π (sometimes written
pi) 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.
Fundamentals
Lower-case π is used to symbolize the
constant.
The letter π
Circumference = π × diameter
The 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 , "pie" in
English, which is the conventional
English pronunciation of the Greek letter. In Greek, the name of
this letter is .
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 square
In
Euclidean plane geometry, π is
defined as the
ratio of a
circle's
circumference
to its
diameter:
- \pi = \frac{C}{d}.
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:
- \pi = \frac{A}{r^2}.
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
π is an
irrational number, meaning
that it cannot be written as the
ratio
of two integers. The belief in the irrationality of π is
mentioned by
Muhammad ibn
Mūsā al-Khwārizmī in the 9th century.
Maimonides also mentions with certainty the
irrationality of π in the 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.
π is also a
transcendental
number, meaning that there is no
polynomial with
rational coefficients for which π is a
root. This was proved by
Ferdinand von Lindemann in 1882. 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
The decimal representation 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 in OEIS for more
digits.
While the decimal representation of π has been computed to more
than a
trillion
(10
^{12}) 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
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,
P_{n} , of
a
regular polygon with
n
sides
circumscribed around a circle
with diameter
d. Then
- \pi = \lim_{n \to \infty}\frac{P_{n}}{d}.
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: π .
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:
- \pi = 4\sum^\infty_{k=0} \frac{(-1)^k}{2k+1} =
\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\frac{4}{11}\cdots.\!
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
- \pi_{0,1} = \frac{4}{1},\ \pi_{0,2} =\frac{4}{1}-\frac{4}{3},\
\pi_{0,3} =\frac{4}{1}-\frac{4}{3}+\frac{4}{5},\ \pi_{0,4}
=\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}, \cdots\!
and then define
- \pi_{i,j} = \frac{\pi_{i-1,j}+\pi_{i-1,j+1}}{2}\text{ for all
}i,j\ge 1
then computing \pi_{10,10} will take similar computation time to
computing 150 terms of the original series in a brute-force manner,
and \pi_{10,10}=3.141592653\ldots, correct to 9 decimal places.
This computation is an example of the
van Wijngaarden
transformation.
History
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 ≈ 2π. 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 algorithm
Archimedes (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 <&NBSP;π&NBSP;<&NBSP;3&NBSP;+&NBSP;1></&NBSP;π&NBSP;<&NBSP;3&NBSP;+&NBSP;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:
- \pi \approx A_{3072} = 3 \cdot 2^{8} \cdot
\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+1}}}}}}}}}
\approx 3.14159
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 <&NBSP;π&NBSP;<&NBSP;3.1415927
using="" Liu="" Hui's="" algorithm="" applied="" to="" a=""
12288-gon.="" This="" value="" was="" the="" most="" accurate=""
approximation="" of="" π="" available="" for="" next="" 900=""
years.=""></&NBSP;π&NBSP;<&NBSP;3.1415927>
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:
- {\pi} = 4\sum^\infty_{k=0} \frac{(-1)^k}{2k+1} = \frac{4}{1} -
\frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \cdots\!
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
- \pi = \sqrt{12}\sum^\infty_{k=0} \frac{(-3)^{-k}}{2k+1} =
\sqrt{12}\sum^\infty_{k=0} \frac{(-\frac{1}{3})^k}{2k+1} =
\sqrt{12}\left(1-{1\over 3\cdot3}+{1\over5\cdot 3^2}-{1\over7\cdot
3^3}+\cdots\right)
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,
- \frac2\pi = \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot
\frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdot \cdots\!
found by
François Viète in
1593. Another famous result is
Wallis'
product,
- \frac{\pi}{2} = \prod^\infty_{k=1} \frac{(2k)^2}{(2k)^2-1} =
\frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5}
\cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot
\frac{8}{9} \cdots\ = \frac{4}{3} \cdot \frac{16}{15} \cdot
\frac{36}{35} \cdot \frac{64}{63} \cdots\!
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
- \frac{\pi}{4} = 4 \, \arctan \frac{1}{5} - \arctan
\frac{1}{239}\!
with
- \arctan \, x = \sum^\infty_{k=0} \frac{(-1)^k x^{2k+1}}{2k+1} =
x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots\!
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 π
^{2} to be irrational. When
Leonhard Euler in 1735 solved the famous
Basel problem – finding the exact
value of
- \sum^\infty_{k=1} \frac{1}{k^2} = \frac{1}{1^2} + \frac{1}{2^2}
+ \frac{1}{3^2} + \frac{1}{4^2} + \cdots\!
which is π
^{2}/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(16/5^{3} − 4/239^{3}) + ... = 3.14159... = π}}
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,
- \frac{1}{\pi} = \frac{2 \sqrt 2}{9801} \sum_{k=0}^\infty
\frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!
and the related one found by the
Chudnovsky brothers in 1987,
- \frac{426880 \sqrt{10005}}{\pi} = \sum_{k=0}^\infty \frac{(6k)!
(13591409 + 545140134k)}{(3k)!(k!)^3 (-640320)^{3k}}\!
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
- a_0 = 1 \quad \quad \quad b_0 = \frac{1}{\sqrt 2} \quad \quad
\quad t_0 = \frac{1}{4} \quad \quad \quad p_0 = 1\!
and iterating
- a_{n+1} = \frac{a_n+b_n}{2} \quad \quad \quad b_{n+1} =
\sqrt{a_n b_n}\!
- t_{n+1} = t_n - p_n (a_n-a_{n+1})^2 \quad \quad \quad p_{n+1} =
2 p_n\!
until
a_{n} and
b_{n} are close
enough. Then the estimate for π is given by
- \pi \approx \frac{(a_n + b_n)^2}{4 t_n}.\!
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,
- \pi = \sum_{k=0}^\infty \frac{1}{16^k} \left( \frac{4}{8k + 1}
- \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k +
6}\right),
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
- \pi = \sum_{k=0}^\infty \frac{1}{b^{ck}}
\frac{p(k)}{q(k)},
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
b^{c} 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 =
e^{π}
(Gelfond's constant), then
- \frac{\pi}{24} = \sum_{n=1}^\infty \frac{1}{n}
\left(\frac{3}{q^n-1} - \frac{4}{q^{2n}-1} +
\frac{1}{q^{4n}-1}\right)
- \frac{\pi^3}{180} = \sum_{n=1}^\infty \frac{1}{n^3}
\left(\frac{4}{q^n-1} - \frac{5}{q^{2n}-1} +
\frac{1}{q^{4n}-1}\right)
and others of form,
- \pi^k = \sum_{n=1}^\infty \frac{1}{n^k} \left(\frac{a}{q^n-1} +
\frac{b}{q^{2n}-1} + \frac{c}{q^{4n}-1}\right)
where
k is an
odd number, and
a,
b,
c are
rational numbers.
In the previous formula, if
k is of the form
4
m + 3, then the formula has the particularly
simple form,
- p\pi^k = \sum_{n=1}^\infty \frac{1}{n^k}
\left(\frac{2^{k-1}}{q^n-1} - \frac{2^{k-1}+1}{q^{2n}-1} +
\frac{1}{q^{4n}-1}\right)
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:\pi=[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,1,84,\cdots]
- :or
{\pi=3+{}}\cfrac{1}{7+\cfrac{1}{15+\cfrac{1}{1+\cfrac{1}{292+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{3+\cfrac{1}{1+\cfrac{1}{14+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}}}}}}}}}}}}}
However, there are
generalized continued
fractions for π with a perfectly regular structure, such
as:
\pi=\cfrac{4}{1+\cfrac{1^2}{2+\cfrac{3^2}{2+\cfrac{5^2}{2+\cfrac{7^2}{2+\cfrac{9^2}{2+\ddots}}}}}}=3+\cfrac{1^2}{6+\cfrac{3^2}{6+\cfrac{5^2}{6+\cfrac{7^2}{6+\cfrac{9^2}{6+\ddots}}}}}=\cfrac{4}{1+\cfrac{1^2}{3+\cfrac{2^2}{5+\cfrac{3^2}{7+\cfrac{4^2}{9+\ddots}}}}}
Memorizing digits
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.
However, this claim is doubted by some skeptics. .
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
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 . 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,
e^{π},
Γ(1/4)} in 1996.
Use in mathematics and science
π is ubiquitous in mathematics, appearing even in places that lack
an obvious connection to the circles of Euclidean geometry.
Geometry and trigonometry
For any circle with radius
r and diameter
d =
2
r, the circumference is π
d and the area is
π
r^{2}. 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:
- \int_{-1}^1 \sqrt{1-x^2}\,dx = \frac{\pi}{2}
and
- \int_{-1}^1\frac{1}{\sqrt{1-x^2}}\,dx = \pi
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 + 2π
n) and cos(
x) = cos(
x
+ 2π
n). Because sin(0) = 0, sin(2π
n) = 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
A
complex number z can be expressed
in
polar
coordinates as follows:
- z = r\cdot(\cos\varphi + i\sin\varphi)
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
- e^{i\varphi} = \cos \varphi + i\sin \varphi \!
where
i is the
imaginary
unit satisfying
i^{2} = −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
- e^{i \pi} = -1.\!
e^{i \pi} + 1 = 0.\!
Euler's
identity is famous for linking several basic mathematical
constants and operators.
There are
n different
n-th
roots of unity
- e^{2 \pi i k/n} \qquad (k = 0, 1, 2, \dots, n - 1).
The
Gaussian integral
- \int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi}.
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.
- :\Lambda = {{8\pi G} \over {3c^2}} \rho
- : \Delta x\, \Delta p \ge \frac{h}{4\pi}
- : R_{ik} - {g_{ik} R \over 2} + \Lambda g_{ik} = {8 \pi G \over
c^4} T_{ik}
- : F = \frac{\left|q_1q_2\right|}{4 \pi \varepsilon_0 r^2}
- : \mu_0 = 4 \pi \cdot 10^{-7}\,\mathrm{N/A^2}\,
- :\frac{P^2}{a^3}={(2\pi)^2 \over G (M+m)}
Probability and statistics
In
probability and
statistics, there are many
distributions whose formulas
contain π, including:
- f(x) = {1 \over \sigma\sqrt{2\pi} }\,e^{-(x-\mu
)^2/(2\sigma^2)}
- f(x) = \frac{1}{\pi (1 + x^2)}.
Note that since \int_{-\infty}^{\infty} f(x)\,dx = 1 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:
- \pi \approx \frac{2nL}{xS}.
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.
In popular culture
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!"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
References
- : Continued fraction for Pi, On-Line Encyclopedia
of Integer Sequences
- Glimpses in the history of a great number: Pi in
Arabic mathematicsby Mustafa Mawaldi
- Commentary to Mishna,
beginning of Eruvin
- : Decimal expansion of Pi, On-Line Encyclopedia
of Integer Sequences
- A. van Wijngaarden, in: Cursus: Wetenschappelijk Rekenen B,
Process Analyse, Stichting Mathematisch Centrum, (Amsterdam, 1965)
pp. 51–60.
- Petrie, W.M.F. 1940 Wisdom of the Egyptians: pp 27
- Edwards. I.E.S. 1979: The Pyramids of Egypt: pp269
- Jackson and Stamp (2002) Pyramid: Beyond Imagination.
pp153
- Lightbody, D.I. 2008. Egyptian Tomb Architecture: The
Archaeological Facts of Pharaonic Circular Symbolism. British
Archaeological Reports. pp36
- The New York Times: Even Mathematicians Can Get
Carried Away
- "An {ENIAC} Determination of pi and e to more than 2000 Decimal
Places", Mathematical Tables and Other Aids to Computation, 4 (29),
pp. 11–15. (January,1950)
- "Statistical Treatment of Values of First 2,000 Decimal Digits
of e and of pi Calculated on the ENIAC", Mathematical Tables and
Other Aids to Computation, 4 (30), pp. 109–111. (April,1950)
- http://news.cnet.com/8301-17938_105-10313808-1.html
- Weisstein, Eric W. "Pi
Wordplay." From MathWorld--A Wolfram Web Resource. Retrieved on
2009-03-12.
- Raz A, Packard MG, Alexander GM, Buhle JT, Zhu H, Yu S,
Peterson BS. (2009). A slice of pi : An exploratory neuroimaging
study of digit encoding and retrieval in a superior memorist.
Neurocase. 6:1-12. PMID 19585350
- See, e.g, Lennart Berggren, Jonathan M. Borwein, and
Peter B. Borwein (eds.), Pi: A Source Book. Springer, 1999
(2nd ed.). ISBN 978-0-387-98946-4.
- See Alfred S. Posamentier and Ingmar Lehmann, Pi: A
Biography of the World's Most Mysterious Number. Prometheus
Books, 2004. ISBN 978-1-59102-200-8.
- E.g., MSNBC, Man recites
pi from memory to 83,431 places July 3, 2005; Matt Schudel,
Obituaries: "John W. Wrench, Jr.: Mathematician Had a Taste for Pi"
The Washington Post, March 25, 2009, p. B5.
- Pi Day activities.
- MIT, E to the U.
- http://news.bbc.co.uk/1/hi/magazine/7296224.stm
External links