A little knowledge about pi 1 Little is known about pi.
Pi, generally expressed by π, is a common mathematical constant in mathematics and physics. It is defined as the ratio of the circumference to the diameter of a circle. It is also equal to the ratio of the area of a circle to the square of its radius. Accurate calculation of geometric shapes such as circle perimeter, circle area and sphere volume is the key value. In the analysis, π can be strictly defined as the smallest positive real number x satisfying sin(x) = 0.
A long time ago, people saw that the ratio of the circumference of a circle to the meridian was a constant independent of the size of the circle, which was called pi. 1600. William Autolante in Britain first used pi to express pi, because pi is the first letter of "circumference" in Greek, and δ is the first letter of "diameter". When δ= 1,
In 200 BC, the ancient Greek mathematician Archimedes first gave the correct solution of π in theory. He used the perimeter of circumscribed and inscribed polygons to approach the perimeter of a circle from large and small directions, and skillfully obtained π.
Around 150 BC, Ptolemy, another ancient Greek mathematician, used the chord table method (the chord length of the central angle 1 multiplied by 360 and then divided by the diameter of the circle) to give the approximate value of π 3. 14 16.
In 200 AD, Liu Hui, a mathematician from China, provided a scientific method to find pi, which embodied extreme views. Liu Hui's method is different from Archimedes's. He only takes "internal connection" instead of "external connection". Using the inequality of circular area, he gets twice the result with half the effort. Later, Zu Chongzhi took the lead in calculating pi and got the "reduction rate". ; π& lt； 3. 14 15927. Unfortunately, Zu Chongzhi's calculation method was later lost. It is speculated that he used Liu Hui's cyclotomy, but what method he used is still a mystery.
15th century, Al Cassie, a mathematician of * *, calculated the perimeters of three regular 2-sided polygons inscribed in and circumscribed by the circle respectively, and pushed the π value to16th place after the decimal point, breaking the record kept by Zu Chongzhi for thousands of years.
1579 A relational expression was found in Veda, France. For the first time, we got rid of the old method of geometry and found the analytic expression of π.
In 1650, Varis expressed π as the product of finite elements.
Later, Leibniz discovered it, and then Euler proved that although these formulas are simple in form, they require a lot of calculation. The biggest breakthrough in the calculation method of π value is to find its expression of arc tangent function.
167 1 year, discovered by Scottish mathematician Gregory.
1706, the British mathematician Laura Mai first discovered that its calculation speed far exceeded the classical algorithm.
1777, the French mathematician Buffon put forward his famous problem of throwing needles. Through it, we can get the over-similarity value by probability method. Suppose we draw a set of parallel lines with a distance of, and throw a needle with a length of at will on this plane. If the number of throwing needles is, the number of times that any parallel line crosses is, many people have done experiments, 190 1 year.
Legendre proved that π is an irrational number in 1794, that is, it cannot be expressed by the ratio of two integers.
1882, the German mathematician Linmand proved that π is a transcendental number, that is, it cannot be the root of an algebraic equation with integral coefficients.
After 1950s, the calculation of pi began with the help of computer, which led to a new breakthrough. At present, some people claim that pi has been calculated to hundreds of millions or even billions of significant digits.
People try to know statistically whether the number of π digits has some regularity. The game is still on. As someone said, the process of mathematicians' exploration is also like π: endless and endless. ...
2. Who knows anything about pi?
Since ancient times, I don't know how many mathematicians have racked their brains for the value of pi.
During the Wei and Jin Dynasties, China mathematician Liu Hui calculated the area of the circle inscribed with the regular polygon of 192 by secant technique, and the pi was 3. 14. Later, he calculated the area of the 3072 polygon inscribed in the circle, and got a more accurate pi value of 3. 14 16.
Zu Chongzhi, a scientist in the Southern and Northern Dynasties in China, accurately miscalculated the value of pi between 3. 14 15926 and 3. 14 15927. After the establishment of calculus theory, the calculation of pi has entered a new field.
After 1947, on the eve of the advent of electronic computers, the value of pi has been calculated to 808 decimal places. After the invention of the electronic computer, the decimal places of pi calculated by the electronic computer increased at an alarming rate.
After 1989, the value of pi has been calculated to more than 10 billion digits after the decimal point. I hope to adopt.
3. Who knows anything about pi?
Since ancient times, I don't know how many mathematicians have racked their brains for the value of pi. During the Wei and Jin Dynasties, China mathematician Liu Hui calculated the area of the circle inscribed with the regular polygon of 192 by secant technique, and the pi was 3. 14. Later, he calculated the area of the 3072 polygon inscribed in the circle, and got a more accurate pi value of 3. 14 16. Zu Chongzhi, a scientist in the Southern and Northern Dynasties in China, accurately miscalculated the value of pi between 3. 14 15926 and 3. 14 15927. After the establishment of calculus theory, the calculation of pi has entered a new field. After 1947, on the eve of the advent of electronic computers, the value of pi has been calculated to 808 decimal places. After the invention of the electronic computer, the decimal places of pi calculated by the electronic computer increased at an alarming rate. After 1989, the value of pi has been calculated to more than 10 billion digits after the decimal point.
Hope to adopt
4. PI knowledge
A long time ago, people saw that the ratio of the circumference of a circle to the meridian was a constant independent of the size of the circle, which was called pi. 1600. William Autolante in Britain first used pi to express pi, because pi is the first letter of "circumference" in Greek, and δ is the first letter of "diameter". When δ= 1,
In 200 BC, the ancient Greek mathematician Archimedes first gave the correct solution of π in theory. He used the perimeter of circumscribed and inscribed polygons to approach the perimeter of a circle from large and small directions, and skillfully obtained π.
Around 150 BC, Ptolemy, another ancient Greek mathematician, used the chord table method (the chord length of the central angle 1 multiplied by 360 and then divided by the diameter of the circle) to give the approximate value of π 3. 14 16.
In 200 AD, Liu Hui, a mathematician from China, provided a scientific method to find pi, which embodied extreme views. Liu Hui's method is different from Archimedes's. He only takes "internal connection" instead of "external connection". Using the inequality of circular area, he gets twice the result with half the effort. Later, Zu Chongzhi took the lead in calculating pi and got the "reduction rate". ; π& lt； 3. 14 15927. Unfortunately, Zu Chongzhi's calculation method was later lost. It is speculated that he used Liu Hui's cyclotomy, but what method he used is still a mystery.
15th century, Al Cassie, a mathematician of * *, calculated the perimeters of three regular 2-sided polygons inscribed in and circumscribed by the circle respectively, and pushed the π value to16th place after the decimal point, breaking the record kept by Zu Chongzhi for thousands of years.
1579 A relational expression was found in Veda, France. For the first time, we got rid of the old method of geometry and found the analytic expression of π.
In 1650, Varis expressed π as the product of finite elements.
Later, Leibniz discovered it, and then Euler proved that although these formulas are simple in form, they require a lot of calculation. The biggest breakthrough in the calculation method of π value is to find its expression of arc tangent function.
167 1 year, discovered by Scottish mathematician Gregory.
1706, the British mathematician Laura Mai first discovered that its calculation speed far exceeded the classical algorithm.
1777, the French mathematician Buffon put forward his famous problem of throwing needles. Through it, we can get the over-similarity value by probability method. Suppose we draw a set of parallel lines with a distance of, and throw a needle with a length of at will on this plane. If the number of throwing needles is, the number of times that any parallel line crosses is, many people have done experiments, 190 1 year.
Legendre proved that π is an irrational number in 1794, that is, it cannot be expressed by the ratio of two integers.
1882, the German mathematician Linmand proved that π is a transcendental number, that is, it cannot be the root of an algebraic equation with integral coefficients.
After 1950s, the calculation of pi began with the help of computer, which led to a new breakthrough. At present, some people claim that pi has been calculated to hundreds of millions or even billions of significant digits.
People try to know statistically whether the number of π digits has some regularity. The game is still on. As someone said, the process of mathematicians' exploration is also like π: endless and endless. ...
5. Who knows anything about pi?
Since ancient times, I don't know how many mathematicians have racked their brains for the value of pi. During the Wei and Jin Dynasties, China mathematician Liu Hui calculated the area of the circle inscribed with the regular 192 polygon by secant technique, and the pi value was 3. 14. Later, he calculated the area of the circle inscribed with the regular 3072 polygon. The more accurate value of pi is 3. 14 16. Zu Chongzhi, a scientist in the Southern and Northern Dynasties in China, accurately miscalculated the value of pi between 3. 14 15926 and 3. 14 15927. After the establishment of calculus theory, the calculation of pi has entered a new field. The value of pi has been calculated to 808 decimal places. After the invention of the electronic computer, the decimal places of pi calculated by the electronic computer increased at an alarming rate. After 1989, the value of pi has been calculated to more than 10 billion digits after the decimal point.
6. A little knowledge of pi
1, π (pronounced "pi") is the 6th Greek letter/kloc-0, which has nothing to do with pi, but the great mathematician Euler began to express pi in letters and papers in 1736. Because he is a great mathematician, people have followed suit and used pi to express pi.
2. Archimedes was the first person to find pi by scientific method. In The Measurement of a Circle (3rd century BC), he used the perimeter of a regular polygon inscribed and circumscribed by a circle, starting from a regular hexagon and multiplying it by a regular 96-sided polygon, and got (3+( 10/7 1)), thus determining the upper and lower bounds of the perimeter of a circle.
3. Why continue to calculate π? First of all, this method can be used to test computer faults. If the calculated value is wrong, it means that there is something wrong with the hardware or software and it needs to be replaced. At the same time, using computers to calculate pi can also create healthy competition and improve technology, thus improving human life. Even calculus and higher trigonometric identities are developed by studying pi. Second, mathematicians calculate π so long because they want to study whether the decimal of π is regular. For example, the π value starts from the 700th100th decimal place, and there are 7 3s in succession, that is, 333333, and starts from the 3204765th decimal place, there are 7 3s in succession. Now people will ask, does π have only such a special property? That's not true.
7. Tell me everything about pi.
Pi refers to the ratio of the circumference to the diameter of a circle on a plane. It is represented by the Greek letter π (pronounced "Pài"). In ancient China, there were names such as cycle, cycle rate and week. In the general calculation of π, people convert π, an infinite acyclic decimal, into 3. 14.
Edit this history of pi.
Euclid's Elements of Geometry in ancient Greece (about the beginning of the 3rd century BC) mentioned that pi was a constant, and China's ancient calculation book Zhou Bi Shu Jing (about the 2nd century BC) recorded that pi was a constant. Many approximations of pi have been used in history, most of which were obtained by experiments in the early days, such as π = (4/3) 4 ≒ 3. 1604 (about BC 1700) in ancient Egyptian papyrus. The first person to find pi scientifically was Archimedes. In The Measurement of a Circle (3rd century BC), he determined the upper and lower bounds of the circumference of a circle by using the circumference of a regular polygon inscribed and circumscribed by the circle. Starting with a regular hexagon, he multiplied it by a regular 96-hexagon and got (3+( 10/7 1)).
When Liu Hui, a mathematician in China, annotated Nine Chapters Arithmetic (263), he got the approximate value of π only by inscribing a regular polygon into a circle, and also got the value of π accurate to two decimal places. His method is called the secant circle method by later generations. He used secant technique until the circle inscribed the regular polygon of 192.
Zu Chongzhi, a mathematician in the Northern and Southern Dynasties, further obtained the π value accurate to 7 decimal places (about the second half of the 5th century), gave the insufficient approximation of 3. 14 15926 and the excessive approximation of 3. 14 15927, and also got two approximate fractional values, namely 355//. In the west, the secret rate was not obtained by German Otto until 1573, and was published in the work of Dutch engineer Antoine in 1625, which is called Antoine rate in Europe.
* * * Mathematician Kathy got the exact decimal value of pi 17 at the beginning of15th century, which broke the record kept by Zu Chongzhi for nearly a thousand years.
1596, the German mathematician Curran calculated the π value to 20 decimal places, and then spent his whole life calculating it to 35 decimal places of 16 10. This value is named Rudolph number after him.
Various π expressions such as infinite product formula, infinite continued fraction and infinite series have appeared one after another, and the calculation accuracy of π value has also improved rapidly. 1706, the British mathematician Mackin calculated the π value, which broke through the decimal mark of 100. 1873, another British mathematician Jean-Jacques calculated π to 707 decimal places, but his result was wrong from 528 decimal places. By 1948, Ferguson in Britain and Ronchi in the United States announced the 808-bit decimal value of π, which became the highest record of manual calculation of pi.
The appearance of electronic computer makes the calculation of π value develop by leaps and bounds. From 65438 to 0949, the Army Ballistics Research Laboratory in Aberdeen, Maryland, USA used a computer (ENIAC) to calculate π value for the first time, and it suddenly reached 2037 decimal places, exceeding thousands of digits. 1989, researchers at Columbia University in the United States used Cray -2 and IBM-VF supercomputers to calculate 480 million digits after the decimal point, and then continued to calculate to 10 1 100 million digits after the decimal point, setting a new record. So far, the latest record is 124 1 1 billion digits after the decimal point.
Besides the numerical calculation of π, its properties have also attracted many mathematicians. 176 1 year, Swiss mathematician Lambert first proved that π is an irrational number. 1794 French mathematician Legendre proved that π2 is also an irrational number. By 1882, German mathematician Lin Deman proved for the first time that π is a transcendental number, thus denying the problem of drawing ruler and ruler that has puzzled people for more than two thousand years. Others study the characteristics of π and its connection with other numbers. For example, 1929, the Soviet mathematician Gelfond proved that eπ is a transcendental number and so on.
8. What is the knowledge about pi?
From 65438 to 0777, Buffon, a French scientist, put forward a method to calculate pi, which is called Buffon's needle problem.
The steps of this method are: 1) Take a piece of white paper and draw many parallel lines with a distance of d on it. 2) Take a root with the length of l(l
Buffon himself proved that the probability is p=2l/(πd) π is pi, and the approximate value of pi can be obtained by probability method with this formula. The following are some data: the estimated number of times the experimenter throws in a year: Wolf1850 5000 25313.1596 Smith1855 320412193./Kloc. 137 Fox 1884 1030 489 3. 1595 Lazzerini 190 13408 1808 3. 14 15929 Lai Na 1929。 The needle throwing experiment is the first example of expressing probability problems in geometric form. He used random experiments to deal with deterministic mathematical problems for the first time, which promoted the development of probability theory to some extent.
Like the needle throwing experiment, we use the probability obtained through the probability experiment to estimate a quantity we are interested in. This method is called Monte Carlo method. Monte Carlo method rose and developed with the birth of computer during the Second World War.
This method is widely used in applied physics, atomic energy, solid state physics, chemistry, ecology, sociology and economic behavior. The French mathematician Buffon (1707- 1788) first designed the needle throwing experiment.
1777 gives a formula for calculating the intersection probability of a needle and a parallel line, P=2L/πd (where l is the length of the needle, d is the distance between parallel lines, and π is π). Because it is related to π, people think of using the throwing needle test to estimate the value of π.
In addition, the probability p that three positive numbers can be randomly named to form an obtuse triangle is also related to π. It is worth noting that the method adopted here is to design a suitable experiment, the probability of which is related to a quantity we are interested in (such as π), and then estimate this quantity with the experimental results. With the development of modern technology such as computer, this method has developed into a widely used Monte Carlo method.
Needle throwing experiment-one of the strangest methods to calculate π. One of the strangest ways to calculate π is18th century French naturalist C Buffon and his needle throwing experiment: on a plane, draw a set of parallel lines with a distance of d with a ruler; Throw a needle with a length less than d on the drawn plane; If the needle intersects the line, throwing is considered favorable, otherwise it is unfavorable. Buffon was surprised to find that the ratio of the number of favorable shots to unfavorable shots is an expression containing π. If the length of the needle is equal to d, the probability of favorable throwing is 2/π. The more times you throw, the more accurate π value you can get. A.D. 19065438. Italian mathematician Laslini made 3408 injections and gave the value of π as 3.1415929-accurate to six decimal places. However, whether Laslini really injected the needle or not, his experiment was questioned by L Badger of the National Weber University in Ogden, Utah, USA. π is discovered through a wide range of channels such as geometry, calculus and probability.
9. Little information about pi
circumference ratio
Pi refers to the ratio of the circumference of a circle to the diameter on a plane. Represented by the symbol π. In ancient China, there were names such as Yuan, Yuan and Zhou. (π≈3. 14)
Euclid's Elements of Geometry in ancient Greece (about the beginning of the 3rd century BC) mentioned that pi was a constant, and China's ancient calculation book Zhou Bi Shu Jing (about the 2nd century BC) recorded that pi was a constant. Many approximations of pi have been used in history, most of which were obtained by experiments in the early days. For example, π = (4/3) 4 ≈ 3. 1604 is taken from ancient Egyptian papyrus (about BC 1700). The first person to find pi scientifically was Archimedes. In The Measurement of a Circle (3rd century BC), he determined the upper and lower bounds of the circumference of a circle by using the circumference of a regular polygon inscribed and circumscribed by the circle. Starting with a regular hexagon, he multiplied it by a regular 96-hexagon and got (3+( 10/7 1)).
When Chinese mathematician Liu Hui annotated Nine Chapters Arithmetic (AD 263), he only used a circle inscribed with a regular polygon to find the approximate value of π, and also got the π value accurate to two decimal places. His method is called the secant circle method by later generations. Zu Chongzhi, a mathematician in the Northern and Southern Dynasties, further obtained the π value accurate to 7 decimal places (about the second half of the 5th century), gave the insufficient approximation of 3. 14 15926 and the excessive approximation of 3. 14 15927, and also got two approximate fractional values with a density of 355//. In the west, the secret rate was not obtained by German Otto until 1573, and was published in the work of Dutch engineer Antoine in 1625, which is called Antoine rate in Europe. * * * Mathematician Kathy got the exact decimal value of pi 17 at the beginning of15th century, which broke the record kept by Zu Chongzhi for nearly a thousand years. 1596, the German mathematician Curran calculated the π value to 20 decimal places, and then spent his whole life calculating it to 35 decimal places of 16 10. This value is named Rudolph number after him.
1579 The French mathematician Veda gave the first analytical expression of π.
Since then, various expressions of π value such as infinite product, infinite continued fraction and infinite series have appeared one after another, and the calculation accuracy of π value has also improved rapidly. 1706, the British mathematician Mackin calculated the π value, which broke through the decimal mark of 100. 1873, another British mathematician Jean-Jacques calculated π to 707 decimal places, but his result was wrong from 528 decimal places. By 1948, Ferguson in Britain and Ronchi in the United States announced the 808-bit decimal value of π, which became the highest record of manual calculation of pi.
The appearance of electronic computer makes the calculation of π value develop by leaps and bounds. From 65438 to 0949, the Army Ballistics Research Laboratory in Aberdeen, Maryland, USA used a computer (ENIAC) to calculate π value for the first time, and it suddenly reached 2037 decimal places, exceeding thousands of digits. 1989, researchers at Columbia University in the United States used Cray -2 and IBM-VF supercomputers to calculate 480 million digits after the decimal point, and then continued to calculate to 10 1 100 million digits after the decimal point, setting a new record.
Besides the numerical calculation of π, its properties have also attracted many mathematicians. 176 1 year, Swiss mathematician Lambert first proved that π is an irrational number. 1794 French mathematician Legendre proved that π2 is also an irrational number. By 1882, German mathematician Lin Deman proved that π is a transcendental number for the first time, thus denying the problem of "turning a circle into a square" that has puzzled people for more than two thousand years. Others study the characteristics of π and its connection with other numbers. For example, 1929, the Soviet mathematician Gelfond proved that eπ is a transcendental number and so on.
10. knowledge of pi
▲ What is pi? Pi is a constant representing the ratio of circumference to diameter.
It is an irrational number, that is, an infinite cycle decimal. But in daily life, pi is usually calculated as 3. 14. Even if engineers or physicists want to calculate more accurately, they only take the value to about 20 decimal places.
▲ What is π? π is the sixteenth Greek letter. Originally, it had nothing to do with pi, but the great mathematician Euler began to express pi in letters and papers in 1736. Because he is a great mathematician, people have followed suit and used pi to express pi.
But π can be used to represent something other than pi, which can also be seen in statistics. ▲ History of the development of pi In history, many mathematicians have studied pi, among which Archimedes, Claudius Ptolemy, Zhang Heng and Zu Chongzhi in Syracuse are famous.
They try to calculate the value of pi in their own country by their own methods. The following are the research results of pi around the world.
China, Asia: 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 (that is, the method of dividing circles), and obtained the approximate value of π 3. 14 16. Zhang Heng of Han Dynasty concluded that π divided by the square of 16 equals 5/8, that is, π equals the root of 10 (about 3. 162).
Although this value is not accurate, it is easy to understand, so it has been popular in Asia for some time. Wang Fan (229-267) discovered another value of pi, which is 3. 156, but no one knows how he got it.
In the 5th century, Zu Chongzhi and his son calculated a pi of about 355/ 1 13 with a regular polygon of 24576. Compared with the real value, the error is less than one in eight hundred million. This record was not broken until 1000 years later.
India: Around 530 AD, the mathematician aryabhata used the perimeter of a 384-sided polygon to calculate the pi of about √9.8684. Brahma Gupta used another method to derive the square root of pi equal to 10.
Fibonacci in Europe calculates pi of about 3. 14 18. The Vedas use Archimedes' method to calculate 3. 14 1596535.
Rudolf Wang Keren calculated the pi with 35 digits after the decimal point from a polygon with more than 32000000000 sides.