# Boxing Pythagoras

## Math is Really Weird: On Strange Sums and Counterintuitive Results

Whenever you add a finite integer to another finite integer, you always get a sum which is, itself, a finite integer. This, by itself, is not very shocking. When you add 1 to 1, you get 2. When you add 5 and -9, you get -4. When you add 0 and 299,792,458, you get 299,792,458. This is all rather unsurprising.

However, math can get weird once you start adding up an infinite collection of numbers. Take Zeno’s Dichotomy Paradox, for example. Numerically, we can represent this problem as an infinite summation: $S=\sum\limits _{n=1}^\infty \frac{1}{2^n}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+...+\frac{1}{2^n}+...$ Even though we are adding up an infinite quantity of numbers, we arrive at a finite value– in this case, $S=1$. Arguably the most famous philosopher in history, Aristotle, would have vehemently objected to this formulation– and, in fact, did object rather loudly in his book Physics, when discussing this particular paradox. However, it has been over three centuries, now, since mathematicians would have found this problem to be controversial; and, in fact, similar cases of infinite summation form the entire basis of integral calculus. High schoolers are introduced to these concepts in their Pre-Calculus classes, nowadays, and you might even remember evaluating some of these limits of convergent sums from your own schoolwork.

But math can get far stranger, still. One of the most peculiar things in all mathematics occurs when you attempt to sum all of the Natural numbers. As absolutely insane as this might sound, today I’m going to demonstrate for you that $1+2+3+4+...=-\frac{1}{12}$.