Everyone learns in primary school that you can’t divide by zero, but few of us ever learn (or fully understand) why. The stock answer is that it gives you an answer of infinity. The truth is a bit more nuanced than that, and an old mechanical calculator offers the perfect illustration.
Try to divide a number by zero with a standard calculator, and you’ll just get an error message. But check out what happens when someone tries to do this on a Facit ESA-01 mechanical calculator, with the cover helpfully removed so all the internal working are exposed:
The machine pretty much goes insane, its gears and cogs hammering away in a nonstop frenzy of clacking. WTF?
It has to do with how such machines work, and also with the nature of division — it’s basically just “glorified subtraction,” as the folks at Numberphile will tell you. To divide 20 by 4, the mechanical calculator perform a sequence of steps: 20 minus 4 is 16, 16 minus 4 is 12, 12 minus 4 is 8, 8 minus 4 is 4, and 4 minus 4 is 0. Voila! It takes five steps, so 20 divided by 4 equals 5.
But 20 divided by zero is just subtracting nothing from 20 over and over again. The calculator never arrives at an answer because it would take an infinite number of steps. And infinity is really more of a concept than an actual number.
So as Eric Limer explains over at Popular Mechanics, “The underlying cause for the freakout is that the poor machine is trying to hammer out an infinite sequence of commands, one by one. A computer will always do exactly what you tell it to do, even if that will take literally forever.” [Twisted Sifter]