γ (gamma) is defined as the limit of (1 + 1/2 + 1/3 + … + 1/n − ln n) as n → ∞. The harmonic series grows without bound, and so does ln n, but their difference converges to exactly γ. It measures the gap between discrete summation and continuous integration.
Whether γ is irrational is one of the oldest open problems in mathematics. Virtually every mathematician believes it is: and probably transcendental: but no proof exists. It has been computed to over 600 billion decimal places.
γ appears throughout analysis, number theory, and physics: in the Gamma function, in the distribution of prime gaps, in Bessel functions, and in the quantum Hall effect.