1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 2 1 35 35 2 1 7 1
… … … … …
The most essential feature of Yang Hui Triangle is that its two hypotenuses are all composed of the number 1, and the other numbers are equal to the sum of the two numbers on its shoulders. In fact, ancient mathematicians in China were far ahead in many important mathematical fields. The history of ancient mathematics in China once had its own glorious chapter, and the discovery of Yang Hui's triangle was a wonderful one. Yang Hui was born in Hangzhou in the Northern Song Dynasty. In his book "Detailed Explanation of Algorithms in Nine Chapters" written by 126 1, he compiled a triangle table as shown above, which is called an "open root" diagram. And such triangles are often used in our Olympic Games. The simplest thing is to ask you to find a way. Now we are required to output such a table through programming.
At the same time, this is also the law of quadratic coefficient of each term after polynomial (A+B) n is opened, that is
0(a+b)^0 0 NCR 0)
1(a+b)^ 1 1 NCR 0)( 1 NCR 1)
2(a+b)^2(2 NCR 0)(2 NCR 1)(2 NCR 2)
3 votes (A+B) 3 (3 abstentions) (3 abstentions 1) (3 abstentions and 2 abstentions) (3 abstentions and 3 abstentions)
. ...............
Therefore, the Y term of X layer of Yang Hui triangle is directly (y nCr x).
It is not difficult for us to get that the sum of all terms in layer X is 2 x (that is, when both A and B in (A+B) x are 1).
[the above y x refers to the x power of y; (a nCr b) refers to the number of combinations]
In fact, ancient mathematicians in China were far ahead in many important mathematical fields. The history of ancient mathematics in China once had its own glorious chapter, and the discovery of Yang Hui's triangle was a wonderful one.
Yang Hui was born in Hangzhou in the Northern Song Dynasty. In his book "Detailed Explanation of Algorithms in Nine Chapters" written by 126 1, he compiled a triangle table as shown above, which is called an "open root" diagram.
And such triangles are often used in our Olympic Games. The simplest thing is to ask you to find a way. Specific usage will be taught in the teaching content.
In foreign countries, this is also called Pascal Triangle. There is also a short story: (1) A miss is thousands of miles away.
1 On August 23, 967, the Soviet Union1spacecraft suddenly had a vicious accident when it returned to the atmosphere-the parachute could not be opened. After studying it, the central leadership of the Soviet Union decided to broadcast the accident live to the whole country. When the TV announcer announced in a heavy tone that the spaceship would crash in two hours and the audience would witness the martyrdom of astronaut Vladimir Komarov, the whole country was immediately shocked and people were immersed in great grief.
On TV, the audience saw the calm image of astronaut komarov. He smiled and said to his mother, "Mom, I can clearly see your image here, including every gray hair on your head. Can you see me clearly? " "Yes, you see very clearly. Son, everything is fine, don't worry! " At this time, komarov's daughter also appeared on the TV screen. She is only 12 years old. Komarov said, "Daughter, don't cry." "I don't cry ..." My daughter broke down in tears, but she fought back her grief and said, "Dad, you are a Soviet hero. I want to tell you that the hero's daughter will live like a hero! " Komarov told her daughter, "When you study, you should take every decimal point seriously. What happened today on Alliance 1 is because a decimal point was ignored during ground inspection ... "
Time is running out. There are only seven minutes before the spaceship crashes. Komalov waved to the national television audience and said, "Fellow citizens, please allow me to say goodbye to you in this vast space."
Even a decimal point error will lead to a tragic farewell that can never be remedied.
Julius Caesar of ancient Rome famously said, "In war, great things are often the result of small things." In China's epigram, it's probably "a short trip makes a long regret".
(B) A mathematician triggered by a story
Chen Jingrun, a famous mathematician, made great contributions to overcoming Goldbach's conjecture and founded the famous "Chen Theorem", so many people affectionately called him "the prince of mathematics". But who would have thought that his achievement originated from a story? 1937, diligent Chen Jingrun was admitted to Huaying College in Fuzhou. At this time, during the period of War of Resistance against Japanese Aggression, Professor Shen Yuan, director of the Department of Aeronautical Engineering in Tsinghua University, returned to Fujian to attend the funeral, unwilling to stay in his hometown because of the war. Several universities got the news and wanted to invite Professor Shen to give lectures. He declined the invitation. As he is an alumnus of Huaying, he came to this middle school to teach mathematics to his classmates in order to report to his alma mater. One day, Teacher Shen Yuan told us a story in math class: "A Frenchman discovered an interesting phenomenon 200 years ago: 6=3+3, 8=5+3, 10=5+5, 12=5+7, 28=5+23. Every even number greater than 4 can be expressed as the sum of two odd numbers. Because this conclusion has not been proved, it is still a guess. Euler said: Although I can't prove it, I am sure this conclusion is correct. It is like a beautiful light ring, shining with dazzling brilliance in front of us not far away. ..... "Chen Jingrun stare eyes, absorbed.
From then on, Chen Jingrun became interested in this wonderful question. In his spare time, he likes going to the library. He not only read the counseling books in middle schools, but also eagerly read the textbooks of mathematics and physics courses in these universities. Therefore, he got the nickname "bookworm". Interest is the first teacher. It is such a mathematical story that aroused Chen Jingrun's interest and his diligence and made him a great mathematician.
(3) people who are fascinated by science
Because the study of infinity often leads to some logical but absurd results (called "paradox"), many great mathematicians are afraid of falling into it and adopt an evasive attitude. During the period of 1874- 1876, Cantor, a young German mathematician less than 30 years old, declared war on the mysterious infinity. With hard sweat, he successfully proved that points on a straight line can correspond to points on a plane one by one, and can also correspond to points in space one by one. In this way, it seems that there are as many points on the 1 cm long line segment as there are points in the Pacific Ocean and the whole earth. In the following years, Cantor published a series of articles about this kind of "infinite set" and drew many amazing conclusions through strict proof.
Cantor's creative work has formed a sharp conflict with the traditional mathematical concept, which has been opposed, attacked and even abused by some people. Some people say that Cantor's set theory is a kind of "disease", Cantor's concept is "fog in fog", and even Cantor is a "madman". Great mental pressure from the authority of mathematics finally destroyed Cantor, making him exhausted, suffering from schizophrenia and being sent to a mental hospital.
True gold is not afraid of fire, and Cantor's thought finally shines. At the first international congress of mathematicians held in 1897, his achievements were recognized, and Russell, a great philosopher and mathematician, praised Cantor's work as "probably the greatest work that can be boasted in this era." But at this time, Cantor was still in a trance, unable to get comfort and joy from people's reverence. 1918 65438+1October 6th, Cantor died in a mental hospital.
Cantor (1845- 19 18) was born in a wealthy family of Danish Jewish descent in Petersburg, Russia. /kloc-moved to Germany with his family at the age of 0/0, and was interested in mathematics since childhood. He received his doctorate at the age of 23 and has been engaged in mathematics teaching and research ever since. The set theory he founded has been recognized as the basis of all mathematics.
Mathematicians' "forgetfulness"
On the 60th birthday of Professor Wu Wenjun, a mathematician in China, as usual, he got up at dawn and buried himself in calculations and formulas all day.
Someone specially chose to visit at home this evening. After greeting, he explained his purpose: "I heard from your wife that today is your sixtieth birthday, and I came to congratulate you." Wu Wenjun seemed to hear a message and suddenly said, "Oh, really? I forgot. " The bearer was secretly surprised and thought, how can a mathematician not even remember his birthday because his mind is full of numbers?
In fact, Wu Wenjun has a good memory for dates. Nearly sixty years old, he conquered a difficult problem for the first time-"machine certificate". This is to change the working mode of "a pen, a piece of paper, a head" for mathematicians, and realize mathematical proof with electronic computers, so that mathematicians have more time to do creative work. In the course of his research on this subject, he clearly remembers the date of installing the electronic computer and compiling more than 300 "instruction" programs for the computer.
Later, a birthday visitor asked him in a chat why he didn't even remember his birthday, and he knew to answer:
"I never remember those meaningless numbers. In my opinion, what does it matter if the birthday is one day earlier and one day later? So, I don't remember my birthday, my wife's birthday, my child's birthday. He never wants to celebrate his or his family's birthday, even my wedding day. However, some figures must be remembered, and it is easy to remember ... "
(5) Routine walking under the apple tree
1884 In the spring, Adolf leonid hurwicz, a young mathematician, came to Koenigsberg from G? ttingen as an associate professor. He is less than 25 years old and has made outstanding research results in function theory. Hilbert and Minkowski soon established a close relationship with their new teacher. They three young people must meet at 5 o'clock every afternoon and go for a walk under the apple tree. Hilbert later recalled: "In the day-to-day walk, we were all immersed in discussion." Exchange our newly acquired understanding of the problem with each other, and exchange ideas and research plans with each other. "Among them, leonid hurwicz has a wide range of" solid basic knowledge, which has been well organized ",so he is a natural leader and persuaded the other two. At that time, Hilbert found that this learning method was many times better than boring in a dark classroom or library. This routine walk lasted for eight and a half years. In this leisurely and interesting way of learning, they explored "every corner" of mathematics and inspected every kingdom in the mathematical world. Hilbert later recalled: "At that time, I never thought that we would take ourselves so far!" " In this way, the three people "formed a lifelong friendship." "
(6) The ambition of serving the motherland-the story of China
As we all know, Hua is a self-taught world-class mathematician. He only has a junior high school diploma. Because a paper was published in Science magazine, it was appreciated by mathematician Xiong Qinglai. From then on, North China went to Tsinghua and began his mathematics career. 65438-0936, recommended by Professor Xiong Qinglai, Hua went to Cambridge University in England to study. Hardy, a famous mathematician in the 20th century, has long heard that China is brilliant. He said, "You can get a doctorate in two years." But Hua said, "I don't want to get a doctorate." I just want to be a tourist. " "I came to Cambridge to study, not to get a degree." In recent two years, he has devoted himself to studying the theory of heap prime numbers, published 18 papers about Willing and odd Goldbach, obtained the famous Fahrenheit theorem, and showed the outstanding wisdom and ability of China mathematicians to the whole world.
From 65438 to 0946, Hua was invited to give lectures in the United States and was hired as a tenured professor by the University of Illinois. His family also settled in the United States, with a house and a car, and their life was very comfortable. At that time, many people thought that Hua would never come back. The birth of the new China touched China's love for the motherland. 1950, he resolutely gave up his comfortable life in America and returned to his motherland. He also wrote an open letter to China students studying in the United States, urging them to return to China to participate in socialist construction. In his letter, he revealed a childlike heart that loves China: "Friends! Although Liangyuan is good, it is not the hometown of longevity. Go back and come back ... For the sake of the nation, we should go back ... "Although mathematics has no national boundaries, mathematicians have their own motherland.
Hua returned from overseas and was warmly welcomed by the party and people. He returned to Tsinghua campus, was appointed as the head of the Department of Mathematics, and was soon appointed as the director of the Institute of Mathematics of China Academy of Sciences. From then on, the real golden age of his mathematical research began. He has not only made remarkable achievements that have attracted worldwide attention, but also enthusiastically cared for and trained a large number of mathematical talents. He devoted a lot of efforts to the research, experiment and popularization of applied mathematics.
According to incomplete statistics, Hua * * * has published important mathematical papers 152, 9 mathematical works and1/mathematical popular science works for decades. He was also elected as a foreign academician of the Academy of Sciences and an academician of third world scientists.
(7) Advocates of cultural exchanges between China and the West.
Leibniz attached great importance to China's scientific, cultural and philosophical thoughts, and was the first German to study China culture and China philosophy. He learned a lot about China from Grimal Di, a Jesuit missionary in China, including sericulture, textile, papermaking, printing and dyeing, metallurgy and minerals, astronomy and geography, mathematics and writing, and edited and published these materials. He believes that a new relationship should be established between China and the West. Leibniz wrote in the introduction of China: "The greatest culture and the most developed civilization of all mankind seem to gather at the two ends of our continent today, that is, Europe and Eastern Europe-China on the other side of the globe." "Compared with Europe, China, an ancient civilization, has a similar area and a population of more than." "We are neck and neck in our daily life and our skills in dealing with nature. We all have the skills to benefit each other through mutual communication. Obviously, we should be slightly better at careful thinking and rational thinking, but "in the philosophy of time, that is, in the ethics of life and human reality and the theory of governing the country, we are really dwarfed." Here, Leibniz not only shows the spirit of being open-minded and eager to learn without the color of "Eurocentrism", but also depicts the grand blueprint for the two-way exchange between Chinese and Western cultures, which has effectively promoted the in-depth development of this exchange. People in the East and the West should learn from each other, learn from each other's strengths and make progress together. Leibniz devoted his life to promoting cultural exchanges between China and the West, which had a wide and far-reaching influence. He is open-minded and eager to learn, treats China culture equally, and his spirit of not containing the prejudice of Eurocentrism is particularly commendable, which is worthy of eternal admiration and imitation by future generations.