# Dictionary Definition

pi

### Noun

1 the ratio of the circumference to the diameter
of a circle [syn: 3.14159265358979323846...]

2 someone who can be employed as a detective to
collect information [syn: private
detective, private eye,
private
investigator, operative, shamus, sherlock]

3 the scientist in charge of an experiment or
research project [syn: principal
investigator]

4 the 16th letter of the Greek alphabet

5 an antiviral drug used against HIV; interrupts
HIV replication by binding and blocking HIV protease; often used in
combination with other drugs [syn: protease
inhibitor]

# Extensive Definition

Pi or π is one of the most important mathematical
constants, approximately equal to 3.14159. It represents the
ratio of any circle's
circumference to its diameter in Euclidean
geometry, which is the same as the ratio of a circle's area to
the square of its radius. Many formulae from mathematics, science, and engineering involve π.

It is an irrational
number, which means that it cannot be expressed as a fraction m/n, where m and n are
integers. Consequently
its decimal
representation never ends or repeats. Beyond being irrational,
it is a transcendental
number, which means that no finite sequence of algebraic
operations on integers
(powers, roots, sums, etc.) could ever produce it. Throughout the
history of mathematics, much effort has been made to determine π
more accurately and understand its nature; fascination with the
number has even carried over into culture at large.

The Greek letter π, often spelled out pi in text,
was adopted for the number from the Greek word for perimeter
"περίμετρος", probably by
William Jones in 1706, and popularized by Leonhard
Euler some years later. 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.

## Fundamentals

### The letter π

The name of the Greek letter π is pi, and this spelling is used in typographical contexts where the Greek letter is not available or where its usage could be problematic. When referring to this constant, the symbol π is always pronounced like "pie" in English, the conventional English pronunciation of the letter. In Greek, the name of this letter is pronounced /pi/.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

In Euclidean plane geometry, π is defined as the ratio of a circle's circumference to its diameter:- \pi = \frac.

The constant π may be defined in other ways that
avoid the concepts of arc length
and area, for example, as twice the smallest positive x for which
cos(x) = 0.
The formulas below illustrate other (equivalent) definitions.

### Irrationality and transcendence

The constant π is an irrational number; that is, it cannot be written as the ratio of two integers. This was proven in 1761 by Johann Heinrich Lambert. A somewhat earlier similar proof is by Mary Cartwright.Furthermore, π is also transcendental,
as was proven 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.

- 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 pi 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
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 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, due to Archimedes, is to calculate the perimeter, Pn , of a regular polygon with n sides circumscribed around a circle with diameter d. Then- \pi = \lim_\frac

i.e., the more sides the polygon has, the closer
the approximation.

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.

π can also be calculated using purely
mathematical methods. Most formulas 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 = \frac-\frac+\frac-\frac+\frac-\frac\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 300 terms are not sufficient 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_ = \frac, \pi_ =\frac-\frac, \pi_
=\frac-\frac+\frac, \pi_ =\frac-\frac+\frac-\frac, \cdots\!

and then define

\pi_ = \frac for all i,j\ge 1

then computing \pi_ will take similar computation
time to computing 150 terms of the original series in a brute force
manner, and \pi_=3.141592653\cdots, correct to 9 decimal places.
This computation is an example of the
Van Wijngaarden transformation.

## History

The history of π 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 approximations date
from around 1900 BC; they are 25/8 (Babylonia) and 256/81 (Egypt),
both within 1% of the true value. as some believe the ratio of 3:1
is of an exterior circumference to an interior diameter of a thinly
walled basin, which could indeed be an accurate ratio, depending on
the thickness of the walls. See:
Biblical value of Pi.

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:

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 = \frac2 \cdot \frac2 \cdot \frac2 \cdot \cdots\!

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

- \frac = \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdots\!

written down 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 = 4 \, \arctan \frac - \arctan \frac\!

with

- \arctan \, x = x - \frac + \frac - \frac + \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. 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 proved in 1794 that also π2 is irrational. When
Leonhard
Euler in 1735 solved the famous Basel
problem – finding the exact value of

- \frac + \frac + \frac + \frac + \cdots\!

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

William Jones' book A New Introduction to Mathematics from
1706 is cited
as the first text where the Greek letter
π was used for this constant, but this notation became
particularly popular after Leonhard
Euler adopted it in 1737. He wrote:

# Synonyms, Antonyms and Related Words

ascender, back, ball up, bastard type,
beard, belly, bevel, black letter, body, bollix up, cap, capital, case, compose, confound, confuse, counter, descender, em, en, face, fat-faced type, feet, font, foul up, fumble, garble, groove, impose, italic, jumble, justify, letter, ligature, logotype, lower case, majuscule, make up, minuscule, mix up, muck up,
muddle, nick, overrun, pi a form, pica, play hob with, point, print, riffle, roman, sans serif, scramble, screw up, script, set, set in print, shank, shoulder, shuffle, small cap, small
capital, snafu, snarl up,
stamp, stem, tumble, type, type body, type class, type
lice, typecase,
typeface, typefounders, typefoundry, upper
case