Mathematical allusions, figures, interesting calculations, little knowledge 1 knowledge learned in grade 5 and extracurricular knowledge.

◆ The story of pi

1. Zu Chongzhi, ranked seventh, ranked first in the world, maintained for a thousand years; "The accuracy of a country's calculation of pi in history can be used as a sign to measure the level of mathematical development in this country at that time."

2. 1427, Arab mathematician Alkasi,16;

1596, Dutch mathematician Rudolf, 35 years old;

1990, with 480 million computers;

65438+February 6, 2002, Tokyo University, 124 1 1 billion.

◆"0"

Roman numerals have no 0;

In the fifth century, "0" spread from the East to Rome. At that time, the Pope was very conservative and thought that Roman numerals could be used to remember any number, so "0" was forbidden. A Roman scholar's handbook introduced some usages of 0 and 0, and the Pope tortured them after discovering it.

◆ Measure the Fiona Fang of the world with "rules" and "moments"

In the stone statue of an ancient building in Jiaxiang County, Shandong Province, there are two images of our ancient ancestors deified in ancient times, one is Fuxi and the other is Nu Wa. The object in Fuxi's hand is a compass, similar to a compass; The object in Nu Wa's hand is called Moment, which is in the shape of a square.

The Pigeon Cage Principle in Ancient China

In the ancient literature of China, there are many successful examples of applying the pigeon hole principle to analyze problems. For example, in the Song Dynasty Fei Zhou's Records of Liang Man, the pigeon hole principle was used to refute the fallacy of superstitious activities such as "fortune telling". Fei Zhou pointed out that the year, month, day and hour (eight characters) of a person's birth are used as the basis for fortune telling, and the eight characters are used as the "drawer". Only 12×360×60 = 259200 different drawers. Taking people in the world as "things", the person who enters the same drawer must be Qian Qian, so the conclusion is that there are many people born at the same time. But since the "eight characters" are the same, "What's the difference between rich and poor?"

There are similar words in Qian Daxin's Collection of Thousand Words in Qing Dynasty, Ruan Kuisheng's Tea Guest Talk and Chen Qiyuan's Notes on Yongxianzhai. But unfortunately, although Chinese scholars have used the pigeon hole principle for the analysis of specific problems for a long time, there is no universal text about the pigeon hole principle in ancient literature, and no one abstracts it as a universal principle. Finally, they had to name this principle after hundreds of years as Dirichlet, a western scholar.

Application of pigeon hole principle

1947, Hungarian mathematicians introduced this principle into middle school students' mathematics competition. At that time, there was a question in the Hungarian National Mathematics Competition: "Prove that among any six people, you can find three people who know each other or three people who don't know each other."

At first glance, this question seems incredible. But if you know the principle of pigeon hole, it is very simple to prove this problem. We use a, b, c, d, e and f to represent six people. Let's choose one of them, such as A, and put the other five people in two drawers: "Know A" and "Don't Know A". According to the pigeon hole principle, there are at least three people in a drawer. Suppose there are three people in the drawer of "Know A". They are B, C and D. If B, C and D don't know each other, then we have found three people we don't know. If two of B, C and D know each other, for example, B and C know each other, then A, B and C are three people who know each other. In either case, the conclusion of this question is valid.

Because of its novel form and ingenious solution, this test quickly spread all over the world, making many people know this principle. In fact, the pigeon hole principle is not only useful in mathematics, but also plays a role everywhere in real life, such as enrollment, employment arrangement, resource allocation, job title evaluation and so on. It is not difficult to see the function of pigeon cage principle.

rabbit hutch

Have you ever heard of the problem of "chickens and rabbits in the same cage" This question is one of the famous and interesting questions in ancient China. About 1500 years ago, this interesting question was recorded in Sun Tzu's calculation. The book describes it like this: "There are chickens and rabbits in the same cage today, with 35 heads on the top and 94 feet on the bottom. The geometry of chicken and rabbit? These four sentences mean: there are several chickens and rabbits in a cage, counting from the top, there are 35 heads; It's 94 feet from the bottom. How many chickens and rabbits are there in each cage?

Can you answer this question? Do you want to know how to answer this question in Sunzi Suanjing?

The answer is this: If you cut off the feet of every chicken and rabbit in half, then every chicken will become a "one-horned chicken" and every rabbit will become a "two-legged rabbit". In this way, the total number of feet of (1) chickens and rabbits changed from 94 to 47. (2) If there is a rabbit in the cage, the total number of feet is more than the total number of heads 1. So the difference between the total number of feet 47 and the total number of heads 35 is the number of rabbits, that is, 47-35 = 12 (only). Obviously, the number of chickens is 35- 12 = 23.

This idea is novel and strange, and its "foot-cutting method" has also amazed mathematicians at home and abroad. This way of thinking is called reduction. Reduction method means that when solving a problem, we do not directly analyze the problem first, but deform and transform the conditions or problems in the problem until it is finally classified as a solved problem.

Pucsok's interesting question.

Puccio is a famous mathematician in the former Soviet Union. 195 1 wrote a book, Math Teaching Methods in Primary Schools. There is an interesting question in this book.

This shop sold 1026 meter cloth in three days. The sales volume of the second day is twice that of the first day; On the third day, it sold three times as much as the next day. How much rice cloth do you want to sell in three days?

The problem can be thought of this way: the number of meters sold on the first day is regarded as 1 serving. You can draw the following line segment diagram:

The first day 1 serving; The second day is twice as much as the first day; The third grade is three times that of the second grade, and the first grade is 2×3 times.

Comprehensive calculation shows the number of meters of cloth sold on the first day:

1026÷(L+2+6)= 1026÷9 = 1 14(m)

And 1 14× 2 = 228 (m).

228× 3 = 684m

So the fabrics sold in three days are: 1 14m, 228m and 684m respectively.

Please do the problem in this way.

Four people donated money for disaster relief. B donated twice as much as A, C donated three times as much as B, D donated four times as much as C, and they donated 132 yuan. How much do you want each of the four people to donate?

Guigu garlic

There was a general named Han Xin in the Han Dynasty in China. Every time he counts soldiers, he only asks his men to count off at L ~ 3, 1 ~ 5, 1 ~ 7, and then reports the remainder of each team's count, so he knows how many people have arrived. His ingenious algorithm is called "Ghost Valley Calculation", "Partition Calculation" or "Han Xin's Point Force", and foreigners call it "Chinese remainder theorem". Cheng Dawei, a mathematician in the Ming Dynasty, summed up this algorithm in his poem. He wrote:

Three people walk seventy, five trees and twenty-one sticks,

Seven sons reunited in the middle of the month and didn't know until 105.

The meaning of this poem is: multiply the remainder obtained by dividing 3 by 70, the remainder obtained by dividing 5 by 2 1, and the remainder obtained by dividing 7 by 15. If the result is greater than 105, subtract the multiple of 105, and you will know the number you want.

For example, if there are 52 eggs in a basket, if there are three more than 1, five more than five than two, and seven more than seven than three. The formula is:

1×70+2×2 1+3× 15= 157

157- 105 = 52 (pieces)

Please calculate the following questions according to this algorithm.

Xinhua primary school subscribed to a batch of China Youth Daily. If there are three digits, the remainder is 1. Five plots of land, and the rest are two; Seven pieces of land, and the rest is two pieces. How many copies of China Youth Daily did xinhua primary school subscribe to?