The Number We Can't Prove Is A Fraction Or Not

Описание: Add up the simplest fractions there are—one, plus one half, plus one third, plus one quarter, on and on. This is the harmonic series, and it climbs forever, but it climbs slowly, shadowing the smooth curve of the natural logarithm the whole way. The two rise together out to infinity, and the space between them locks onto a single fixed number: 0.5772156649..., the Euler-Mascheroni constant, gamma.

Euler found it around 1734; Mascheroni pushed the digits further in 1790. Today we know gamma to trillions of decimal places, and it turns up all over mathematics—in the Riemann zeta function near s = 1, and as the slope of the Gamma function at 1. And yet, nearly three hundred years on, nobody has been able to prove the most basic thing about it: no one knows whether gamma is even irrational. If it were a fraction, its denominator would have to exceed 10^242080. e fell to Hermite in 1873, pi to Lindemann in 1882—gamma, sitting in the simplest sum in all of mathematics, still refuses to say whether it is a fraction.

Chapters:
0:00 The sum that climbs forever
1:39 Staircase over a curve
3:09 The gap that refuses to move
4:26 A number with a name
5:37 Where it actually lives
7:29 It keeps turning up
8:45 We know it to the ends of the earth
9:48 The question nobody can answer
11:04 Sitting in plain sight

Music by Vincent Rubinetti
Download the music on Bandcamp:
https://vincerubinetti.bandcamp.com/a...