Department of Mathematics - University of Utah

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A Remarkable Formula

The complete formula at the upper left reads like this:

It features four basic constants: e, pi, i, which is the square root of -1, and -1 itself. Pi is the oldest of the constants for which we have historical references. Both the Babylonians and the Egyptians, ca. 1500 BC, used 3 as an approximate values, although there a few documents in which 3 1/8 appears. Archimedes, born 287 BC, showed in in his article "On the Measurement of a Circle" that

3 10/71 < pi < 3 1/7.

Archimedes' method could, of course, easily produce much more precise inequalities. Not much progress was made, however, until late the 16th century, when François Viète showed that

3.1415926535 < pi < 3.1415926537 .

Subsequently Newton, Gauss, and Euler made substantial contributions, all introducing new methods.

The numbers -1, i, and e were all discovered much later. Even late in the sixteenth century equations such as x^3 + x = 2 and x^3 = 2 + x were considered different enough to deserve separate treatment: certain proof that negative numbers were not an accepted notion. Curiously, it was the difficulties encountered in solving such cubic equations that lead to the discovery of imaginary numbers.

The first published reference to e is aparently in Leonhard Euler's Mechanica, published in 1727. Euler knew that e is described by the limit formula which gives continuously compounded interest, and that it is given by the infinite series

1 + 1/1! + 1/2! + 1/3! + 1/4! + ...

where 4! = 1x2x3x4, etc. From this it follows that e is approximately 2.7128, and it is using infinite series that Euler proved the remarkable formula above.


  1. Petr Beckman: A History of Pi --- offbeat, entertaining, and informative.

  2. Willam Dunham: Journey through Genius --- a short, brilliantly written history of mathematics. See chapter 6 for comments on negative and imaginary numbers.

  3. Eli Maor: e: The Story of a Number --- see Chapter 13.

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Department of Mathematics
University of Utah
155 South 1400 East, JWB 233
Salt Lake City, Utah 84112-0090
Tel: 801 581 6851, Fax: 801 581 4148

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