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(B A)
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 y-axis.
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 one-to-one, onto, and invertible!
The Problems Associated with Division by Zero • by Darryl J. Engler • © April 13, 1972
