There is no formula for pi that middle school students can understand

Thank you! The formula derived from the "circumcision" used by Zu Chongzhi is probably the best example of both "geometric intuition" and "formal primary".

Let a(1) sqrt(2), a(n 1) sqrt(2 a(n)), n 1, 2 ,......Here sqrt stands for square root.

b(n) 2 2 a(1) 2 a(2) 2 a(n), n 1, 2 ,......

then when n, b(n).

The geometric fact referred to by this formula is that the circumference of the circle tends to be the circumference of the circle, starting from the circle with a square and gradually doubling the number of sides of the circle with a regular polygon. The use of advanced mathematics has been avoided in formulas, but when it comes to the problem of accurate representation of pi, it is impossible not to use limits. You only have to think about the example of finding the surface area of a sphere in solid geometry to see that there is no escaping the limit.

This question is relatively clear to Zu Chongzhi.

c d= i.e. c= d=2 r

The formula for area is derived by dividing a circle into equal parts from the center to the circumference and then joining them into an approximate rectangle. Approximately the area of the rectangular row of the destroyed book was found.

Amount. Length multiplied by width.

Approximately equal to. Half of the circumference of the circle is multiplied by half of the price of the round fiber, i.e., s= r r= r

The ratio of the circumference to the diameter of the circle. The first person to use the scientific method to find the value of pi was Archimedes, who in "The Measurement of Circles" (3rd century BC) used the circumference of the circumference of the circle inscribed and the circumscribed regular polygons to determine the upper and lower bounds of the circumference of the circle, starting from the regular hexagon and doubling it one by one to the regular 96 sides, resulting in (3+(10 71))<3+(1 7)) He pioneered the geometric method of calculating pi (also known as the classical method, or the Archimedes method), which yielded a value accurate to two decimal places. Pi.

The Chinese mathematician Liu Hui used only the approximation of the circle with a regular polygon in the annotation of the Nine Chapters of Arithmetic (263), and also obtained a value accurate to two decimal places, and his method was later called circumcision. He used circumcision until the circle was inscribed with a regular 192 polygonal shape, and obtained the root number 10 (approx. Zu Chongzhi, a famous mathematician in the Northern and Southern Dynasties, further obtained a value accurate to 7 decimal places (about the second half of the 5th century), gave an under-approximation and an excess approximation, and also obtained two approximate fractional values, a dense ratio of 355 113 and an approximate rate of 22 7.

His brilliant achievements predate Europe by at least 1,000 years. The density rate was not obtained by the German Otto until 1573 in the West, and it was published in the work of the Dutch engineer Antonis in 1625. At the beginning of the 15th century, the Arab mathematician Qasi obtained the exact decimal value of 17 digits of pi, breaking the record held by Zu Chongzhi for nearly a thousand years.

Circle into equilateral triangles to find the area.

What is Pi?

Pi is a constant and represents the ratio of circumference and diameter. It is an irrational number, i.e. an infinite non-cyclic decimal. However, in daily life, it is usually used to represent pi to make calculations, and even if engineers or physicists want to make more precise calculations, the value is only about 20 decimal places.

What is ?It is the sixteenth Greek letter, which originally had nothing to do with pi, but the great mathematician Euler began to use in 1736 to represent pi in letters and **. Since he was a great mathematician, people used it in the same way to express pi. But in addition to representing pi, it can also be used to denote other things, and it can also be seen in statistics.

The history of the development of pi.

In history, many mathematicians have studied pi, including Archimedes of Syracuse, Ptolemy, Zhang Heng, Zu Chongzhi and so on. In their own countries, they used their own methods to calculate the value of pi. The following is the research results of pi in various parts of the world.

Asia China:

During the Wei and Jin dynasties, Liu Hui used the method of gradually increasing the number of sides of a regular polygon to approximate the circumference (i.e., circumcision) to obtain an approximate value.

During the Han Dynasty, Zhang Heng derived the square divided by 16 equal to 5 8, that is, equal to the square of 10 (approximately. Although this value is not very accurate, it is easy to understand, so it has also become popular in Asia for a while.

Wang Fan (229-267) discovered another value of pi, which is, but no one knows how he found it.

In the 5th century AD, Zu Chongzhi and his son used a positive 24576 polygonal shape to find pi about 355 113, which is less than 1 in 800 million compared with the true value. It took a thousand years to break this record.

India: Around 530 A.D., the mathematician Ayebodo used the circumference of a 384-sided polygon to calculate that the rate of pi was about .

Brahmanguma used another method to deduce the square root of pi equal to 10.

European Fibonacci calculates that pi is about.

Veda used Archimedes' method to calculate < <

He was also the first to describe pi in terms of an infinite product.

Rudolf Vankoren calculates pi with 35 decimal places from a polygon with more than 320000000000 sides.

1. Pi is calculated by dividing the circumference of a circle by its diameter.

2. "Pi" is the ratio between the circumference of a circle and its diameter. The problem of its calculation has always been of great interest to Chinese and foreign mathematicians. A mathematician in Germany once said:

The accuracy of pi calculated by a country in history can be used as a measure of the development of mathematics in that country at that time. ”

3. In ancient times, China was ahead of the world level in the calculation of pi for a long time, which should be attributed to the new method created by the mathematician Liu Hui in the Wei and Jin dynasties - "circumcision".

4. The so-called "circumcision" is a method of using the circumference of a regular polygon to infinitely approximate the circumference of the circle and obtain pi from it. This method is a brand-new method created by Liu Hui after criticizing and summarizing various old calculation methods in the history of mathematics.

5. Pi is represented by the Greek letter (pronounced pài) and is a constant (approximately equal to, which represents the ratio of the circumference and diameter of the circle. It is an irrational number, i.e., an infinite non-cyclic decimal.

6. In daily life, it is usually used to approximate the calculation of pi. Ten decimal places is sufficient for general calculations. Even the most sophisticated calculations for engineers or physicists can be taken to a few hundred decimal places.