|Did you know . . .?|
has a long and interesting history!
(Ancient history--More pi history--A novel way to compute pi--The symbol for pi--Having fun with pi--For more information)
That the ratio of the circumference to the diameter of a circle is constant (namely, pi) has been recognized for as long as we have written records.
A ratio of 3:1 appears in the following biblical verse:
And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about. (I Kings 7, 23; II Chronicles 4, 2.)
The ancient Babylonians generally calculated the area of a circle by taking 3 times the square of its radius (=3), but one Old Babylonian tablet (from ca. 1900-1680 BCE) indicates a value of 3.125 for pi.
Ancient Egyptians calculated the area of a circle by the following formula (where d is the diameter of the circle):
This yields an approximate value of 3.1605 for pi.
The first theoretical calculation of a value of pi was that of Archimedes of Syracuse (287-212 BCE), one of the most brilliant mathematicians of the ancient world. Archimedes worked out that 223/71 < < 22/7. Archimedes's results rested upon approximating the area of a circle based on the area of a regular polygon inscribed within the circle and the area of a regular polygon within which the circle was circumscribed.
Beginning with a hexagon, he worked all the way up to a ploygon with 96 sides!
Archimedes's method for approximating the value of pi.
The approximate area of the circle lies between the areas of the circumscribed and the inscribed hexagons.
More pi history:
European mathematicians in the early modern period developed new arithmetical formulae to approximate the value of pi, such as that of James Gregory (1638-1675), which was taken up by Leibniz:
/4 = 1 - 1/3 + 1/5 - 1/7 + . . . . . . . . . . .
One problem with using this formula to calculate the value of pi is that you would have to add 5 million terms to work out a value of /4 that extends to 6 or 7 decimal places!
In 1706, another mathematician named John Machin developed a refinement on Gregory's formula, yielding the formula still used today by computer programmers to compute pi:
Using this formula, an Englishman named William Shanks calculated pi to 707 places, a labor of many years, which he published in 1873. (Only 527 places were correct, however!)
A novel way to compute pi:
An eighteenth-century French mathematician named Georges Buffon devised a way to calculate pi based on probability. Buffon's method begins with a uniform grid of parallel lines, a unit distance apart. If you drop a needle of length k < 1 on the grid, the probability that the needle falls across a line is 2k/. Various people have tried to calculate pi by throwing needles. Depending on when you stop the experiment, you can obtain a reasonably accurate estimate of pi.
You can try Buffon's needle experiment for yourself (virtually) at http://www.angelfire.com/wa/hurben/buff.html
Some ants apparently actually use this algorithm to measure the size of potential nest sites!
The symbol for pi:
was introduced by the British mathematician William Jones in 1706, who wrote:
This symbol was adopted by Euler in 1737 and became the standard symbol for pi.
Having fun with pi:
Some people are just crazy about pi!
There are pi poems . . .
There are pieces of music based on the digits of pi . . .
There is a web site where you can find your birthday in pi . . .
There are people who have memorized 1000+ digits of pi . . .
In 1897 the Indiana Legislature tried to legally establish the value of pi . . .
Pi has earned a spot in "The Useless Pages" . . .
For more information:
By Laura Smoller, Department of History,
University of Arkansas at Little Rock.
N.B.: This site is no longer maintained.