Zu Chongzhi s information about pi, Zu Chongzhi s knowledge of pi

Zu Chongzhi (429-500 AD) was a native of Laiyuan County, Hebei Province during the Northern and Southern Dynasties of China He read many books on astronomy and mathematics since he was a child, and he was diligent and studious, and practiced hard, which finally made him an outstanding mathematician and astronomer in ancient China

Zu Chongzhi's outstanding achievement in mathematics is about the calculation of pi Before the Qin and Han dynasties, people to"Trail three times a week"As pi, this is"Ancient rate"Later, it was found that the error of the paleorate was too large, and the pi should be"The circle diameter is more than three days", but how much is left, opinions differ Until the Three Kingdoms period, Liu Hui proposed a scientific method for calculating pi"Circumcision", use the circumference of the circle inscribed regular polygon to approximate the circumference of the circle Liu Hui calculates that the circle is inscribed with 96 polygons, and obtains =, and points out that the more sides of the inscribed regular polygon, the more accurate the value obtained Zu Chongzhi on the basis of the achievements of his predecessors, after hard work and repeated calculations, found In between and and obtained the approximate value in the form of fractions, take it as the approximate rate, take it as the density rate, where take six decimal places is, it is the fraction of the closest value of the numerator denominator within 1000 What method did Zu Chongzhi use to get this result, Now there is no way to examine if it is assumed that he will press Liu Hui's"Circumcision"If you want to find this method, you have to calculate that the circle is connected with 16,384 polygons, which requires a lot of time and labor! It can be seen that his tenacious perseverance and intelligence in his studies are admirable Zu Chongzhi's calculation of the dense rate, foreign mathematicians obtained the same result, it is more than a thousand years later In order to commemorate Zu Chongzhi's outstanding contributions, some foreign historians of mathematics suggest that = be called"Ancestral rate"．

Zu Chongzhi read the famous classics at that time, insisted on seeking truth from facts, he compared and analyzed a large number of materials from his own measurement and calculation, found the serious errors of the past calendar, and had the courage to improve, and at the age of 33, he successfully compiled the "Ming Calendar", opening up a new era in the history of the calendar

Zu Chongzhi also worked with his son Zu Xuan (also a famous mathematician in China) to solve the calculation of the volume of the sphere with ingenious methods One of the principles they adopted at that time was:"If the power potential is the same, the product cannot be different"That is, the two three-dimensional planes located between the two parallel planes are truncated by any plane parallel to these two planes, and if the areas of the two cross-sections are always equal, then the volume of the two three-dimensional dimensions is equal This principle is called the Cavalelli principle in Western language, but it was discovered by the Cavalelli principle more than a thousand years after Zu's In order to commemorate the great contribution of Zu's father and son in discovering this principle, everyone also calls this principle "Zu's principle".

Zu Chongzhi's knowledge of pi is that Zu Chongzhi calculated that the true value of pi is between and is equivalent to the 7th decimal place.

1. On the basis of his predecessors, Zu Chongzhi calculated pi to 7 decimal places (i.e., between and) after hard study and repeated calculations, and obtained an approximate value in the form of pi fractions.

2. Zu Chongzhi calculated pi accurate to the 7th place in the number of small slag shelters, simplified into, 3. And used the latest pi results to correct the calculation of the volume of the ancient measuring instrument. In ancient times, there was a measuring instrument called a kettle, which was generally one foot deep and cylindrical in shape, and Zu Chongzhi used his pi research to find the exact value.

4. Zu Chongzhi also recalculated the amount of law made by Liu Xin in the Han Dynasty, and corrected the value by using the ancestral rate. Later, when people made measuring instruments, they used Zu Chongzhi's ancestral rate value.

Zu Chongzhi introduced:

1. Zu Chongzhi was born in Jiankang, his ancestral home is Fanyang County, Luoxian County (now Laishui County, Hebei), and he was an outstanding mathematician and astronomer during the Northern and Southern Dynasties of China. He devoted his life to the natural sciences, and his main contributions were mathematics, astronomical calendars, and mechanical engineering.

2. Zu Chongzhi also observed and calculated the orbits of the five planets in the sky and the time required to orbit for one week, and gave a more accurate five-star conjunction period.

3. Zu Chongzhi re-measured and came to the conclusion that Jupiter is supercelerated once every 84 years, that is, Jupiter's orbital period is determined to be years (now measured as years).

4. For example, Pizhou Zu Chongzhi designed and manufactured guide cars, thousand-mile boats, timers, etc., which have been driven by water mills and copper parts. In ancient China, the name of the guide car has been around for a long time, but its mechanism and structure have not been circulated.

Three Kingdoms period, Liu Hui.

Calculation of pi was proposed.

The scientific method --"Circumcision"to approximate the circumference of the circle by incorporating the perimeter of the regular polygon.

Liu Hui calculates that the circle is inscribed with 96 polygons, and obtains =, and points out that the more sides of the inscribed regular polygon, the more accurate the value obtained Zu Chongzhi.

On the basis of the achievements of our predecessors, we have worked hard and calculated repeatedly to find out.

And an approximation in the form of a fraction is obtained, which is taken as the approximate rate and is taken as the density rate, where six decimal places are taken, which is the fraction with the closest value within 1000 The method Zu Chongzhi used to arrive at this result is now impossible to examine If he is conceived according to Liu Hui's"Circumcision"If you want to find the method, you have to calculate that the book is surrounded by 16,384 polygons, which requires a lot of time and labor! It can be seen that his tenacious perseverance and intelligence in governing the state are admirable Zu Chongzhi's calculation of the dense rate, foreign mathematics bureau and scholars to obtain the same result, it is more than a thousand years later In order to commemorate Zu Chongzhi's outstanding contributions, some foreign historians of mathematics suggest that = be called"Ancestral rate"．

Zu Chongzhi has been fond of mathematics since he was a child, and he learned a lot of mathematics under the guidance of his father and grandfather. Once, his father brought him a copy of the Sutra of Calculations from the bookshelf, which was a famous mathematics book from the Western Han Dynasty or earlier. It is said that the circumference of a circle is three times its diameter.

So, he measured the wheel with a rope and checked it, only to find that the circumference of the wheel was a little more than three times the diameter of the wheel. He went to measure the basin again, and the result was the same. He thought that the circumference is not exactly 3 times the diameter, so how much longer is the circumference than the 3 diameters?

Before the Han Dynasty, China generally used three as the value of pi, that is, "three diameter one". This is a large error when calculating the circumference and area of a circle. On the basis of the scientific method of using "circumcision" to find pi created by Liu Hui, Zu Chongzhi used the method of opening and density, and after repeated calculations, found that pi is:

>。This was the most accurate value in the world at the time, and he became the first person in the world to calculate the exact value of the circumferential nuclear rent conversion rate to the seventh decimal place. It wasn't until more than 1,000 years later that this record was broken by the Europeans.

The calculation of pi is an outstanding contribution of Zu Chongzhi in mathematics, and some foreign historians of mathematics call it "Zu rate."

The method of circumcision to find pi is as follows:

Starting from the circle with a regular hexagon, as shown in the figure, gradually double the number of sides, and draw the inscribed circle with regular 12 sides, regular 24 sides, regular 48 sides, regular 96 sides, and regular 192 sides in turn, ......The area of these polygons gradually approaches the area of the circle.

If the area of the circumscribed 2n side of the circle is S2N, then S2N gradually approaches the circle area r with the increase of n, and if R=1, then S2N gradually approximates .

<> this gives the recursive formula for the edges, starting with n=6:

According to this line of thinking, Liu Hui calculated the circumference of the regular polygon in the circle until the regular 3072 polygon, and obtained an approximation of pi, which was the most accurate pi in the world at that time.

Other ways to find pi:

1. Even scores:

There are very few people who use even fractions to calculate pi, probably because of the large amount of calculation. For example, Brown Cairo's Lian Senheng score.

2. Series method:

The series method is an analytical method that obtains an analytical formula about pi through a power series. Leibniz first obtained an analytic formula, and then mathematicians such as Euler and Martin obtained a large number of such analytic formulas, and their convergence speed is fast and slow.