Despite some confusion, Euler's constant is not the same as Euler's number,
a.k.a. e.
Euler's constant, commonly symbolized by the lower-case gamma (which you might see here, if your browser and OS display "symbol" font:
g)
was described by mathematician Leonhard Euler.
It is sometimes called instead the Euler-Mascheroni constant.
It is thought to be an irrational (transcendental) number, but mathematicians don't know for sure (yet).
The value of Euler's constant is approximately 0.57721566490153286061.
Mathematically, it is the limit of
(1 + 1/2 + 1/3 + 1/4 + 1/5 + ... + 1/n) - ln n
as
n goes to
infinity.
The series being summed, {1; 1/2; 1/3; 1/4; 1/5; ... }, is called the harmonic series. It is interesting because the series does not converge as n becomes arbitrarily large, even though the difference between consecutive harmonic numbers becomes arbitrarily small. In other words, the sum approaches infinity, but barely, and in no great hurry.
Calculating the digits of Euler's constant is a problem that has occupied Donald Knuth, among others. It's interesting to computer scientists because analysis of algorithms often involves the sum of harmonic series, and so Euler's constant often pops up. It occurs in study and use of the Gamma function, Bessel functions, and Mersenne primes, among other places.
I was a math major, long ago, but I may have erred here. If you think so, /msg me, please.