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Above, we defined as the particular real number satisfying

which gave us when . From this expression, we have, as ,

or

This is one way to define . Another way to arrive at the same definition is to ask what logarithmic base gives that the derivative of is . We denote by .

Numerically, is a transcendental number (a type of irrational number^3.5), so its decimal expansion never repeats. The initial decimal expansion of is given by^3.6

Any number of digits can be computed from the formula by making sufficiently small.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.