The Problems Associated with Division by Zero • by Darryl J. Engler • © April 13, 1972

# Division by Zero Undefined

In the development of a field, we note one combination of symbols
which is undefined, namely x 0, where x is any
real number. To be sure x 0 is undefined, we
need only check the two equivalent definitions of division. Definition
[46] states that if there is a unique q such that x = yq, then x
y = q. Let y = 0 in this definition. If there
exists a unique q such that x = 0 q, then x 0
= q.
Case I: x 0 If x 0, then
q such that x = 0 q,
for 0 q = 0 q .

Since there is __no__ q, the definition does not apply.

Case II: x = 0 If x = 0, then every q satisfies
the equation x = y q, for 0 = 0 q
q .

Since there is no __unique__
q, the definition does not apply.

Definition [47] states that if y 0, then
x y = xy*. Since y = 0,
this definition does not apply either.

The Problems Associated with Division by Zero • by Darryl J. Engler • © April 13, 1972