If you had to pick the most famous number in the world, you would probably go for pi, right? But why? Despite being crucial to our understanding of circles, it's not a particularly easy number to work with, because it's literally impossible to know its exact value, and with no discernible pattern to its digits, we could continue calculating each digit of pi to infinity.
But in spite of its unwieldy nature, pi has earned its fame by popping up everywhere, in both nature and maths - and even in places that have no clear connection to the circles. And it's not the only number that has a rather eerie ubiquitiousness - for some reason, 0.577 keeps cropping up everywhere too.
Euler's constant, also called the Euler-Mascheroni constant, is defined as the limiting difference between the natural logarithm the harmonic series. (The harmonic series is the infinite series of numbers you get if you start with 1 + 1/2 + 1/3 +1/4... and continue on in that pattern indefinitely.) The difference between these two values is approaches a finite number called Euler's constant, and it's equal to about 0.577.
This is pretty interesting by itself. But Euler's constant shows up pretty much everywhere. It's in all sorts of mathematical equations, like integrals and transforms. It shows up in all kinds of problems in physics, including several quantum mechanics equations. It's even in the equations used to find the Higgs boson.
The purple curve is the graph of the natural log function. The Blue bars are the values of the harmonic series. The difference between them is Euler's constant. (William Demchick)
Despite its usefulness, Euler's constant is still pretty much a mystery. Mathematicians can't say for sure whether it's a rational number or not. (Remember, a rational number can be expressed as a fraction, while an irrational number like pi has digits that go on forever). Even though Euler's constant has been calculated to more than a trillion digits, its mystery endures.
Take a few minutes and let Numberphile explain all this for you in the video below. (And soak in the mind-blowing idea about the ant around the 5-minute mark.)