Pi refers tocircularOfPerimeterAnddiameterOfratio, i.ePi=circumference ÷ diameter, for general useGreek alphabetπIt is a universal mathematics in mathematics and physicsconstant。π is also equal to the ratio of the area of a circle to the square of its radius, that isPi=area of circle ÷ radiustwoExact calculationCircumference、Circular area, spherical volume, etc.stayAnalyticsWhere π can be strictly defined as satisfying sinx=Minimum of 0Positive numberx。
Pi is expressed by the Greek letter π (pronounced [pa ɪ]), which is a constant (approximately equal to 3.141592654) and represents the ratio of circumference to diameter.It is aIrrational number, that is, infinite non recurring decimal.In daily life, 3.14 is usually used to represent pi for approximate calculation.The nine decimal places 3.141592654 are enough for general calculation.Even if engineers or physicists want to carry out more precise calculations, they only need to take values to hundreds of decimal places at best.[1]
English mathematician in 1665John Wallis (John Wallis) published a monograph on mathematics[24]He deduced a formula and found that pi is equal to the product of infinite fractions.In 2015,university of rochester Scientists in[2]。
On March 14, 2019, Google announced that Pi has now reached 31.4 trillion decimal places.[3]
On August 17, 2021, the US interesting science website reported that Swiss researchers used a supercomputer to calculate the famous mathematical constant pi to 62.8 trillion decimal places after 108 days, setting the record for the most accurate value of the constant so far.[22]
A stone plaque of ancient Babylon (produced from about 1900 BC to 1600 BC) clearly records that Pi=25/8=3.125.[4]ContemporaneousAncient EgyptThe cultural relic, the Rhind Mathematical Papyrus, also shows that pi is equal to the square of the fraction 16/9, about 3.1605.[4]Egyptians seem to have known pi even earlier.John Taylor (1781-1864), a British writer, pointed out in his famous work The Great Pyramid:Khufu PyramidIt is related to pi.For example, the ratio of the perimeter to the height of a pyramid is equal to twice the circumference, which is exactly equal to the ratio of the circumference to the radius of a circle.Written between 800 and 600 BCancient IndiaThe religious masterpiece Satapatha Brahmana shows that pi is equal to the fraction 339/108, about 3.139.[5]
Geometric period
ancient GreekAs an ancient geometric kingdom, it made outstanding contributions to pi.Ancient Greek mathematicianArchimedes(287 BC - 212 BC) pioneered the theoretical calculation of the approximate value of pi in human history.Archimedes fromUnit circleStart by using the inscribed regular hexagon to find the piLower boundIs 3, then use the external regular hexagon and use thePythagorean theoremFinding piupper boundLess than 4.Then, he doubled the number of sides of the inscribed regular hexagon and the circumscribed regular hexagon respectively, changed them into the inscribed regular 12 polygon and the circumscribed regular 12 polygon, and improved the lower and upper bounds of pi with the help of Pythagorean theorem.He gradually doubled the number of sides of inscribed regular polygons and circumscribed regular polygons until they were inscribed and circumscribed regular 96 polygons.Finally, he calculated the lower and upper bounds of pi as 223/71 and 22/7 respectively, and took their average value 3.141851 as the approximate value of pi.Archimedes usediterationThe concept of algorithm and bilateral numerical approximation can be called“Computational mathematics”The ancestor of.
Chinese ancient calculation book《Zhou Bi Suanjing》In (about the second century BC), there is a record of "one track and three weeks", which means
(about 3.162).This value is not very accurate, but it is simple and easy to understand.[7]
In 263 AD, Chinese mathematicianLiu HuiUse“Cyclotomy”To calculate pi, he first connected the regular hexagon inside the circle, and then divided it gradually until it was inscribed with the regular 192 polygon.He said: "If you cut it carefully, you will lose less. If you cut it again, you will not be able to cut it. Then you will integrate it with the circumference without losing anything."limitThe idea of.Liu Hui gave the approximate value of pi=3.141024. After obtaining pi=3.14, Liu Hui compared this value with that of the Han Dynasty in the Jin Dynastyusurper who founded the Xin dynastyCopper volume made in the eraWeights and measuresstandardJialiangThe diameter and volume of dendrobium were tested, and it was found that the value of 3.14 was still small.Then continue to cut the circle to 1536 polygons, calculate the area of 3072 polygons, and get the pi that satisfies you
。
Around 480 AD, mathematicians in the Northern and Southern DynastiesZu ChongzhiFurther get the result accurate to 7 decimal places, give the insufficient approximation value 3.1415926 and the excess approximation value 3.1415927, and also get two approximate fractional values, the density ratio
Contract rate
。The density ratio is a good fractional approximation
In the next 800 years, the π value calculated by Zu Chongzhi was the most accurate.The secret rate was not obtained by the German Valentinus Otho until 1573 in the West. It was published in 1625 in the work of the Dutch engineer Metius, which is called Metius' number in Europe.
Around 530 AD, the Indian master of mathematicsAyeportCalculate pi as
Arab mathematicianKashiAt the beginning of the 15th century, we obtained 17 precise decimal values of Pi, which broke the record of Zu Chongzhi for nearly a thousand years.German mathematicianLudolph van Ceulen (Ludolph van Ceulen) calculated the value of π to 20 decimal places in 1596, and then devoted his life to calculating it to 35 decimal places in 1610. This value is called Rudolph number in his name.
Analytical period
During this period, people began to useInfinite seriesOr finding π by infinitely continuous product, getting rid of the complicated calculation of secant circle technique.Infinite product, infinityContinued fraction, infinite series and other π value expressions have emerged, which makes the calculation accuracy of π value increase rapidly.
The first fast algorithm was proposed by John Machin, a British mathematician. In 1706, Machin's π value exceeded the 100 decimal mark. He used the following formula:[9]
Where arctanxMay be determined bytaylor series Calculate.The similar method is called "Mechin type formula".
SloveniaMathematician Jurij Vega obtained the first 140 digits after the decimal point of π in 1789, of which only 137 digits are correct.The world record is maintained for 50 years.He used the number formula proposed by Meiqin in 1706.
By 1948, D. F. Ferguson of Britain and Lunch of the United States jointly published the 808 decimal value of π, which became the highest record for manual calculation of pi.
Computer Age
Pi
The appearance of electronic computer makes π value calculation develop rapidly.The world's first computer made in the United States in 1949——ENIAC(ElectronicNumerical Integrator And Computer)Aberdeen Proving GroundEnabled.The next year, Ritvesner, von Neumann and Metzhuplis used this computer to calculate 2037 decimal places of π.This computer only took 70 hours to complete the workPunch cardThe time taken is equal to the average two minutes to calculate a single digit.Five years later, IBM NORC (Naval Weapon Research Computer) only took 13 minutes to calculate 3089 decimal places of π.With the continuous progress of science and technology, the computing speed of computers has become faster and faster. From the 1960s to the 1970s, with the continuous competition of computer scientists in the United States, Britain and France, the value of π has become more and more accurate.In 1973, Jean Guilloud and Martin Bouyer discovered the millionth decimal place of π with the computer CDC 7600.
In 1976, a new breakthrough occurred.Eugene Salamin published a new formula, which is a quadratic convergence algorithm, that is, every calculation,Significant figuresWill multiply.Gauss also found a similar formula before, but it was very complicated, and it was not feasible in the age without computers.This algorithm is called the Brent Salaming (or Salaming Brent) algorithm, also known as the Gauss Legendre algorithm.
1989 United StatesColumbia UniversityResearchers used Cray-2 and IBM-3090/VF supercomputers to calculate 480 million digits after the decimal point of π, and then continued to calculate to 1.01 billion digits after the decimal point.January 7, 2010 - French engineersfabrice bellard Calculate pi to 2700 billion decimal places.August 30, 2010 - Japanese computer wizard Shigeru Kondo used home computers andcloud computingCombined, calculate the pi to 5 trillion digits after the decimal point.
October 16, 2011, JapanNagano CountyIita CityThe company employee, Maoli Kondo, used his home computer to calculate the pi to 10 trillion digits after the decimal point, breaking the 5 trillion digits set by himself in August 2010Guinness World Records。Shigeru Kondo, 56, uses a computer assembled by himself. It has taken about a year since October to set a new record.
March 14, 2022 is the International Pi Day.Certified by Guinness World Records, the most accurate value of π at present exceeds 62831853071796 decimal places.[25]
On March 15, 2024, according to the report of American Fun Science website, on the International Pi Day, Solidigm, a computer storage company headquartered in California, USA, issued a statement saying that,The company has calculated Pi (π) to about 105 trillion digits after the decimal point, breaking the previous world record of 100 trillion digits.[28]
This symbol is also the first letter of the Greek πε ρ→→→→→ε →α (meaning periphery, region, circle, etc.).In 1706, the British mathematician William Jones (1675-1749) first used "π" to represent pi[13]。In 1736,SwitzerlandMacromathematicianEulerAlso started using
Represents pi.Since then,
It has become a synonym for pi.[14]
Be careful not to
Mixed with its capital Π, the latter means continuous multiplication.
formula
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Pi
Pi（
）Generally defined as the perimeter of a circle（
）And diameter（
）Ratio:
, or directly defined asUnit circleHalf of the perimeter of.According to the nature of similar figures, for any circle,
The values of are the same, so the constant is defined
。
Note: Set
It is meaningful to define it as half of the perimeter of the unit circle, because from the perspective of modern mathematics, the perimeter C of a circle with diameter d and radius r is given by the following integral:
I.e
The middle order of the above formula
, from the substitution method of definite integral, we can get:
among
Is the perimeter of the unit circumference (r=1 in the expression of C).If defined
, then
, consistent with the perimeter formula.
The area S of a circle with radius r is given by the following integral:
order
, from the substitution method of definite integral, we can get:
among
Is the area of the unit circle (R=1 in the expression of S).Using the method of integration by parts,
therefore,
Therefore, the formula is obtained:
In this way, the formula of circle area is also obtained
The second method is to make a square with a circular radius as the side length, and then set the ratio of the circular area to the square area as
Is the ratio of the area of a circle to the square of its radius.
It is not necessary to use geometric concepts to define pi, for example, defining
It is of little practical significance to calculate the value of pi so accurately.More than a dozen pi values used in modern science and technology are enough.If we calculate the size of the observable universe with 39 bit precision pi, the error is less than oneatomVolume of[1]。Previous people calculated pi to explore whether piRecurring decimal。Since Lambert proved in 1761 that pi isIrrational numberIn 1882, Linderman proved that pi isTranscendental numberAfter that, the mystery of pi was unveiled.
π plays a very important role in many mathematical fields.
geometry
Plane figure
Perimeter
the measure of area
circular
ring
sector
Note: ①
Is perimeter,
Is the area,
Is the arc length;
Is the diameter,
Is the radius (inner circle radius),
Is the outer circle radius,
Is the degree of center angle.
② The perimeter and arc length are in length units, and the area is in area units.
Stereoscopic figure
Surface area
volume
cylinder
cone
Note: ①
Is the perimeter of the bottom surface,
Is the bottom area,
Is the side area,
Is the surface area,
Is the volume;
Is the bottom diameter,
Is the bottom radius,
Is high.
② The perimeter of the bottom is in length unit, the surface area (including the bottom area and side area) is in area unit, and the volume is in volume unit or volume unit.
Algebra
π is an irrational number, that is, the ratio of two integers can not be expressedJohann Heinrich Lambert Certified in 1761[15]。In 1882, Ferdinand von Lindemann further proved that π is a transcendental number, that is, π cannot be anyIntegral coefficient polynomialThe root of.
The transcendence of pi negatesRound to SquareThis is ancientDrawing with ruler and gaugeThe possibility of the problem, because all ruler and gauge drawings can only be obtainedAlgebraic numberThe transcendental number is not an algebraic number.
Draw a group of parallel lines on the plane with spacing d, and find a length l (l＜d）The needle of is thrown on the plane at will, and the probability of intersection between the needle and any parallel line is calculated.This is it.Bufeng needleQuestion.In 1777,BuffonI have proved that the probability is
In 2011, the International Mathematical Association officially announced that March 14 of each year will be set asInternational Mathematics FestivalThe source is the pi of ancient Chinese mathematician Zu Chongzhi.[16]
The International Pi Day can be traced back to March 14, 1988. Larry Shaw, a physicist at the San Francisco Science Museum, organized the staff and participants of the museum to make 3 and 1/7 circles around the museum monument (22/7, one of the approximate values of π) and eat fruit pie together.Later, the Museum of Science in San Francisco inherited this tradition and held celebrations on this day every year.
In 2009,United States House of RepresentativesA non binding resolution was formally adopted, setting March 14 of each year as the "Pi Day".According to the resolution, "Since mathematics and natural science are an interesting and indispensable part of education, and learning about π is a fascinating way to teach children geometry and attract them to learn natural science and mathematics... π is about 3.14, so March 14 is the most appropriate day to commemorate the circumference day."
Anecdotes
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The most marathon human hand π value calculation in history, one of which is from GermanyLudolph van Ceulen (Ludolph van Ceulen), who spent almost his whole life getting the 35 digit precision value of pi in 1609, so that pi is called Ludolphine number in Germany;The second is William Shanks of England, who spent 15 years calculating 707 decimal places of pi in 1874 and engraved it on the tombstone as a lifelong honor.Unfortunately, later generations found that he was wrong from 528.[17]
stayGoogleIn a public offering in 2005, the company raised more than US $4 billion in total, and the number of A-shares issued was 14159265 shares, which is of course derived from the digits after the decimal point of π.(By the way, Google's 2004 IPO raised $2718281828, related to the mathematical constant e[18]）
Typesetting softwareTeXThe version number after the third version is to increase one decimal place one by one to make it more and more close to the value of π: 3.1, 3.14,... The current latest version number is 3.1415926.[19-20]
On March 14 every yearPi day 。The "ultimate pi day" is 6:54 on March 14, 1592 (because its British notation is "3/14/15926.54") and 2:6:5 on May 9, 3141 (from front to back, 3.14159265).
July 22 is the approximate day of pi (English date is recorded as 22/7, which is regarded as the approximate fraction of pi).
Some mathematicians believe that the "true pi" should be defined as 2 π and recorded as τ (tau).[21]
On March 14, 2019, Google announced that Emma Haruka Iwao, a Japanese former Google engineer, with the help of Google's cloud platform, had calculated 31.4 trillion digits after the pi decimal point, or 31415926535897 digits to be precise, an increase of several trillion digits over the record set in 2016.It is understood that Emma's team used a program called ycruncher, which can calculate π to trillions of decimal places.The program is driven by 25 virtual machines running on the Google Cloud platform computing engine.Peter Trueb, the creator of the record in 2016, was calculated by a computer.This calculation requires 170TB of data, which is roughly the same as the data volume of the printed collections of the United States Library of Congress. It took Emma about four months to break the previous world record.[3]
In 2020, Timothy Muliken, the founder of a non-profit organization called North Alabama Charitable Computing, used a personal computer to calculate the value to 50 trillion decimal places, which took 303 days.[23]
On August 17, 2021, the US interesting science website reported that Swiss researchers used a supercomputer to calculate the famous mathematical constant pi to 62.8 trillion decimal places after 108 days, setting the record for the most accurate value of the constant so far.[22]
In June 2022, Google Cloud said in a press release that Google's cloud service had broken the record it set in 2019 and calculated 100 trillion digits of pi. Last time, Google Cloud accurately calculated piIt reached 31.4 trillion.[26]
