This is part of a series on common misconceptions.
Why some people say it's 0: Zero divided by any number is 0.
Why some people say it's 1: A number divided by itself is 1.
Only one of these explanations is valid, and choosing the other explanations can lead to serious contradictions.
The expression is .
Remember that means "the number which when multiplied by gives " For example, the reason is undefined is because there is no number such that
The situation with is strange, because every number satisfies Because there's no single choice of that works, there's no obvious way to define , so by convention it is left undefined.
Of course, there are many possible counterarguments to this. Here are a few common ones:
Rebuttal: Any number divided by itself is
Reply: This is true for any nonzero number, but dividing by is not allowed.
Rebuttal: divided by any number is
Reply: This is true for any nonzero denominator, but dividing by is not allowed no matter what the numerator is.
Rebuttal: Any number divided by is
Reply: Even for nonzero writing is not entirely accurate: see 1/0 for a discussion. But this reasoning only makes sense for a nonzero numerator.
Rebuttal: If we choose to set or it is not inconsistent with other laws of arithmetic, and it makes one of the rules in the above rebuttals true in all cases.
Reply: This is a combination of the first two rebuttals, so here is a "big-picture" reply. Any specific choice of value for will allow some function to be extended continuously. For instance, if we mandate then the function becomes continuous at If the function becomes continuous at
But this is not satisfactory in all cases, and the arbitrariness of the choice will break other laws of arithmetic. For instance,
which doesn't make any sense for any (finite) choice of
Introduction of terms like in otherwise sound arguments can break them down. See if you can spot the error in the problem below:
I will attempt to prove that . In which of these steps did I first make a mistake by using flawed logic?
Step 1: We can rewrite 15 as or .
Step 2: This means that .
Step 3: If we move one term from each side of the equation to the other side, we will get
Step 4: Dividing both sides by gives .
Step 5: Since and , .