##
Never Divide by Zero

Division by zero is **undefined**, and for good reason. If we
assigned a number to the result of dividing by zero we'd run into
contradictions, and mathematics would become useless.

Let's approach the issue from several angles:

**Definition.**
Division is the inverse process of multiplication. For example, the
unique (one and only) solution of
for , is
When we get the equation
This
equation has no solution if , and any number will be a
solution if . In neither case do we have a unique solution.
This argument only shows that our usual way of defining division fails.
The next argument shows that there is no way of extending the definition in a way that does not give rise to contradictions.

**A Contradiction.** Suppose we define
for some real number . Multiplying on both sides of the equation
gives
which is a contradiction (to and 0 being different numbers).

**A more subtle contradiction.** Try to figure out what's
wrong with the following "proof" that , before reading the answer
at the bottom of this page.
Let

Then

**Another Fallacy.**
Temptations to divide by zero arise in subtle ways. For example,
consider the equation
The
routine approach to solving this equation begins by taking reciprocals
on both sides, giving
Multiplying with
on both sides gives
Finally, adding and
subtracting on both sides gives what might appear to be a
"solution":
.
However, when we set in we divide by zero on both
sides of the equation and obtain nonsense. The equation has
no solution!

Of course, you would have discovered that fact by
following another one of our basic principles:

### The Moral of the Story

Whenever you divide by anything you should think about the possibility
of the divisor being zero. This should become a reflex like checking
your blind spot when you change lanes on the freeway. Usually there
is no problem, but if in fact there is somebody in your blind spot then the
consequences can be devastating, and you want to avoid the resulting
carnage!

### The Answer to the Puzzle

In going from (3) to (4) we divide by zero since implies that .