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 Did you know . . .?

 

 

Pi (symbol)

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 (pi=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):

formula:  [(8d)/9] squared

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 < pi < 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!

 


Circle with inscribed and circumscribed hexagons.

Archimedes's method for approximating the value of pi.

(Source: http://www.math.psu.edu/dna/graphics.html#archimedes)

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:

 

pi/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 pi/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:

 

Machin's formula:  pi/4=4arctan(1/5)-arctan(1/239)

 

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/pi. 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:

3.14159 =pi

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:

http://www-history.mcs.st-and.ac.uk/~history/HistTopics/Pi_through_the_ages.html

http://www.cs.unb.ca/~alopez-o/math-faq/mathtext/node12.html

http://www.cecm.sfu.ca/personal/jborwein/Kanada_50b.html

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibpi.html

http://ic.net/~jnbohr/java/Machin.html

http://www.mste.uiuc.edu/reese/buffon/buffon.html

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Shanks.html

http://www.escape.com/~paulg53/math/pi/greg/

 


By Laura Smoller, Department of History,

University of Arkansas at Little Rock.

February 2001.

N.B.:  This site is no longer maintained.