How is pi calculated?

Born in Jixian, Fanyang, in the northern and Southern Dynasties (429-500 A.D.), Zu Chongzhi once calculated that the moon's orbit around the earth is 27.21223 days, which is now recognized as 27.21222 days. It is no wonder that Western scientists named a crater on the moon as "Zu Chongzhi", which is the only one on the moon with Chinese characteristics

How is the PI calculated?

Calculation method of PI
Archimedes used the regular polygon to approximate the circumference of a circle. Archimedes used the positive 96 polygon to get the accuracy of 3 decimal places of PI. Liu Hui got 5-digit precision with 3072 polygon. Ludolph van ceulen got 35 bit accuracy by using positive 262 polygon. This geometry based algorithm has a large amount of calculation and slow speed, With the development of mathematics, mathematicians have found many formulas for calculating pi. Here are some classical formulas to introduce. In addition to these classical formulas, there are many other formulas and formulas derived from these classical formulas, which will not be listed one by one
This formula was discovered by John machin, a British astronomy Professor, in 1706. He used this formula to calculate the Pi of 100 bits. Each term of machin's formula can get 1.4 decimal precision. Because the multiplier and divisor in the calculation process are not larger than long integers, so it can be easily programmed on the computer
Machine. C source program
There are many arctangent formulas similar to machin's formula. Among all these formulas, machin's formula seems to be the fastest. However, if you want to calculate more digits, such as tens of millions of bits, the machine formula will not be able to do it. The algorithm described below takes about a day to calculate on a PC, These algorithms are more complicated to implement by program. Because the calculation process involves the multiplication and division of two large numbers, FFT (fast Fourier transform) algorithm is used. FFT can shorten the operation time of multiplication and division of two large numbers from O (N2) to o (NLog (n))
In our country, it is the mathematician Liu Hui who first got the more accurate PI. Around 263 ad, Liu Hui put forward the famous cyclotomy, and obtained π = 3.14, which is usually called "Hui rate". He pointed out that this is not an approximate value. Although he proposed the cyclotomy later than Archimedes, Liu Hui proposed the cyclotomy, In addition, some people think that Liu Hui provides a wonderful method of finishing, As a result, Liu Hui pointed out that if we can get this result by simply weighted average of several rough approximations cut to 192 polygon, we can get the PI with four significant figures π = 3927 / 1250 = 3.1416, It needs to be cut to 3072 polygon. The effect of this finishing method is wonderful. This magical finishing technology is the most wonderful part of circle cutting. Unfortunately, it has been buried for a long time due to people's lack of understanding of it
I'm afraid that Xu's contribution to this book is as follows: the first half of a second of the Song Dynasty's record is as follows, A positive number is between the two limits of profit. Density: diameter 113, circumference 355. Reduction rate, circle diameter 7, Tuesday 12. "
This record points out that Zu Chongzhi has made two great contributions to the PI. One is to obtain the PI
3.1415926 ＜ π ＜ 3.1415927
Second, two approximate fractions of π are obtained, that is, the reduction rate is 22 / 7 and the density ratio is 355 / 113
The 8-digit reliable figure of π calculated by him was not only the most precise PI at that time, but also kept the world record for more than 900 years. As a result, some mathematicians suggested that the result be named Zulu
How did this result come about? It is precisely based on the inheritance and development of Liu Hui's circle cutting technique that Zu Chongzhi got this extraordinary achievement. Therefore, when we praise the achievements of Zu Chongzhi, we should not forget that his achievement is due to his standing on the shoulder of Liu Hui, a great mathematician, In order to get this result, we need to calculate the inscribed 12288 polygon in the circle to get such an accurate value. Whether Zu Chongzhi used other ingenious methods to simplify the calculation? It is not known, because the book "Jieshu", which recorded his research results, has long been lost. This is a very deplorable thing in the history of Chinese mathematics development
Commemorative stamps of Zu Chongzhi issued in China
This research achievement of Zu Chongzhi enjoys a world reputation: articles on the wall of Science Museum of discovery palace in Paris introduce the PI obtained by Zu Chongzhi; the corridor of Moscow University auditorium is inlaid with marble statues of Zu Chongzhi; there are craters named after Zu Chongzhi on the moon
People usually don't pay much attention to Zu Chongzhi's second contribution to PI, that is, he chooses two simple fractions, especially the density ratio to approximate π. However, in fact, the latter has more important mathematical significance
Professor Liang Zongju, a historian of mathematics, has proved that among all the scores whose denominator is less than 16604, there is no fraction closer to π than that of the fraction whose denominator is less than 16604
It can be seen that the proposal of MI rate is not simple. People naturally have to investigate what method he used to get this result? What method did he use to convert the approximate value of pi from decimal to approximate fraction? This problem has always been concerned by mathematical historians. Due to the loss of literature, Zu Chongzhi's method has not been known. Many people have speculated about it

PI represents the multiple relationship of () and () in the same circle?

The ratio of circumference to diameter is a multiple of the circumference and diameter of a circle

PI represents the multiple relationship between () and () in the same circle

PI represents the multiple relationship between (circumference) and (diameter) in the same circle

The PI represents the same circle And The multiple relation of, using letters__ Represents

PI is the relationship between the circumference and diameter of a circle
So the answer is: circumference, diameter, π

PI represents the same circle And Multiple relation of

PI is the relationship between the circumference and diameter of a circle
So the answer is: circumference, diameter

PI represents the same circle And Multiple relation of

PI is the relationship between the circumference and diameter of a circle
So the answer is: circumference, diameter

The PI represents the multiple relationship of () and () in the same circle. It is represented by the letter () with two decimal places reserved, and the approximate value is ()

The ratio of circumference to diameter of a circle on the plane is expressed by π (PAI)
three point one four

The function f (x) = SiNx cos (x + π) 6) The value range of is () A. [-2，2] B. [- 3， 3] C. [-1，1] D. [- Three 2， Three 2]

The function f (x) = SiNx cos (x + π)
6）=sinx-
Three
2cosx+1
2sinx
= -
Three
2cosx+3
2sinx
=
3sin（x-π
6）∈[−
3，
3]．
Therefore, B

The function f (x) = SiNx cos (x + π) 6) The value range of is () A. [-2，2] B. [- 3， 3] C. [-1，1] D. [- Three 2， Three 2]

The function f (x) = SiNx cos (x + π)
6）=sinx-
Three
2cosx+1
2sinx
= -
Three
2cosx+3
2sinx
=
3sin（x-π
6）∈[−
3，
3]．
Therefore, B

How is pi calculated?