March 14, 2011

Happy Pi Day!

Love is like pi – natural, irrational, and very important. -Lisa Hoffman

For those who don’t know, pi, or π, is a mathematical constant which is approximately equal to 3.14. Throughout the history of mathematics, there has been much effort to determine pi more accurately and to understand its nature. The concept of pi has become entrenched in popular culture to a degree far greater than almost any other mathematical construct.

Today is National Pi Day, being the fourteenth day of the third month (3/14) — a day to recognize pi. In celebration, I bring to you some facts about pi which you may not be aware!

  • Pi’s value is the ratio of any circle’s circumference to its diameter.

  • It is also the ratio of a circle’s area to the square of its radius.

  • Pi may be the most important constant in math.
  • Pi is an irrational number, which means that its value cannot be expressed exactly as a fraction.
  • 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. 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.
  • It is, perhaps, the most common ground between mathematicians and non-mathematicians. Reports on the latest, most-precise calculation of pi are common news items.
  • The current record for the decimal expansion of pi, if verified, stands at 5 trillion digits.
  • Elementary applications, such as estimating the circumference of a circle, will rarely require more than a dozen decimal places. For example, the decimal representation of π truncated to 11 decimal places is good enough to estimate the circumference of any circle that fits inside the Earth with an error of less than one millimetre, and the decimal representation of pi truncated to 39 decimal places is sufficient to estimate the circumference of any circle that fits in the observable universe with precision comparable to the radius of a hydrogen atom.
  • The Greek letter π was first adopted for the number as an abbreviation of the Greek word for perimeter “περίμετρος” (or as an abbreviation for “perimeter/diameter”) by William Jones in 1706.
  • The constant is also known as Archimedes’ Constant, after Archimedes of Syracuse who provided an approximation of the number, although this name for the constant is uncommon in modern English-speaking contexts.
  • The Great Pyramid at Giza, constructed c.2589–2566 BC, was built with a perimeter of 1760 cubits and a height of 280 cubits; the ratio 1760/280 ≈ 2π.
  • One open question about pi 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.
  • On November 7, 2005, alternative musician Kate Bush released the album Aerial. The album contains the song “π” whose lyrics consist principally of Bush singing the digits of pi to music, beginning with “3.14”
  • In Carl Sagan‘s novel Contact, pi played a key role in the story. The novel suggested that there was a message buried deep within the digits of π placed there by the creator of the universe. This part of the story was omitted from the film adaptation of the novel.
  • In the Star Trek: The Original Series episode Wolf in the Fold, after a murderous alien entity (which had once been Jack the Ripper) takes over the Enterprise’s main computer with the intention of using it to slowly kill the crew, Kirk and Spock draw the entity out of the computer by forcing it to compute pi to the nonexistent last digit, causing the creature to abandon the computer, allowing it to be beamed into space.
  • The Wheel of Dublin (Ferris wheel) has been nicknamed “the pi in the sky”.
  • It makes beautiful music.
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