6931471805599453

6931471805599453 is the sequence of digits of the natural logarithm of 2 with the decimal point removed: 0.6931471805599453... In mathematics, this value is ln 2, the natural logarithm of the base e evaluated at 2.

ln 2 is irrational and transcendental, meaning it cannot be expressed as a simple fraction and it is not a root of any nonzero polynomial with integer coefficients. Its non-repeating decimal expansion and its transcendence are well established consequences of transcendence theory.

Approximate value and appearances: ln 2 ≈ 0.6931471805599453, with the digits continuing indefinitely. This constant arises naturally in various areas of science and mathematics and is often encountered when converting between natural units (nats) and information units (bits).

Representations and identities: ln 2 can be written as the alternating harmonic series ln 2 = 1 − 1/2 + 1/3 − 1/4 + 1/5 − ..., which is a special case of the series for ln(1+x) with x = 1. It also equals the integral ∫ from 1 to 2 of (1/x) dx. These forms illustrate its role as the area under 1/x between 1 and 2 and as a natural log value arising from exponential growth.

Applications and context: The constant links natural logarithms to logarithms base 2, and in information theory, one nat equals roughly 1/ln 2 bits. The string 6931471805599453 is often cited as a curiosity reflecting the initial digits of ln 2.