Problem: Show 1 = 2.
“Solution”: Let . Then , and . Factoring gives us . Canceling both sides, we have , but remember that , so . Since is nonzero, we may divide both sides to obtain , as desired.
Explanation: This statement, had we actually proved it, would imply that all numbers are equal, since subtracting 1 from both sides gives and hence for all real numbers . Obviously this is ridiculous.
Digging into the algebraic mess, we see that the division by is invalid, because and hence .
Division by zero, although meaningless, is nevertheless interesting to think about. Much advanced mathematics deals with it on a very deep and fundamental level, either by extending the number system to include such values as (which still gives rise to other problems, such as and ), or by sidestepping the problem by inventing “pseudo” operations (linear algebra) and limiting calculations (calculus).