Math is Really Weird: On Strange Sums and Counterintuitive Results

The latest bit of weird math is going the rounds. Here is boxingpythagoras’s presentation of:
“So, now we’ve arrived at the strangest result of the day: S=1+2+3+4+5+6+…=-\frac{1}{12}”. (or -1/12)
together with comments from Joseph Nebus and myself.

Boxing Pythagoras

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: $latex S=sumlimits _{n=1}^infty frac{1}{2^n}=frac{1}{2}+frac{1}{4}+frac{1}{8}+frac{1}{16}+…+frac{1}{2^n}+…$ Even though we are adding up an infinite quantity of numbers, we arrive at a finite value– in this case, $latex S=1$. 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…

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