TITLE: Has there ever been an application of dividing by $0$?
QUESTION [59 upvotes]: Regarding the expression $a/0$, according to Wikipedia: 

In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by $0$, gives $a$ (assuming $a\not= 0$), and so division by zero is undefined.

Is there some other kind of mathematics that is not "ordinary", where the expression $a/0$ has meaning? Or is the word "ordinary" being used superfluously in the quoted statement?
Is there any abstract application of $a/0$?

REPLY [3 votes]: In two dimensional space, if we want a vector corresponding to a given angle $t$, we can do:
$(x, y) = ( \cos(t), \sin(t) )$
But what about the reverse?  We have a vector, and want to know the angle.  A common solution is to use arctan:
$t = \arctan( \frac{y}{x} )$
Here $\frac{y}{x}$ represents the gradient of the vector. So it has a meaningful value even when $x = 0$.

Trivia: Many program languages will fail to perform that calculation, because division by zero will produce an error.  They usual address this by offering an alternative function which takes two arguments: atan2(y, x).
But a notable exception is Javascript, which has the concept of +Infinity and -Infinity as numbers, and can perform this calculation even when x = 0.
$ node
> Math.atan( 1 / 0 )
1.5707963267948966           (Pi/2)
> Math.atan( -1 / 0 )
-1.5707963267948966          (-Pi/2)

Another language which can perform this calculation is Haskell.

Inspired by Qiaochu Yuan's comment.