Division By Zero

(Image by trekkyandy)

I’m reading a book on the history of zero which I’m sure I’ll talk more about later, but for the time being I want to reiterate a proof as to why division by zero is a mathematical no no. This isn’t a complete answer at all but it’s a quick little proof by contradiction that is easily grasped. Assume for the time being that division by zero is okay.

Take, as an assumption, that 3 does not equal 11.
We know that both 3 * 0 = 0 and 11 * 0 = 0.
So 3 * 0 = 11 * 0.
Then (3 * 0) / 0 = (11 * 0) / 0.
On both the left and right sides of the equal sign cancel the zeroes in the fraction.
So 3 = 11.
But we know from our initial assumption that 3 does not equal 11.
Contradiction.

The only other assumption we made was that division by zero was possible. Since this assumption led to a contradiction it is a false assumption. So division by zero is a questionable practice. I used 3 and 11 as example numbers, but the same argument holds for arbitrary numbers x and y, where x does not equal y.