The Problems Associated with Division by Zero • by Darryl J. Engler • © April 13, 1972
Simplifications
There are some simplifications we can make if the definition
a/0=0 is accepted.
For example, the Signum function now defined to be:
f(x)

= 
x
x


if x 0



= 
0


if x = 0

could be defined as:
Conditional probability of B occurring, given that A
has occurred is now defined to be:
P(BA)
P(A)


if P(A) 0


0 

if P(A) = 0.

Conditional probability could be defined to be:
P(BA)
P(A)


events A, B.

The family of curves y=cx fills the plane except
for (0,y) : y 0.
Notice that c=y/x represents the same family of curves for x 0.
Let x = 0, apply the definition a/0=0, and the rest of the plane is filled in, including all
of the yaxis.
Finally consider the function y=1/x. The domain of this function
is  0.
The range is also  0.
If we add the one missing point to the range and the one missing
point to the domain, we have exactly 0=1/0. The graph of this
function would then be:
Notice that the function is completely defined on all real numbers;
it is onetoone, onto, and invertible!
The Problems Associated with Division by Zero • by Darryl J. Engler • © April 13, 1972