How pi is calculated and derived

Analytically, it can be strictly defined as the smallest positive real number satisfying sin(x) = 0 that can be solved by computer in series. This is my guess, I think you're a good question, I haven't thought about it before.

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.

The German mathematician Curran calculated the value to 20 decimal places in 1596 and devoted his life to the last 35 decimal places in 1610, which was called the Rudolf number after him. Various value expressions, such as infinite product formulas, infinite continuous fractions, and infinite series, have appeared, and the accuracy of value calculation has also increased rapidly. In 1706, the English mathematician Machin broke the 100-decimal place mark in his calculations.

In 1873, another British mathematician, Jeanccos, calculated the value to 707 decimal places, but unfortunately his result was wrong from 528 places. By 1948, Ferguson of the United Kingdom and Lench of the United States jointly published the 808-digit decimal value, which became the highest record for manual calculation of pi value. Textbook on pi for the sixth grade of primary school.

The advent of electronic computers has led to the rapid development of value calculation. In 1949, the Military Ballistics Research Laboratory in Aberdeen, Maryland, USA, used a computer (ENIAC) to calculate the value for the first time, and it was calculated to 2,037 decimal places, exceeding the thousands. In 1989, researchers at Columbia University in the United States used the Cray 2 and IBM VF giant electronic computers to calculate the value of 100 million decimal places, and then continued to calculate to 100 million decimal places, setting a new record.

January 7, 2010 – A French engineer counts pi to 2,700 billion decimal places. August 30, 2010 – Japanese computer wizard Shigeru Kondo has used a combination of home computers and cloud computing to calculate pi to 5 trillion decimal places.

In ancient and modern times, many people have devoted themselves to the study and calculation of pi. In order to calculate an increasingly good approximation of pi, generations of mathematicians have devoted countless hours and efforts to this mysterious number. Before the 19th century, the calculation of pi progressed quite slowly, and after the 19th century, world records for calculating pi were frequently updated.

The entire nineteenth century can be said to be the century with the greatest amount of manual calculations of pi. In the twentieth century, with the invention of the computer, the calculation of pi advanced by leaps and bounds. With the help of supercomputers, people have obtained an accuracy of 206.1 billion bits of pi.

One of the most marathon calculations in history was Ludolph van Ceulen in Germany, who spent almost his entire life calculating the inscribed regular 262 sides of a circle, and obtained a 35-bit precision value of pi in 1609, so much so that pi is called Ludolph number in Germany; The second was William Shanks of England, who spent 15 years calculating 707 decimal places of pi in 1874. Unfortunately, later generations found out that he was wrong from the 528th place. It doesn't really make much sense to calculate the value of pi so precisely.

The value of pi used in the field of modern science and technology, a dozen digits is sufficient. If you use the 35-bit precision pi value calculated by Ludolph van Ceulen to calculate the circumference of a circle that encloses the solar system, the error is less than one millionth of the diameter of a proton. In the past, people calculated pi to ** whether pi is a cyclic decimal.

The mystery of pi has been unveiled since Lambert proved that pi is an irrational number in 1761 and Lindemann proved that pi is a transcendent number in 1882. Nowadays, people calculate pi, mostly to verify the computing power of computers, and also, for interest.

It seems to be in elementary school (the math teacher will tell you).

The domestic version is that Zu Chongzhi learned to use the knowledge of calculus to find the top 4 places of pi long ago.

The calculation of pi is made using calculus.

The simplest formula for calculating pi:

s=180 sin when infinitely close to zero but not equal to zero, s=

Make an inscribed hexagon in a circle of radius r (as shown in the figure). At this time, the side length of the regular hexagon is equal to the radius r of the circle, therefore, the circumference of the regular hexagon is equal to 6r. If the circumference of the circumference of the circle is regarded as an approximation of the circumference of the circle, and then the ratio of the circumference of the circumference of the circumference of the circle to the diameter of the circle is regarded as the ratio of the circumference of the circle to the diameter of the circle, so that the pi is 3, this is obviously inaccurate.

If you double the number of sides of the circle with a regular hexagon, you can get the circle with a regular dodecagonal and a twenty-four ......It is not difficult to see that as the number of sides of the regular polygon of the circle continues to multiply, their circumference becomes closer and closer to the circumference of the circle.

That is to say, the ratio of their circumference to the diameter of the circle is also getting closer and closer to the ratio of the circumference of the circle to the diameter of the circle, so that we get an approximation of pi.

In a circle with radius r, make an inscribed regular hexagon. At this time, the side length of the regular hexagon is equal to the radius r of the circle, therefore, the circumference of the regular hexagon is equal to 6r. If the circumference of the circumference of the circumference of the circle is considered as an approximation of the circumference of the circle, and then the ratio of the circumference of the circumference of the circle to the diameter of the circle is regarded as the ratio of the circumference of the circle to the diameter of the circle, so that the pi obtained is 3, which is obviously inaccurate.

We get an approximation of pi.

As early as more than 1,700 years ago, Liu Hui, an ancient mathematician in China, used circumcision to find pi. Following Liu Hui, the ancient Chinese mathematician Zu Chongzhi made important developments in the study of pi. The result of his calculations is a total of two numbers:

One is the surplus (i.e., the approximation of the excess), which is; The other is the (nǜ) number (i.e., an approximation of insufficient), for. The true value of pi is exactly between the two numbers. Zu Chongzhi also used two fractional values:

One is 22 7 (approximately equal to, called "approximate rate"; The other is 355 113 (approximately equal to, called "density"). Zu Chongzhi obtained the density rate at least a thousand years earlier than foreign mathematicians to obtain this value.

4 4arctg(1 5)-arctg(1 239) (Note: tgx=.......)

426880√10005∕（∑6n）！*545140134n+13591409))

（n！）*3n)！*640320）＾（3n)))

0≤n→∞）

The quotient of the circumference of a circle divided by its diameter is called pi.

Pi. In 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 equals 5 8, that is, equal to 10 square (approx.

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 regular 24576 polygon 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.

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

Brahmangumpta used a different method to deduce the arithmetic square root of pi equal to 10.

Fibonacci calculated that pi was 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.

In 1655, Wallace came up with a formula 2=2 2 2 4 4 6 6 8 8....3×3×5×5×7×7×9×9...

Euler's discovery that the i power of e plus 1 is equal to 0 becomes an important basis for proving that transcendence is an important basis.

Pi is 100 digits.