Pi
From Teach And Discover Wiki
| List of numbers – Irrational numbers ζ(3) – √2 – √3 – √5 – φ – α – e – π – δ | |
| Binary | 11.00100100001111110110… |
| Decimal | 3.14159265358979323846… |
| Hexadecimal | 3.243F6A8885A308D31319… |
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 formulas 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.
Contents |
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.
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.
Definition
In Euclidean plane geometry, π is defined as the ratio of a circle's circumference to its diameter:
Note that the ratio c/d does not depend on the size of the circle. 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. This fact is a consequence of the similarity of all circles.
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:
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. In the 20th century, proofs were found that require no prerequisite knowledge beyond integral calculus. One of those, due to Ivan M. Niven, is widely known. 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.
Numerical value
The numerical value of π truncated to 50 decimal places is:
- 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
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.
Here's a sound clip from an old 80's tune that highlights PI to 500 decimal places (I think... I lost count somewhere)!
