What is Pi?
- Webster's Collegiate Dictionary defines π as "1: the 16th letter of the Greek alphabet... 2 a: the symbol pi denoting the ratio of the circumference of a circle to its diameter b: the ratio itself: a transcendental number having a value to eight decimal places of 3.14159265" A number can be placed into several categories based on its properties. Is it prime or composite? Is it imaginary or real? Is it transcendental or algebraic? These questions help define a number's behavior in different situations. In order to understand where π fits in to the world of mathematics, one must understand several of its properties: π is irrational and π is transcendental.
- π (sometimes written pi) is a mathematical constant whose value is the ratio of any circle's circumference to its diameter; this is the same value as the ratio of a circle's area to the square of its radius. π is approximately equal to 3.14159 in the usual decimal positional notation. Many formulae from mathematics, science, and engineering involve π, which makes it one of the most important mathematical constants.
- 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. π 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.
- Probably because of the simplicity of its definition, the concept of π has 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 are common news items. The current record for the decimal expansion of π, if verified, stands at 5 trillion digits.
The Greek letter
The Latin name of the Greek letter π is pi. When referring to the constant, the symbol π is pronounced like the English word "pie", which is also the conventional English pronunciation of the Greek letter. The constant is named "π" because "π" is the first letter of the Greek word περιφέρεια "periphery" (or perhaps περίμετρος "perimeter", referring to the ratio of the perimeter to the diameter, which is constant for all circles). William Jones was the first to use the Greek letter in this way, in 1706, and it was later popularized by Leonhard Euler in 1737. William Jones 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 ... 3.14159, etc. = π ...
When used as a symbol for the mathematical constant, the Greek letter (π) is not capitalized at the beginning of a sentence. The capital letter Π (Pi) has a completely different mathematical meaning; it is used for expressing the product of a sequence.
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 ... 3.14159, etc. = π ...
When used as a symbol for the mathematical constant, the Greek letter (π) is not capitalized at the beginning of a sentence. The capital letter Π (Pi) has a completely different mathematical meaning; it is used for expressing the product of a sequence.
Geometric definition
When a circle's diameter is 1, its circumference is π.
In Euclidean plane geometry, π is defined as the ratio of a circle's circumference C 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 defined as the ratio of a circle's area A to the area of a square whose side is equal to the radius r of the circle.
These definitions depend on results of Euclidean geometry, such as the fact that all circles are similar, and the fact that the right-hand-sides of these two equations are equal to each other (i.e. the area of a disk is Cr/2). These two geometric definitions 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 the trigonometric function cos(x) equals zero.
In Euclidean plane geometry, π is defined as the ratio of a circle's circumference C 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 defined as the ratio of a circle's area A to the area of a square whose side is equal to the radius r of the circle.
These definitions depend on results of Euclidean geometry, such as the fact that all circles are similar, and the fact that the right-hand-sides of these two equations are equal to each other (i.e. the area of a disk is Cr/2). These two geometric definitions 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 the trigonometric function cos(x) equals zero.