Unveiling the Enigma of $\mathrm{-0.08(3)}$

Let's start saying that $\mathrm{-0.08(3)}$ is a different way of writing $-0.083$ ( $\mathrm{-0.08333...}$ the number goes on forever), that is the same as $\mathrm{-}\frac{1}{\mathrm{12}}$ . Now we have a new question:

What is $\mathrm{-}\frac{1}{\mathrm{12}}$ ?

The easy answer is that the sum of all natural numbers is $\mathrm{-}\frac{1}{\mathrm{12}}$ .

This can be proved in a couple of different ways. The first one is a little easier to understand and it's explanation can be found in this video . I won't explain it here because it's a little long and I don't want to copy it. The second one is definitely more complicated and it's based on the Riemann Zeta function (you can find more about it in this video ).

The best part is that this is not true! The sum of all natural numbers is not equal to $\mathrm{-}\frac{1}{\mathrm{12}}$ , or at least it's not that simple.

The first video I linked is a little misleading. It uses a trick to get to the result, an infinite series property: the "infinite distributive law", it exists and can be used, but only for convergent series! If you felled for it sorry, but otherwise we would be able to explain that everything is equal to everything else. You can see more about it in this video .

If you think about it that makes sense. If you sum all the natural numbers you will get a number that goes on forever, so it cannot be a finite number. It's not a number, it's a divergent series. This means that it's not possible to assign a value to it. The sum of all natural numbers is infinite!

Well no, not so simple as I said. Using the Riemann Zeta function we can prove that the sum of all natural numbers is equal to $\mathrm{-}\frac{1}{\mathrm{12}}$ , but the Riemann Zeta function is defined only for $\mathrm{Re}(s)>1$ , so it's not defined for $\mathrm{s}=1$ . To get around this problem we can use the analytic continuation of the Riemann Zeta function, that is a way to extend the domain of the function to the whole complex plane. This is a little complicated and I don't want to explain it here, but you can find more about it in this video .

This means that the sum of all natural numbers is not equal to $\mathrm{-}\frac{1}{\mathrm{12}}$ even if... it is.

So why am I telling you all this?

This intriguing puzzle serves as a reminder of the profound beauty and complexity of mathematics. It challenges our preconceived notions, tests our mental agility, and unveils the intricate fabric of the mathematical universe. Mathematics is not just about numbers; it's about exploring the world around us, uncovering hidden patterns, and finding connections that were previously unseen. I hope this journey through mathematical labyrinths has sparked your curiosity and encouraged you to explore the world with a fresh perspective.

Remember, there are endless fascinating things to discover, waiting to be unraveled by minds like yours. And as we delve deeper into the mysteries of the universe, let us forever embrace the enigmatic number -0.08(3) as a symbol of endless possibilities.