On cars, old, new and future; science & technology; vintage airplanes, computer flight simulation of them; Sherlockiana; our English language; travel; and other stuff
TODAY, MARCH 14th, at 53 seconds past 9:26 a.m., can be represented 3/14/15 9:26:53, which displays the first ten digits of the mathematical constant π, 3.14159265358979323846…
Also, given that I have a feline friend named Πwacket, there’s all the more reason to celebrate the lore of this fascinating number. Among its virtues, π is the ratio of a circle’s circumference to its diameter, it’s an irrational member of the real number line and it’s transcendental.
These three facts are quite enough to fill a book (Piece of Pi: Wit-Sharpening, Brain-bruising, Number-Crunching Activities with Pi (Grades 6-8); The Number Pi for an advanced look; and Pi: A Biography of the World’s Most Mysterious Number for those in between). Today I offer favorite tidbits of π. Yum.
It’s useful, of course, to be able to compute the distance around a circle, its circumference, knowing only its diameter or, half of this, its radius. Ancient Babylonians and Egyptians had values within 1 percent of π; 25/8 = 3.1250 for the Babylonians; (16/9)2 = 3.1605 for the Egyptians, both around 1900 – 1600 B.C.
Around 250 B.C., the Greek mathematician Archimedes improved estimates of π by computing the perimeters of inscribed and circumscribed polygons of increasingly many sides. He proved the value lay between 223/71 (3.1408) and 22/7 (3.1429).
In the 16th and 17th centuries, the concept of infinite series led to increasingly fine approximations. Isaac Newton used this technique to compute π to 15 digits.
The challenge of π’s decimal representation lies in its irrational nature. Here, this doesn’t mean “illogical” or “unreasonable.” Rather, it’s a mathematical term: A number is rational if it can be represented as a quotient, A/B, where both A and B are whole numbers. A number is irrational if no such rational representation exists. That is, 22/7 is rational. And (a much more challenging thing to show), π is not.
Any rational number, A/B, has a decimal representation that eventually repeats digits. Some of these repeated digits may be zeros: 1/2 = 0.500…. Other ratonals repeat single non-zero digits: 5/3 = 1.666…. Still others repeat blocks of digits: 2/7 = 0.285714285714….; 5/7 = 0.714285714285….
Guess the length of 3/11’s repeating blocks.
Because π is not rational, its decimal representation never repeats. In fact, number theorists ask questions such as “does the sequence 1234567890 ever appear in π’s decimal representation?” This is currently unknown.
Iterative methods with computers of increasingly higher power have led to decimal representations of π to quadrillions of digits. Such endeavors are more than examples of human compulsiveness. They test supercomputers, numerical algorithms and searches for pseudorandom numbers (all three of which have applications in cryptology—and hence computer security).
The number π displays another mathematical curiosity: It is not algebraic; in a few words, it’s never the solution of an equation. More precisely, not a root of any non-zero polynomial equation with rational coefficients, something of the form anxn + an-1xn-1 + ··· a1 x = 0, where the a’s are all rational numbers.
Such non-algebraic numbers are called transcendental. It’s known that there are uncountably many of these transcendental numbers, though most people know of only one, π. To confirm a number’s transcendental nature, you apply the Lindemann-Weierstrass Theorem. (Or take a mathematician’s word for it.)
Having mentioned Lindemann, I conclude with a bizarre example of simplifying π by legislation, the Indiana Pi Bill of 1897. (There are apocryphal tales of other governments fooling with π, but this story appears the most robust.)
An amateur mathematician in Indiana devised several crackpot claims, one of which was “squaring the circle,” solving a geometry problem dating back to the ancients. He finessed these claims into the 1897 Indiana General Assembly’s Bill #246. An implication of his claims was that π = 3.2, hence the name Indiana Pi Bill.
Legislative history of bill #248 was a blend of high comedy, misunderstanding and ignorance. The Indiana House of Representatives sent the bill to its Committee on Education, which reported favorably; a unanimously favorable House vote followed.
By chance, Purdue mathematics professor C.A. Waldo was in town and he saved Indiana’s upper house from embarrassment.
In 1882, German mathematician Ferdinand von Lindemann had already proved the transcendental nature of π, one implication being proof of the impossibility of squaring the circle. On Professor Waldo’s counsel, the Indiana Senate voted to postpone the bill indefinitely.
Happy π-day! ds
© Dennis Simanaitis, SimanaitisSays.com, 2015