by 21 and the remainder of seven by 15, and finally adding those three numbers together, you will keep all the three remainders. Take three as an example, both three times 21 and two times 15 are multiples of three, so the only number that determines the remainder of three is 2 times 70, which will keep the remainder of three the same of the original number. Adding the three number together gets people 233, a number that satisfies the three conditions, but not necessarily the smallest number. To find the smallest, you just have to subtract the least common multiple of 3, 5 and 7, 105, to find the smallest, which is 23. The whole process is long and complicated, so the ancient Chinese people made a poem to make it easier to memorize: “三人同行七十稀，五树梅花廿一支，七子团圆正半月，减百零五便得知”. This poem means, multiply the remainder of three by 70, the remainder of five by 21, the remainder of seven by 15, then subtracting 105 to get the smallest number. However, the ancient way is limited. If the divisor gets bigger, it will be hard to find the multiplier. It requires one as the remainder when it is divided by one of the divisors. The modern way can solve this problem. First, we simply find a number that meets the first two conditions, which is eight. Then, by adding 15, which is the least common multiple of three and five, we can ultimately find a number that meets the third condition. Even the modern way is easier than the ancient Chinese way, it is clear that the ancient Chinese way is much more of a “mathematical” way, and there is more mathematical principles hidden. In modern times, people abstract it and found out one of the four basic theorems of number theory: The Chinese Remainder Theorem. In addition, at the time when functions had not been invented, the ancient Chinese figured out clever ways to do problems in different ways. However, because there is no universal solution like functions, the Chinese had to figure out different ways of doing different problems.
For example, a classical problem, A cage with rabbits and chickens, asks, “An unknown number of rabbits and chickens were locked in a cage, count from top, there was 35 heads; count from bottom, there was 94 feet. How many rabbits and chickens were locked in this cage?” The modern solution is to use functions. Simply setting the number of chickens to x, number of rabbits to (35-x) will get you the answer easily. The ancient Chinese did it differently: they

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