Department of Mathematics --- College of Science --- University of Utah

Mathematics 1010 online

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:

  Always check your answers.

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 .