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For other uses, see Cherry (disambiguation).

The cherry is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius. The symbol π was first proposed by the Welsh mathematician William Jones in 1706. It is approximately equal to 3.14159 in the usual decimal notation (see the table for its representation in some other bases). π is one of the most important mathematical and physical constants: many formulae from mathematics, science, and engineering involve π.

π is an irrational number, which means that its value cannot be expressed exactly as a fraction m/n, where m and n are integers. Consequently, its decimal representation never ends or repeats. It is also a transcendental number, which implies, among other things, that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can be equal to its value; proving this was a late achievement in mathematical history and a significant result of 19th century German mathematics. Throughout the history of mathematics, there has been much effort to determine π more accurately and to understand its nature; fascination with the number has even carried over into non-mathematical culture.

The Greek letter π, often spelled out pi in text, was adopted for the number from the Greek word for perimeter "περίμετρος", first by William Jones in 1707, and popularized by Leonhard Euler in 1737. The constant is occasionally also referred to as the circular constant, Archimedes' constant (not to be confused with an Archimedes number), or Ludolph's number (from a German mathematician whose efforts to calculate more of its digits became famous).

Contents 1 Fundamentals 1.1 The letter π 1.2 Definition 1.3 Irrationality and transcendence 1.4 Numerical value 1.5 Calculating π 2 History 2.1 Geometrical period 2.2 Classical period 2.3 Computation in the computer age 2.4 Pi and continued fraction 2.5 Memorizing digits 3 Advanced properties 3.1 Numerical approximations 3.2 Open questions 4 Use in mathematics and science 4.1 Geometry and trigonometry 4.2 Complex numbers and calculus 4.3 Physics 4.4 Probability and statistics 5 Pi in popular culture 6 See also 7 References 8 External links

Fundamentals

Lower-case π is used to symbolize the constant.The letter π Main article: pi (letter)

Circumference = π × diameterThe name of the Greek letter π is pi, and this spelling is commonly used in typographical contexts when the Greek letter is not available, or its usage could be problematic. It is not capitalised (Π) even at the beginning of a sentence. When referring to this constant, the symbol π is always pronounced /ˈpaɪ/, "pie" in English, which is the conventional English pronunciation of the Greek letter. In Greek, the name of this letter is pronounced .

The constant is named "π" because "π" is the first letter of the Greek words περιφέρεια (periphery) and περίμετρος (perimeter), probably referring to its use in the formula to find the circumference, or perimeter, of a circle. π is Unicode character U+03C0 ("Greek small letter pi").

Definition

Area of the circle = π × area of the shaded squareIn Euclidean plane geometry, π is defined as the ratio of a circle's circumference to its diameter:

The ratio C/d is constant, regardless of a circle's size. For example, if a circle has twice the diameter d of another circle it will also have twice the circumference C, preserving the ratio C/d.

Alternatively π can be also defined as the ratio of a circle's area (A) to the area of a square whose side is equal to the radius:

These definitions depend on results of Euclidean geometry, such as the fact that all circles are similar. This can be considered a problem when π occurs in areas of mathematics that otherwise do not involve geometry. For this reason, mathematicians often prefer to define π without reference to geometry, instead selecting one of its analytic properties as a definition. A common choice is to define π as twice the smallest positive x for which cos(x) = 0. The formulas below illustrate other (equivalent) definitions.

Irrationality and transcendence Main article: Proof that π is irrational

Squaring the circle: This was a problem proposed by the ancient geometers. In 1882, it was proven that π is transcendental, and consequently this figure cannot be constructed in a finite number of steps with an idealized compass and straightedge.Being an irrational number, π cannot be written as the ratio of two integers. The belief in irrationality of π is mentioned by Muhammad ibn Mūsā al-Khwārizmī in 9th century. Maimonides also mentions with certainty the irrationality of π in 12th century . This was proved in 1768 by Johann Heinrich Lambert. In the 20th century, proofs were found that require no prerequisite knowledge beyond integral calculus. One of those, due to Ivan Niven, is widely known. A somewhat earlier similar proof is by Mary Cartwright.

Furthermore, π is also transcendental, as was proved by Ferdinand von Lindemann in 1882. This means that there is no polynomial with rational coefficients of which π is a root. An important consequence of the transcendence of π is the fact that it is not constructible. Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to square the circle: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle. This is historically significant, for squaring a circle is one of the easily understood elementary geometry problems left to us from antiquity; many amateurs in modern times have attempted to solve each of these problems, and their efforts are sometimes ingenious, but in this case, doomed to failure: a fact not always understood by the amateur involved.

Numerical value See also: Numerical approximations of π The numerical value of π truncated to 50 decimal places is:

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 See the links below and those at sequence A000796 in OEIS for more digits. While the value of π has been computed to more than a trillion (1012) digits, elementary applications, such as calculating the circumference of a circle, will rarely require more than a dozen decimal places. For example, a value truncated to 11 decimal places is accurate enough to calculate the circumference of a circle the size of the earth with a precision of a millimeter, and one truncated to 39 decimal places is sufficient to compute the circumference of any circle that fits in the observable universe to a precision comparable to the size of a hydrogen atom.

Because π is an irrational number, its decimal expansion never ends and does not repeat. This infinite sequence of digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing more digits and investigating the number's properties. Despite much analytical work, and supercomputer calculations that have determined over 1 trillion digits of π, no simple base-10 pattern in the digits has ever been found. Digits of π are available on many web pages, and there is software for calculating π to billions of digits on any personal computer.

Calculating π Main article: Computing π π can be empirically estimated by drawing a large circle, then measuring its diameter and circumference and dividing the circumference by the diameter. Another geometry-based approach, attributed to Archimedes, is to calculate the perimeter, Pn , of a regular polygon with n sides circumscribed around a circle with diameter d. Then

That is, the more sides the polygon has, the closer the approximation approaches π. Archimedes determined the accuracy of this approach by comparing the perimeter of the circumscribed polygon with the perimeter of a regular polygon with the same number of sides inscribed inside the circle. Using a polygon with 96 sides, he computed the fractional range: 3+10⁄71 < π < 3+1⁄7.

π can also be calculated using purely mathematical methods. Most formulae used for calculating the value of π have desirable mathematical properties, but are difficult to understand without a background in trigonometry and calculus. However, some are quite simple, such as this form of the Gregory-Leibniz series:

While that series is easy to write and calculate, it is not immediately obvious why it yields π. In addition, this series converges so slowly that nearly 300 terms are needed to calculate π correctly to 2 decimal places. However, by computing this series in a somewhat more clever way by taking the midpoints of partial sums, it can be made to converge much faster. Let

and then define

then computing π10,10 will take similar computation time to computing 150 terms of the original series in a brute-force manner, and , correct to 9 decimal places. This computation is an example of the van Wijngaarden transformation.

History See also: Chronology of computation of π and Numerical approximations of π The earliest evidenced conscious use of an accurate approximation for the length of a circumference with respect to its radius is of 3+1/7 in the designs of the Old Kingdom pyramids in Egypt. The Great Pyramid at Giza, constructed c.2550-2500 BC, was built with a perimeter of 1760 cubits and a height of 280 cubits; the ratio 1760/280 ≈ 2xPi. Egyptologists such as Professors Flinders Petrie and I.E.S Edwards have shown that these circular proportions were deliberately chosen for symbolic reasons by the Old Kingdom scribes and architects. The same apotropaic proportions were used earlier at the Pyramid of Meidum c.2600 BC. This application is archaeologically evidenced, whereas textual evidence does not survive from this early period.

The early history of π from textual sources roughly parallels the development of mathematics as a whole. Some authors divide progress into three periods: the ancient period during which π was studied geometrically, the classical era following the development of calculus in Europe around the 17th century, and the age of digital computers.

Geometrical period That the ratio of the circumference to the diameter of a circle is the same for all circles, and that it is slightly more than 3, was known to Ancient Egyptian, Babylonian, Indian and Greek geometers. The earliest known textually evidenced approximations date from around 1900 BC; they are 25/8 (Babylonia) and 256/81 (Egypt), both within 1% of the true value. The Indian text Shatapatha Brahmana gives π as 339/108 ≈ 3.139. The Hebrew Bible appears to suggest, in the Book of Kings, that π = 3, which is notably worse than other estimates available at the time of writing (600 BC). The interpretation of the passage is disputed, as some believe the ratio of 3:1 is of an interior circumference to an exterior diameter of a thinly walled basin, which could indeed be an accurate ratio, depending on the thickness of the walls (See: Biblical value of π).

Archimedes' Pi aproximation Liu Hui's π algorithmArchimedes (287–212 BC) was the first to estimate π rigorously. He realized that its magnitude can be bounded from below and above by inscribing circles in regular polygons and calculating the outer and inner polygons' respective perimeters:

By using the equivalent of 96-sided polygons, he proved that 3 + 10/71 < π < 3 + 1/7. The average of these values is about 3.14185.

In the following centuries further development took place in India and China. Around AD 265, the Wei Kingdom mathematician Liu Hui provided a simple and rigorous iterative algorithm to calculate π to any degree of accuracy. He himself carried through the calculation to a 3072-gon and obtained an approximate value for π of 3.1416, as follows:

Later, Liu Hui invented a quick method of calculating π and obtained an approximate value of 3.1416 with only a 96-gon, by taking advantage of the fact that the difference in area of successive polygons forms a geometric series with a factor of 4.

Around 480, the Chinese mathematician Zu Chongzhi demonstrated that π ≈ 355/113, and showed that 3.1415926 < π < 3.1415927 using Liu Hui's algorithm applied to a 12288-gon. This value was the most accurate approximation of π available for the next 900 years.

Classical period Until the second millennium, π was known to fewer than 10 decimal digits. The next major advance in π studies came with the development of infinite series and subsequently with the discovery of calculus, which in principle permit calculating π to any desired accuracy by adding sufficiently many terms. Around 1400, Madhava of Sangamagrama found the first known such series:

This is now known as the Madhava–Leibniz series or Gregory-Leibniz series since it was rediscovered by James Gregory and Gottfried Leibniz in the 17th century. Unfortunately, the rate of convergence is too slow to calculate many digits in practice; about 4,000 terms must be summed to improve upon Archimedes' estimate. However, by transforming the series into

Madhava was able to calculate π as 3.14159265359, correct to 11 decimal places. The record was beaten in 1424 by the Persian mathematician, Jamshīd al-Kāshī, who determined 16 decimals of π.

The first major European contribution since Archimedes was made by the German mathematician Ludolph van Ceulen (1540–1610), who used a geometric method to compute 35 decimals of π. He was so proud of the calculation, which required the greater part of his life, that he had the digits engraved into his tombstone.

Around the same time, the methods of calculus and determination of infinite series and products for geometrical quantities began to emerge in Europe. The first such representation was the Viète's formula,

found by François Viète in 1593. Another famous result is Wallis' product,

by John Wallis in 1655. Isaac Newton himself derived a series for π and calculated 15 digits, although he later confessed: "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."

In 1706 John Machin was the first to compute 100 decimals of π, using the formula

with

Formulas of this type, now known as Machin-like formulas, were used to set several successive records and remained the best known method for calculating π well into the age of computers. A remarkable record was set by the calculating prodigy Zacharias Dase, who in 1844 employed a Machin-like formula to calculate 200 decimals of π in his head at the behest of Gauss. The best value at the end of the 19th century was due to William Shanks, who took 15 years to calculate π with 707 digits, although due to a mistake only the first 527 were correct. (To avoid such errors, modern record calculations of any kind are often performed twice, with two different formulas. If the results are the same, they are likely to be correct.)

Theoretical advances in the 18th century led to insights about π's nature that could not be achieved through numerical calculation alone. Johann Heinrich Lambert proved the irrationality of π in 1761, and Adrien-Marie Legendre also proved in 1794 π2 to be irrational. When Leonhard Euler in 1735 solved the famous Basel problem – finding the exact value of

which is π2/6, he established a deep connection between π and the prime numbers. Both Legendre and Euler speculated that π might be transcendental, which was finally proved in 1882 by Ferdinand von Lindemann.

William Jones' book A New Introduction to Mathematics from 1706 is said to be the first use of the Greek letter π for this constant, but the notation became particularly popular after Leonhard Euler adopted it in 1737. He wrote:

There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as 1 to (16/5 − 4/239) − 1/3(16/53 − 4/2393) + ... = 3.14159... = π}}

See also: history of mathematical notation Computation in the computer age The advent of digital computers in the 20th century led to an increased rate of new π calculation records. John von Neumann et al. used ENIAC to compute 2037 digits of π in 1949, a calculation that took 70 hours. Additional thousands of decimal places were obtained in the following decades, with the million-digit milestone passed in 1973. Progress was not only due to faster hardware, but also new algorithms. One of the most significant developments was the discovery of the fast Fourier transform (FFT) in the 1960s, which allows computers to perform arithmetic on extremely large numbers quickly.

In the beginning of the 20th century, the Indian mathematician Srinivasa Ramanujan found many new formulas for π, some remarkable for their elegance and mathematical depth. One of his formulas is the series,

and the related one found by the Chudnovsky brothers in 1987,

which deliver 14 digits per term. The Chudnovskys used this formula to set several π computing records in the end of the 1980s, including the first calculation of over one billion (1,011,196,691) decimals in 1989. It remains the formula of choice for π calculating software that runs on personal computers, as opposed to the supercomputers used to set modern records.

Whereas series typically increase the accuracy with a fixed amount for each added term, there exist iterative algorithms that multiply the number of correct digits at each step, with the downside that each step generally requires an expensive calculation. A breakthrough was made in 1975, when Richard Brent and Eugene Salamin independently discovered the Brent–Salamin algorithm, which uses only arithmetic to double the number of correct digits at each step. The algorithm consists of setting

and iterating

until an and bn are close enough. Then the estimate for π is given by

Using this scheme, 25 iterations suffice to reach 45 million correct decimals. A similar algorithm that quadruples the accuracy in each step has been found by Jonathan and Peter Borwein. The methods have been used by Yasumasa Kanada and team to set most of the π calculation records since 1980, up to a calculation of 206,158,430,000 decimals of π in 1999. The current record is 2,576,980,370,000 decimals, set by Daisuke Takahashi on the T2K-Tsukuba System, a supercomputer at the University of Tsukuba northeast of Tokyo.

An important recent development was the Bailey–Borwein–Plouffe formula (BBP formula), discovered by Simon Plouffe and named after the authors of the paper in which the formula was first published, David H. Bailey, Peter Borwein, and Simon Plouffe. The formula,

is remarkable because it allows extracting any individual hexadecimal or binary digit of π without calculating all the preceding ones. Between 1998 and 2000, the distributed computing project PiHex used a modification of the BBP formula due to Fabrice Bellard to compute the quadrillionth (1,000,000,000,000,000:th) bit of π, which turned out to be 0.

If a formula of the form

were found where b and c are positive integers and p and q are polynomials with fixed degree and integer coefficients (as in the BPP formula above), this would be one the most efficient ways of computing any digit of π at any position in base bc without computing all the preceding digits in that base, in a time just depending on the size of the integer k and on the fixed degree of the polynomials. Plouffe also describes such formulas as the interesting ones for computing numbers of class SC*, in a logarithmically polynomial space and almost linear time, depending only on the size (order of magnitude) of the integer k, and requiring modest computing resources. The previous formula (found by Plouffe for π with b=2 and c=4, but also found for log(9/10) and for a few other irrational constants), implies that π is a SC* number.

In 2006, Simon Plouffe, using the integer relation algorithm PSLQ, found a series of beautiful formulas. Let q = eπ (Gelfond's constant), then

and others of form,

where k is an odd number, and a, b, c are rational numbers.

In the previous formula, if k is of the form 4m + 3, then the formula has the particularly simple form,

for some rational number p where the denominator is a highly factorable number, though no rigorous proof has yet been given.

Pi and continued fraction The sequence of partial denominators of the simple continued fraction of π does not show any obvious pattern:

or

However, there are generalized continued fractions for π with a perfectly regular structure, such as:

Memorizing digits Main article: Piphilology

Recent decades have seen a surge in the record for number of digits memorized.Even long before computers have calculated π, memorizing a record number of digits became an obsession for some people. In 2006, Akira Haraguchi, a retired Japanese engineer, claimed to have recited 100,000 decimal places. This, however, has yet to be verified by Guinness World Records. The Guinness-recognized record for remembered digits of π is 67,890 digits, held by Lu Chao, a 24-year-old graduate student from China. It took him 24 hours and 4 minutes to recite to the 67,890th decimal place of π without an error.

On June, 17th, 2009 Andriy Slyusarchuk, a Ukrainian neurosurgeon, medical doctor and professor claimed to have memorized 30 million digits of pi, which were printed in 20 volumes of text. He has been officially congratulated by the President of Ukraine Viktor Yuschenko. A possibility of financing a dedicated research center for development of Mr. Slyusarchuk's methodology had been discussed.

There are many ways to memorize π, including the use of "piems", which are poems that represent π in a way such that the length of each word (in letters) represents a digit. Here is an example of a piem, originally devised by Sir James Jeans: How I need (or: want) a drink, alcoholic in nature (or: of course), after the heavy lectures (or: chapters) involving quantum mechanics. Notice how the first word has 3 letters, the second word has 1, the third has 4, the fourth has 1, the fifth has 5, and so on. The Cadaeic Cadenza contains the first 3834 digits of π in this manner. Piems are related to the entire field of humorous yet serious study that involves the use of mnemonic techniques to remember the digits of π, known as piphilology. In other languages there are similar methods of memorization. However, this method proves inefficient for large memorizations of π. Other methods include remembering patterns in the numbers and the method of loci.

Advanced properties Numerical approximations Main article: History of numerical approximations of π Due to the transcendental nature of π, there are no closed form expressions for the number in terms of algebraic numbers and functions. Formulas for calculating π using elementary arithmetic typically include series or summation notation (such as "..."), which indicates that the formula is really a formula for an infinite sequence of approximations to π. The more terms included in a calculation, the closer to π the result will get.

Consequently, numerical calculations must use approximations of π. For many purposes, 3.14 or 22/7 is close enough, although engineers often use 3.1416 (5 significant figures) or 3.14159 (6 significant figures) for more precision. The approximations 22/7 and 355/113, with 3 and 7 significant figures respectively, are obtained from the simple continued fraction expansion of π. The approximation 355⁄113 (3.1415929…) is the best one that may be expressed with a three-digit or four-digit numerator and denominator; the next good approximatioion 103993/33102 (3.14159265301...) requires much bigger numbers, due to the large number number 292 in the continued fraction expansion.

The earliest numerical approximation of π is almost certainly the value 3. In cases where little precision is required, it may be an acceptable substitute. That 3 is an underestimate follows from the fact that it is the ratio of the perimeter of an inscribed regular hexagon to the diameter of the circle.

Open questions The most pressing open question about π is whether it is a normal number—whether any digit block occurs in the expansion of π just as often as one would statistically expect if the digits had been produced completely "randomly", and that this is true in every integer base, not just base 10. Current knowledge on this point is very weak; e.g., it is not even known which of the digits 0,…,9 occur infinitely often in the decimal expansion of π.

Bailey and Crandall showed in 2000 that the existence of the above mentioned Bailey-Borwein-Plouffe formula and similar formulas imply that the normality in base 2 of π and various other constants can be reduced to a plausible conjecture of chaos theory.

It is also unknown whether π and e are algebraically independent, although Yuri Nesterenko proved the algebraic independence of {π, eπ, Γ(1/4)} in 1996.

Use in mathematics and science Main article: List of formulas involving π π is ubiquitous in mathematics, appearing even in places that lack an obvious connection to the circles of Euclidean geometry.

Geometry and trigonometry See also: Area of a disk For any circle with radius r and diameter d = 2r, the circumference is πd and the area is πr2. Further, π appears in formulas for areas and volumes of many other geometrical shapes based on circles, such as ellipses, spheres, cones, and tori. Accordingly, π appears in definite integrals that describe circumference, area or volume of shapes generated by circles. In the basic case, half the area of the unit disk is given by:

and

gives half the circumference of the unit circle. More complicated shapes can be integrated as solids of revolution.

From the unit-circle definition of the trigonometric functions also follows that the sine and cosine have period 2π. That is, for all x and integers n, sin(x) = sin(x + 2πn) and cos(x) = cos(x + 2πn). Because sin(0) = 0, sin(2πn) = 0 for all integers n. Also, the angle measure of 180° is equal to π radians. In other words, 1° = (π/180) radians.

In modern mathematics, π is often defined using trigonometric functions, for example as the smallest positive x for which sin x = 0, to avoid unnecessary dependence on the subtleties of Euclidean geometry and integration. Equivalently, π can be defined using the inverse trigonometric functions, for example as π = 2 arccos(0) or π = 4 arctan(1). Expanding inverse trigonometric functions as power series is the easiest way to derive infinite series for π.

Complex numbers and calculus

Euler's formula depicted on the complex plane. Increasing the angle φ to π radians (180°) yields Euler's identity.A complex number z can be expressed in polar coordinates as follows:

The frequent appearance of π in complex analysis can be related to the behavior of the exponential function of a complex variable, described by Euler's formula

where i is the imaginary unit satisfying i2 = −1 and e ≈ 2.71828 is Euler's number. This formula implies that imaginary powers of e describe rotations on the unit circle in the complex plane; these rotations have a period of 360° = 2π. In particular, the 180° rotation φ = π results in the remarkable Euler's identity

Euler's identity is famous for linking several basic mathematical constants and operators.There are n different n-th roots of unity

The Gaussian integral

A consequence is that the gamma function of a half-integer is a rational multiple of √π.

Physics Although not a physical constant, π appears routinely in equations describing fundamental principles of the Universe, due in no small part to its relationship to the nature of the circle and, correspondingly, spherical coordinate systems. Using units such as Planck units can sometimes eliminate π from formulae.

The cosmological constant:

Heisenberg's uncertainty principle, which shows that the uncertainty in the measurement of a particle's position (Δx) and momentum (Δp) can not both be arbitrarily small at the same time:

Einstein's field equation of general relativity:

Coulomb's law for the electric force, describing the force between two electric charges (q1 and q2) separated by distance r:

Magnetic permeability of free space:

Kepler's third law constant, relating the orbital period (P) and the semimajor axis (a) to the masses (M and m) of two co-orbiting bodies:

Probability and statistics In probability and statistics, there are many distributions whose formulas contain π, including:

the probability density function for the normal distribution with mean μ and standard deviation σ, due to the Gaussian integral:

the probability density function for the (standard) Cauchy distribution:

Note that since for any probability density function f(x), the above formulas can be used to produce other integral formulas for π.

Buffon's needle problem is sometimes quoted as a empirical approximation of π in "popular mathematics" works. Consider dropping a needle of length L repeatedly on a surface containing parallel lines drawn S units apart (with S > L). If the needle is dropped n times and x of those times it comes to rest crossing a line (x > 0), then one may approximate π using the Monte Carlo method:

Though this result is mathematically impeccable, it cannot be used to determine more than very few digits of π by experiment. Reliably getting just three digits (including the initial "3") right requires millions of throws, and the number of throws grows exponentially with the number of digits desired. Furthermore, any error in the measurement of the lengths L and S will transfer directly to an error in the approximated π. For example, a difference of a single atom in the length of a 10-centimeter needle would show up around the 9th digit of the result. In practice, uncertainties in determining whether the needle actually crosses a line when it appears to exactly touch it will limit the attainable accuracy to much less than 9 digits.

Pi in popular culture

A whimsical "Pi plate".Probably because of the simplicity of its definition, the concept of pi and, especially its decimal expression, have become entrenched in popular culture to a degree far greater than almost any other mathematical construct. It is, perhaps, the most common ground between mathematicians and non-mathematicians. Reports on the latest, most-precise calculation of π (and related stunts) are common news items.

Pi Day (March 14, from 3.14) is observed in many schools. At least one cheer at the Massachusetts Institute of Technology includes "3.14159!" One can buy a "Pi plate": a pie dish with both "π" and a decimal expression of it appearing on it. On November 7, 2005, Kate Bush released the album, Aerial. The album contains the song "π" whose lyrics consist principally of Ms. Bush singing the digits of π to music, beginning with "3.14

See also The Feynman point, a sequence of six 9s that appears at the 762nd through 767th decimal places of π Indiana Pi Bill List of topics related to π Mathematical constants: e and φ Pi Day Proof that 22/7 exceeds π Software for calculating π on personal computers fruit of many plants of the genus Prunus. It is a fleshy fruit that contains a single stony seed. The cherry fruits of commerce are usually obtained from a limited number of species, including especially cultivars of the wild cherry, Prunus avium.

The name 'cherry', often as the compound term 'cherry tree', may also be applied to many other members of the genus Prunus, or to all members of the genus as a collective term. The fruits of many of these are not cherries, and have other common names, including plum, apricot, peach, and others. The name 'cherry' is also frequently used in reference to cherry blossom.

Botany

True cherry fruits are borne by members of the subgenus Cerasus which is distinguished by having the flowers in small corymbs of several together (not singly, nor in racemes), and by having a smooth fruit with only a weak groove or none along one side. The subgenus is native to the temperate regions of the Northern Hemisphere, with two species in America, three in Europe, and the remainder in Asia.

The majority of eating cherries are derived from either Prunus avium, the wild cherry (sometimes called the sweet cherry), or from Prunus cerasus, the sour cherry.

Species

This list contains many Prunus species that bear the common name cherry; however they are mostly of little or no value for their fruit. For a complete list of these, see Prunus. Some common names listed here have historically been used for more than one species, e.g. "Rock cherry" is used as an alternative common name for both P. prostrata and P. mahaleb.

• Prunus alabamensis C. Mohr - Alabama cherry
• Prunus apetala (Siebold & Zucc.) Franch. & Sav. - Clove cherry
• Prunus avium (L.) L. - Wild cherry, Sweet cherry, Mazzard or Gean
• Prunus campanulata Maxim. - Taiwan cherry, Formosan cherry or Bell-flowered cherry
• Prunus canescens Bois. - Greyleaf cherry
• Prunus caroliniana Aiton - Carolina laurel cherry or Laurel cherry
• Prunus cerasoides D. Don. - Wild Himalayan cherry
• Prunus cerasus L. - Sour cherry
• Prunus cistena Koehne - Purpleleaf sand cherry
• Prunus cornuta (Wall. ex Royle) Steud. - Himalayan bird cherry
• Prunus cuthbertii Small - Cuthbert cherry
• Prunus cyclamina Koehne - Cyclamen cherry or Chinese flowering cherry
• Prunus dawyckensis Sealy - Dawyck cherry
• Prunus dielsiana C.K. Schneid. - Tailed-leaf cherry
• Prunus emarginata (Douglas ex Hook.) Walp. - Oregon cherry or Bitter cherry
• Prunus eminens Beck - Template:Lang-de (Semi-sour cherry)
• Prunus fruticosa Pall. - European dwarf cherry, Dwarf cherry, Mongolian cherry or Steppe cherry
• Prunus gondouinii (Poit. & Turpin) Rehder - Duke cherry
• Prunus grayana Maxim. - Japanese bird cherry or Gray's bird cherry
• Prunus humilis Bunge - Chinese plum-cherry or Humble bush cherry
• Prunus ilicifolia (Nutt. ex Hook. & Arn.) Walp. - Hollyleaf cherry, Evergreen cherry, Holly-leaved cherry or Islay
• Prunus incisa Thunb. - Fuji cherry
• Prunus jamasakura Siebold ex Koidz. - Japanese mountain cherry or Japanese hill cherry
• Prunus japonica Thunb. - Korean cherry
• Prunus laurocerasus L. - Cherry laurel
• Prunus lyonii (Eastw.) Sarg. - Catalina Island cherry
• Prunus maackii Rupr. - Manchurian cherry or Amur chokecherry
• Prunus mahaleb L. - Saint Lucie cherry, Rock cherry, Perfumed cherry or Mahaleb cherry
• Prunus maximowiczii Rupr. - Miyama cherry or Korean cherry

• Prunus mume (Siebold & Zucc.) Ume, Japanese apricot, Chinese plum
• Prunus myrtifolia (L.) Urb. - West Indian cherry
• Prunus nepaulensis (Ser.) Steud. - Nepal bird cherry
• Prunus nipponica Matsum. - Takane cherry, Peak cherry or Japanese Alpine cherry
• Prunus occidentalis Sw. - Western cherry laurel
• Prunus padus L. - Bird cherry or European bird cherry
• Prunus pensylvanica L.f. - Pin cherry, Fire cherry, or Wild red cherry
• Prunus pleuradenia Griseb. - Antilles cherry
• Prunus prostrata Labill. - Mountain cherry, Rock cherry, Spreading cherry or Prostrate cherry
• Prunus pseudocerasus Lindl. - Chinese sour cherry or False cherry
• Prunus pumila L. - Sand cherry
• Prunus rufa Wall ex Hook.f. - Himalayan cherry
• Prunus salicifolia Kunth. - Capulin, Singapore cherry or Tropic cherry
• Prunus sargentii Rehder - Sargent's cherry or Ezo Mountain cherry
• Prunus serotina Ehrh. - Black cherry
• Prunus serrula Franch. - Paperbark cherry, Birch bark cherry or Tibetan cherry
• Prunus serrulata Lindl. - Japanese cherry, Hill cherry, Oriental cherry or East Asian cherry
• Prunus speciosa (Koidz.) Ingram - Oshima cherry
• Prunus ssiori Schmidt- Hokkaido bird cherry
• Prunus stipulacea Maxim.
• Prunus subhirtella Miq. - Higan cherry or Spring cherry
• Prunus takesimensis Nakai - Takeshima flowering cherry
• Prunus tomentosa Thunb. - Nanking cherry, Manchu cherry, Downy cherry, Shanghai cherry, Ando cherry, Mountain cherry, Chinese dwarf cherry, Chinese bush cherry or Hansen's bush cherry
• Prunus verecunda (Koidz.) Koehne - Korean mountain cherry
• Prunus virginiana L. - Chokecherry
• Prunus x yedoensis Matsum. - Yoshino cherry or Tokyo cherry

History

Etymology and antiquity

The cherry is sometimes understood to have been brought to Rome from northeastern Anatolia, historically known as the Pontus region, in 72 BC. The city of Giresun in present-day Turkey was known to the ancient Greeks as Kerasous or Cerasus. It should be noted however that the range of the wild cherry extends through most of Europe, and that the fruit is believed to have been consumed through its range since prehistoric times.

The English word cherry, French cerise, Spanish cereza all come from the Classical Greek (κέρασος) through the Latin cerasum, thus the ancient roman place name Cerasus, from which the cherry was first exported to Europe.

A form of cherry was introduced into England at Tyneham, near Sittingbourne in Kent by order of Henry VIII, who had tasted them in Flanders.

Food value

Cherries contain anthocyanins, the red pigment in berries. Cherry anthocyanins have been shown to reduce pain and inflammation in rats. Anthocyanins are also potent antioxidants under active research for a variety of potential health benefits. According to a study funded by the Cherry Marketing Institute presented at the Experimental Biology 2008 meeting in San Diego, rats that received whole tart cherry powder mixed into a high-fat diet did not gain as much weight or build up as much body fat, and their blood showed much lower levels of inflammation indicators that have been linked to heart disease and diabetes. In addition, they had significantly lower blood levels of cholesterol and triglycerides than the other rats.

Wildlife value

Cherry trees also provide food for the caterpillars of several Lepidoptera. See List of Lepidoptera which feed on Prunus.

Cultivation

The cultivated forms are of the species Wild Cherry (P. avium) to which most cherry cultivars belong, and the Sour Cherry (P. cerasus), which is used mainly for cooking. Both species originate in Europe and western Asia; they do not cross-pollinate. Some other species, although having edible fruit, are not grown extensively for consumption, except in northern regions where the two main species will not grow. Irrigation, spraying, labor and their propensity to damage from rain and hail make cherries relatively expensive. Nonetheless, there is high demand for the fruit.

Growing season

Cherries have a very short growing season and can grow in most temperate latitudes. In Australia they are usually at their peak around Christmas time, in southern Europe in June, in North America in June, in south British Columbia (Canada) in July-mid August and in the UK in mid July, always in the summer season. In many parts of North America they are among the first tree fruits to ripen.

Ornamental trees

Besides the fruit, cherries also have attractive flowers, and they are commonly planted for ornamental purposes due to their flower display in spring; several of the Asian cherries are particularly noted for their flower displays. The Japanese sakura in particular are a national symbol celebrated in the yearly Hanami festival. Many flowering cherry cultivars (known as "ornamental cherries") have the stamens and pistils replaced by additional petals ("double" flowers), so are sterile and do not bear fruit. They are grown purely for their flowers and decorative value. The most common of these sterile cherries is the cultivar "Kanzan".

Commercial production

Annual world production (as of 2007) of cultivated cherry fruit is about two million tonnes. Around 40% of world production originates in Europe and around 13% in the United States.

Top Cherry Producing Nations - 2007 (in thousand metric tons)
Turkey	398.1
United States	310.7
Iran	225.0
Italy	145.1
Russia	100.0
Syria	75.0
Spain	72.6
Ukraine	68.2
Romania	65.2
Greece	62.8
World Total	2,083.1
Source: Food and Agriculture Organization of the United Nations

Europe

Major commercial cherry orchards in Europe extend from the Iberian peninsula east to Asia Minor, and to a smaller extent may also be grown in the Baltic States and southern Scandinavia.

United States

In the United States, most sweet cherries are grown in Washington, California, Oregon, and Northern Michigan. Important sweet cherry cultivars include "Bing", "Brooks", "Tulare", "King" and "Rainier". In addition, the Lambert variety is grown on the eastern side of Flathead Lake in northwestern Montana. Both Oregon and Michigan provide light-colored "Royal Ann" ('Napoleon'; alternately "Queen Anne") cherries for the maraschino cherry process. Most sour (also called tart) cherries are grown in Michigan, followed by Utah, New York, and Washington. Additionally, native and non-native cherries grow well in Canada (Ontario and British Columbia). Sour cherries include Nanking and Evans Cherry. Traverse City, Michigan claims to be the "Cherry Capital of the World", hosting a National Cherry Festival and making the world's largest cherry pie. The specific region of Northern Michigan that is known the world over for tart cherry production is referred to as the "Traverse Bay" region. Farms in this region grown many varieties of cherries, sold through companies in the region.

Australia

In Australia, the New South Wales town of Young is famous as the "Cherry Capital of Australia" and hosts the internationally famous National Cherry Festival. Popular varieties include the "Montmorency", "Morello", "North Star", "Early Richmond", "Titans", and "Lamberts". Cherries come in a variety of different colors, like red as well as yellow.

Gallery

• 
• 
• Stella, Prunus avium

See also

• Cherry blossom
• Cherry pitter
• Marasca cherry
• Griotte de Kleparow
• Dried cherry

Notes

1. Herbermann, Charles, ed. (1913). "Pontus" . Catholic Encyclopedia. New York: Robert Appleton Company.
2. A History of the Vegetable Kingdom, Page 334.
3. The curious antiquary John Aubrey (1626–1697) noted in his memoranda: "Cherries were first brought into Kent tempore H. viii, who being in Flanders, and likeing (sic) the Cherries, ordered his Gardener, brought them hence, and propagated them in England." Oliver Lawson Dick, ed. (1949). Aubrey's Brief Lives. Edited from the Original Manuscripts. p. xxxv. {{cite book}}: |author= has generic name (help)
4. "All the cherry gardens and orchards of Kent are said to have been stocked with the Flemish cherry from a plantation of 105 acres in Teynham, made with foreign cherries, pippins, and golden rennets, done by the fruiterer of Henry VIII." (Kent On-line: Teynham Parish)
5. The civic coat of arms of Sittingbourne with the crest of a "cherry tree fructed proper" were only granted in 1949, however.
6. Tall JM, Seeram NP, Zhao C, Nair MG, Meyer RA, Raja SN, JM (2004). "Tart cherry anthocyanins suppress inflammation-induced pain behavior in rat". Behav. Brain Res. 153 (1): 181�"8. doi:10.1016/j.bbr.2003.11.011. ISSN 0166-4328. PMID 15219719. {{cite journal}}: |first2= missing |last2= (help); |first3= missing |last3= (help); |first4= missing |last4= (help); |first5= missing |last5= (help); |first6= missing |last6= (help); Unknown parameter |month= ignored (help); replacement character in |pages= at position 4 (help)CS1 maint: multiple names: authors list (link)
7. "Tart Cherries May Reduce Heart/Diabetes Risk Factors". Newswise, Retrieved on July 7, 2008.
8. United States Food and Drug Administration (2024). "Daily Value on the Nutrition and Supplement Facts Labels". FDA. Archived from the original on 2024-03-27. Retrieved 2024-03-28.
9. National Academies of Sciences, Engineering, and Medicine; Health and Medicine Division; Food and Nutrition Board; Committee to Review the Dietary Reference Intakes for Sodium and Potassium (2019). "Chapter 4: Potassium: Dietary Reference Intakes for Adequacy". In Oria, Maria; Harrison, Meghan; Stallings, Virginia A. (eds.). Dietary Reference Intakes for Sodium and Potassium. The National Academies Collection: Reports funded by National Institutes of Health. Washington, DC: National Academies Press (US). pp. 120–121. doi:10.17226/25353. ISBN 978-0-309-48834-1. PMID 30844154. Retrieved 2024-12-05.
10. "FAOSTAT: ProdSTAT: Crops". Food and Agriculture Organization. 2007. Retrieved 07-02-2009. {{cite web}}: Check date values in: |accessdate= (help)
11. ^ Cherry Production National Agricultural Statistics Service, USDA, Retrieved on August 19, 2008.
12. Sweet Cherries Of Flathead Lake, Retrieved on August 28, 2009

External links

• "Cherry juice hailed as superfood", Daily Mail, 26 September 2008.
• Phenolic compounds in sweet and sour cherries—Cornell University study.

v t e Cherry cultivars
Sweet (Bigaroon, Mazzard)	Bing Black Republican Black Tartarian Chelan Chinook Emperor Francis Ferrovia Germersdorfer Hudson Lambert Lapins Picota Rainier Regina Royal Ann (Napoleon) Sam Schmidt Skeena Stella Sweetheart Tieton Ulster Van
Sour (Amarelle, Morello)	Amarena Balaton Evans Griotte de Kleparow Marasca Montmorency North Star
Other edible	Nanking
Category

• Cherries
• Medicinal plants
• Prunus
• Symbols of Utah
• Plants of temperate climates