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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.

Known as Euler's constant, or the Euler-Mascheroni
constant, this number is defined as the limiting difference between two classic
mathematical sequences: the natural logarithm and the harmonic series.

The harmonic series is a very famous series of
numbers that you get if you start adding up numbers like this: 1 + 1/2 + 1/3
+1/4. Continue that to infinity, and you've mastered the harmonic series.

The natural logarithm is far more complicated to
explain than that, but the tl:dr version is if you take the difference between
the values of natural logarithm and the harmonic series, you'll end up with a
finite number called Euler's constant, which is 0.577 to three decimal places
(and just like pi, it can go on into many, many decimals - around 100 billion).

What the number 0.577 can explain is something truly
mind-boggling.

Imagine you have a circle with a circumference of 1
metre, and you put an ant at the top, and it starts travelling around the
circle at a constant rate of 1 cm every second. Then imagine that while the
ant's doing its thing, you're expanding the circumference of the circle by 1
metre every second.

So every second, the ant is making 1 cm of progress
around your circle, but you're adding 1 metre to the length of the journey.
There's no way the ant will every make it all the way around the circle, right?

Incredibly, that's wrong. The ant can actually make
it all the way around the circle when travelling at a constant rate, despite
the fact that you keep adding to it, and the reason why is that the distance in
front of it isn't the only thing that's increasing - so is the distance behind.

Of course, by the time our ant makes it all the way
around, the Sun will have burnt out, so we're talking about a series of numbers
that grows almost unfathomably slowly.

That's interesting in itself, but what's perhaps even
weirder is that Euler's constant isn't just involved in explaining that
seemingly paradoxical riddle. 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.

And no one has any clue why. I'll let the Numberphile
video below explain that one to you, but let's just say we've never thought of
numbers as being so uncomfortably spooky as much as we do right now:

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