Wanderer of the whirling world

Chapter 37 "Things Don't Count" and the Remainder Theorem
"A calculation problem in the pre-Qin period?"

As a top student in the Mathematics Department of Yanda University, Xin Ziqiu is very interested in the history of Chinese mathematics in private, and has done a lot of research, but he has never seen any mathematics materials before the Han Dynasty.

Bao Gong's words immediately aroused his interest.

Everyone was puzzled when they saw Xin Ziqiu and Jia Xian's reaction.

Jiang Ping, who is senior, asked for everyone:
"Mr. Jia, you still don't know what the question is about. How could you be so surprised? Could it be something weird?"

Jia Xian smiled and said:

"I'm laughing at you all. I don't mean to belittle the ancients. It's really the old man who has been studying the art for many years, but it's the first time I've seen a calculation problem from the pre-Qin period. I'm bound to lose my composure."

Seeing the people around him still looking at him with a puzzled expression, he thought for a while and continued to explain:

"Actually, as early as after King Wu defeated Zhou, the Duke of Zhou established rites and music. At that time, the study of arithmetic and mathematics was already one of the 'six arts', and it was also a compulsory course for the children of the famous nobles at that time. In the "Ten Books of Suanjing" we studied The "Zhoubi Suanjing" borrowed the question and answer between the Duke of Zhou and Shang Gao to explain the most important principles in many mathematical questions. However, this book was not written by the Duke of Zhou, but was written in the Han Dynasty. But it’s just in the name of Duke Zhou.”

"Besides, there are almost only scattered fragments about the development of the science of mathematics in the pre-Qin countries, and there has never been a truly systematic work handed down to later generations. Therefore, the old man heard that this golden tablet came from the pre-Qin period, and It actually recorded a calculation problem, and I was really surprised."

Listening to his explanation, Xin Ziqiu nodded frequently, agreeing very much.

……

In fact, in the pre-Qin period, mathematics played a very important role in the process of the ancients moving from barbarism to civilized society.

As early as the Yin and Shang Dynasties, the decimal counting method had been recorded in oracle bone inscriptions.

During the Warring States Period, unprecedented changes occurred in the land of China. The vassal states changed from slavery to feudalism, and the production level was greatly improved. Mathematics was widely used, including land surveying, trading and exchange, and even city design and construction. , taxes, etc., and so on.

Coupled with the contention of various schools of thought at that time, a large number of thinkers and doers emerged, and the development of mathematics was very rapid.

It is a pity that due to successive years of wars and Qin Shihuang's burning of books and burying Confucianism, no mathematical works have been handed down to the world.

However, the development of mathematics during this period laid the foundation for the later Western Han Dynasty's masterpiece of mathematics, "Nine Chapters on Arithmetic".

……

Jia Xian is familiar with "Zhou Bi Suan Jing", "Nine Chapters of Suan Shu", "Sun Tzu Suan Jing" and other works of mathematics in previous dynasties, but he has never seen the calculation questions in the pre-Qin period, so he is very yearning for the problems on the gold slips. But he didn't know the characters of the Warring States Period, so he couldn't understand what was written on it.

During the period of the Eastern Zhou Dynasty, each country had its own characters. It was not until Qin Shihuang unified the six countries that the weights, measures and characters were unified.

Therefore, even if he is as erudite as Jia Xian, it is impossible to know everything, even knowing the languages ​​of the six countries thousands of years ago.

But he didn't know it, which doesn't mean that the text was lost. There were a lot of literati and Confucians in the Song Dynasty. Emperor Renzong gathered several old scholars who specialized in the study of pre-Qin works, and spent a lot of time before finally translating the text on the gold tablet.

Bao Zheng took out another piece of paper from the bookcase, with the deciphered arithmetic problems written on it.

When everyone watched, they saw that the title was like this:

Now there are objects whose numbers are not known. When counted with 84, 23 are left, when counted with one hundred and sixty, seven are left, and when counted with 63, there are two left. How many objects are there?
This topic is expressed in modern Chinese as follows:

There is a pile of objects, and the number is unknown. Count every 84, and there are 23 left; count every 160, and there are 7 left; count every 63, and there are 2 left, ask how many objects there are.

换句话说，就是求被84除余23，被160除余7，被63除余2的最小数。

After Jia Xian finished reading the topic, he fell into deep thought.

All the people present, except Jia Xian and Xin Ziqiu, were not experts in arithmetic. Jiang Ping was fluent in writing and ink. After reading the topic, he slapped his thigh and cursed:

"What's the point of this kind of question? It's purely farting with your pants off. Counting by 84, counting by one hundred and sixty, and counting by 63, can't you count one by one?"

Xin Ziqiu and Jia Xian glanced at him at the same time, they didn't say anything, but they were a little bit contemptuous in their hearts.

It's really playing the piano to the cow. This kind of person doesn't understand mathematics at all, and he doesn't understand the meaning of this kind of topic.

This is not a scientific examination. The important thing about mathematics questions is not the answer, but the general idea extracted from solving the problem.

The abstracted problem-solving methods and steps are the essence.

Just like the famous elementary school math topic:
For a pool, it takes 10 minutes to fill it with water, and 10 minutes to fully drain it. How long does it take to fill the pool with water and drain it at the same time?

Many people jokingly criticize that this kind of question is meaningless, why do you need to pour water while pouring water?In reality, there are no such stupid people.

This has become a widely circulated joke.

But in fact mathematics has never existed to solve one or two problems.

Mathematics is philosophy, the study of world outlook and methodology.

By solving these seemingly simple and unrealistic problems, mathematicians extract effective methods from them to solve more complex and practical problems, and at the same time improve people's understanding of the world.

For example, the problem of water filling in a pool seems insignificant and superfluous, but the idea of ​​differential equations formed during the solution process is the basis for the survival of modern industry.

In fact, as a digression, there are actually many examples of this kind of pool filling problem in reality, such as flood control control of dams, water delivery and intake of water towers, etc., so there is no need to repeat them.

In the same way, this counting problem seems to be superfluous, but at the same time it solves it, it also reveals the remainder theorem in number theory, or the solution of linear indefinite equations.

And the remainder theorem is no longer an irrelevant conclusion.

Even in modern industrial production in the 21st century, the remainder theorem is widely used, such as signal processing, radar detection, etc., too numerous to mention.

……

Seeing Jia Xian concentrating on thinking, Bao Gong seemed to have some experience, so he couldn't help asking:

"What's Mr. Jia's opinion? Several scholars from Sitianjian and Guozijian studied this question together. They spent a lot of time experimenting with numbers one by one. From the beginning, they have been experimenting to more than 7000. I didn't find the answer, so I gave up this method for the time being. It seems that the answer should be a very large number. What do you think, old man?"

Jia Xian nodded and said:
"In "Sun Tzu's Suanjing", there is a topic called 'I don't know the number of things'. The title says, 'Today there are things that I don't know the number of. If you count three or three, you will get two, if you count five or five, you will get three, and if you count seven or seven, you will get three. The remaining two. Ask the geometry?' is very similar to this question."

Bao Gongqi said:

"So that's it. Mr. Jia really has a lot of knowledge and memorization. Then, is there any technique for solving the problem in the book?"

……

(End of this chapter)