Four-thousand years ago, people discovered that the ratio of the circumference of a circle
to its diameter was about 3. In nature people saw circles, big and small, and they
realized that this ratio was an important tool.

Undoubtedly, Pi is one of the most famous and most remarkable numbers you have ever met.
The number, which is the ratio of circumference of a circle to its diameter, has a long
story about its value. Even nowadays supercomputers are used to try and find its decimal
expansion to as many places as possible.

Pi is one of those numbers that cannot be evaluated exactly as a decimal. It is in
that class of numbers called irrationals.

The Precomputer History of

That the ratio of circumference to diameter is the same (and roughly equal to
3) for all circles has been accepted as "fact" for centuries; at least 4000
years, as far as I can determine. (But knowing *why* this is true, as
well as knowing the *exact* value of this ratio, is another story). The
"easy" history of concerns the ongoing
story of our attempts to improve upon our estimates of . This page
offers a brief survey of a few of the more famous early approximations to .
The value of given in the Rhynd
Papyrus (c. 2000 BC) is
Various Babylonian and Egyptian writings suggest that each of the values
were used (in different circumstances, of course). The Bible (c. 950 BC, 1
Kings 7:23) and the Talmud both (implicitly) give the value simply as 3.
Archimedes of Syracuse (240 BC), using a 96-sided polygon and his method of
exhaustion, showed that
and so his error was no more than
The important feature of Archimedes' accomplishment is not that he was able
to give such an accurate estimate, but rather that his methods could be used to
obtain any number of digits of . In fact, Archimedes' method of
exhaustion would prove to be the basis for nearly all such calculations for
over 1800 years.
Over 700 years later, Tsu Chung-Chih (c. 480) improved upon Archimedes'
estimate by giving the familiar value
which agrees with the actual value of to 6 places.
Many years later, Ludolph van Ceulen (c. 1610) gave an estimate that was
accurate to 34 decimal places using Archimedes' method (based on a -sided polygon).
The digits were later used to adorn his tombstone.
The next era in the history of the extended calculation of was
ushered in by James Gregory (c. 1671), who provided us with the series
Using Gregory's series in conjunction with the identity
John Machin (c. 1706) calculated 100 decimal digits of .
Methods similar to Machin's would remain in vogue for over 200 years.
William Shanks (c. 1807) churned out the first 707 digits of . This feat took Shanks over 15 years -- in other words, he averaged
only about one decimal digit per week! Sadly, only 527 of Shanks' digits were
correct. In fact, Shanks published his calculations 3 times, each time
correcting errors in the previously published digits, and each time new errors
crept in. As it happened, his first set of values proved to be the most
accurate.
In 1844, Johann Dase (a.k.a., Zacharias Dahse), a calculating prodigy (or
"idiot savant") hired by the Hamburg Academy of Sciences on Gauss's
recommendation, computed to 200 decimal places
in less than two months.
In the era of the desktop calculator (and the early calculators truly
required an entire desktop!), D. F. Ferguson (c. 1947) raised the total to 808
(accurate) decimal digits. In fact, it was Ferguson who discovered the errors in
Shanks' calculations.
Today, of course, in the era of the supercomputer, hundreds of millions of
digits are known. The evolution of the machine-assisted approximations to is summarized on.

Click here for more detail about
Postcomputer History