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The triangular number is the sum of the first n terms of the positive integer: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, ..n (n + 1) / 2, from 1 + 2 + 3 + ... + n talked about stacking many wooden logs on the construction site, and they looked like a triangle when viewed from the side.

The triangular number is the sum of the first n terms of the positive integer: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, ..n (n + 1) / 2, from 1 + 2 + 3 + ... + n talked about stacking many wooden logs on the construction site, and they looked like a triangle when viewed from the side. One on the top, two on the second, and three on the third ... Want to know how many logs there are in this pile of wood? Began to calculate: one, two, three, .... But this calculation is not fast and error-prone. In order to more accurately and quickly obtain the total number of stacked wooden bars, a method is introduced for the working people of ancient China and Greece.

The quotient of N times (N + 1) divided by 2 is the number of N triangles.

An odd square number minus 1 is divided by 8 to be a triangular number.

[(2K + 1) ^ 2-1] / 8 = T (T triangle number).

General term formula

{[(2K + 1) 2 ^ n] ^ 2- (2 ^ N) ^ 2)} / 8 (2 ^ n) ^ 2 = T

Multiplying the triangle number by 9 plus 1 is still a triangle number.

The rounding of the square root of the double of the triangular number is the ordinal of this triangular number.

The sum of the number of digits of the triangle number is only 1, 3, 6, and 9.

The sum of digits is a triangular number of 1, and the sum of its ordinal digits is only 1, 4, and 7.

The ordinal number sum is a triangle number minus 1 and divided by 9 is still a triangle number. Its ordinal number sum is comprehensive.

The sum of ordinal digits is 4 triangles minus 1 divided by 9, which is still the triangle of digits and 1.

The sum of the ordinal digits is a triangular number of 7, minus 1 divided by 9, and the sum of the digits is a triangular number of 3, 6, 9.

The sum of digits is a triangular number of 3, and the sum of ordinal digits is only 2 and 6.

The sum of the ordinal digits is a triangular number of 2, plus 6 divided by 9, and the full number of subtraction of the ordinal number divided by 9 (rounded up, the same applies below), it is still the number of digits and the triangular number of 3, 6, and 9.

The ordinal number sum is a triangle number of 6, minus 3 divided by 9, and the ordinal number divided by a full number of 9 is still the number of bits and the triangle number of 3, 6, and 9.

The sum of the digits is a triangular number of 6, and the sum of the ordinal digits is only 3 and 5.

The ordinal number sum is a triangular number of 3, minus 6 divided by 9, and the ordinal number divided by the full number of 9 is still a triangle number of 1 and 1.

The ordinal number sum is a triangular number of 5, plus 3 divided by 9 and the ordinal number divided by a full number of 9. It is still a triangular number of 1 and 1.

Math theorem found by an eight-year-old:

18th century

How did the ancient Chinese and Greeks calculate this and:

Greek mathematician before 2400

The achievements of Chinese accountants in this area:

Chinese mathematicians have long known about the series of equal difference. In the earliest mathematics book in China, the "Zhou Xuan Jing Jing" talked about the diameter of the "seven balance" (circle of the sun and the moon) increasing by 100 steps × 2 in 19833. This is the progression.

The problems in the two chapters of "Nine Chapters of Arithmetic", an important Chinese mathematical work written around the 1st century AD, are related to the series of equal differences.

At the end of the 5th century, Zhang Qiujian in the North and South Dynasties had three problems in his "Zhangqiu Jianshu Jingjing": the problem of equal series:

[Question 1] There are women who are good at weaving today. The cloth weaving day by day increases by the same number. It is known that weaving five feet on the first day, weaving a total of 39 feet in a month.

[Question 2] There are women who are not good at weaving today. The cloth weaved day by day decreases by the same amount. It is known that weaving five feet on the first day and one foot on the last day, weaving for 30 days. Ask how much weave.

Answer: 9 feet.

[Title 3] Today, a certain monarch gave money to many people. The first person gave three money, the second person gave four money, and the third person gave five money. They continued to increase in order, and the money was given to many others. After giving the money, all the money received by the people was recovered and distributed evenly. As a result, each person received 100 money. How many people?

Answer: 195 people.

Mathematicians in Tang and Song dynasties studied series not just for fun, but for practical needs. At the time, astronomers assumed that the movement of the sun, moon, and stars in the sky was equal acceleration or deceleration, and that the distance traveled by each day was equal progression.

For example, the Tang dynasty astronomer Seng (683-727) was the first astronomer in the world to find that the position of stars in the sky would change. In his "Yanyan Calendar", he calculated the travel of the planets using the sum formula of the series of equal difference.

In the Song Dynasty, Shen Kuo (1031--1095) had the most outstanding contribution to the research on equal difference series and higher-order difference series. He saw hotels, pottery shops, and other things such as pots, jars, and tile basins. Pushing a long square platform, the bottom layer is arranged in a rectangle, and the length and width of each layer above are reduced by one, so he wants to know if there is a simple formula to calculate.

He read the ancient arithmetic book: the original "Shang Gong" chapter of the "Nine Chapters of Arithmetic" formula of the original rectangular table volume (the ancient book is called "mother"). Using this formula to find practical problems is often less than the original number. Therefore, he created a new method, the "gap product method", to supplement "the ancient book can not find." ("Using the Chutong method to find it, it is often lost in a small number, and it is obtained by thinking.")

Assume that the upper and lower sides of the rectangular platform are a × b, and the bottom is a '× b'. There are n layers in total. Since the top and bottom of each layer increase by one each, the a'-a = b'-b = n -1, Shen Kuo's summation formula is:

ab + (a + 1) (b + 1) + (a + 2) (b + 2) + ... + a'b '=

If the reader makes a = b = 1 and a '= b' = n, substituting the above formula can get

12 + 22 + + n2 = n (n + 1) (2n + 1) ÷ 6

"Meng Xi Bi Tan" left by Shen Kuo to future generations is a rich scientific work, which talks about mathematics, astronomy, physics, chemistry, biology, geology, geography, meteorology, medicine and engineering technology, etc. The book is highly rated. ^[3] And Japanese mathematician Mikami Yoshio (1875-1950) highly respected Shen Kuo. He believed that in terms of ancient mathematics: "No Japanese mathematician is as good as Shen Kuo, like Nakone Motoi who is good at medicine , Music, and almanac, but without Shen Kuo's genius; Bentolimine sailing, genius, but not as versatile as Shen Kuo. If you can find a mathematician comparable to Shen Kuo in other countries, then Germany Leibniz and Carlo of France may be compared with Shen Kuo at a certain point, but if one side is far better than Shen Kuo and at the same time versatile, then it is impossible to talk about. Only Architez in Greece, his knowledge Experience is the most comparable to Shen Kuo. In short, such a figure like Shen Kuo can't be found in the history of mathematics around the world. Only China came out of this person. I think Shen Kuo is a model or ideal figure for Chinese mathematicians. "(See" Characteristics of Chinese Mathematics ")

After Shen Kuo, the mathematicians of the Song Dynasty who had better results in series studies should be Yang Hui in the 13th century. He came up with the triangle formula:

1+ (1 + 2) + (1 + 2 + 3) + ... + (1 + 2 + 3 + ... + n) = n (n + 1) (n + 2) ÷ 6

In the Yuan Dynasty, Zhu Shijie was a teacher who taught mathematics everywhere. He wrote a book "Enlightenment in Mathematics" in 1299 and "Siyuan Yujian" written in 1303. In that work, he expanded Yang Hui's triangular stacks and formulas, and established his formula for triangular numbers, as well as more complex formulas. These are also more than 300 years before Fermat.

Zhu Shijie's book spread to Japan in the 17th century, and it had a great influence on the research of series theory by Japanese mathematicians. On the contrary, in the 400 years since Zhu Shijie, the series theory has stopped and has not developed. It is not until the 18th century that Dong Youcheng and Li Shanlan have some opinions.

Series theory is closely related to the generation of calculus. Chinese mathematicians have long used geometric methods to calculate the volume of spheres. At the time of the Song and Yuan Dynasties, China basically had the preparatory conditions for generating calculus, but unfortunately, no one was able to carry forward the work like Leibniz and Newton in Western Europe.