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

Summary

There are several disadvantages to the suggestion of defining division by zero:

The definition defies custom and intuition.
The definition produces an uneasy feeling in some people.
It would be desirable to make the function y=1/x continuous. This definition does not make the function continuous.
The definition does not supply the expected result when examining limits of functions.
This definition would not be solving the problem of division by zero. It would just be giving it a different name.

There are several advantages of defining division by zero:

The field of real numbers is now closed under the operation of division.
Division is a well-defined operation for all of onto .
The function y=1/x is now a one-to-one function from onto .
Many definitions and theorems can be simplified both in statement and proof.
The definition is logically consistent with the postulates used in developing a field.
The definition removes many undefined terms and undecidable expressions from Algebra.
The definition is intuitively obvious to many people.

In summary, I would like to note that the symbols 1/0 are undefined for a different reason than the symbols 0*, where * is the unary operation meaning "multiplicative inverse of." The latter cannot be defined consistent with the field postulates; the former has not yet been defined.

Therefore, since division by zero can be defined consistent with the field postulates, I suggest that we abolish the myth that division by zero cannot be defined, and approach the problem more realistically.