This is part of a series on common misconceptions.
True or False?
For real numbers and if then
Why some people say it's true: Just divide both sides of the equation by .
Why some people say it's false: We cannot simply divide by , right?
The statement is .
We can prove that the statement is false using a simple counter-example:
Proof: Consider . We have and so they are equal. However, .
More generally, by the zero product property, we know that . Always remember that if we want to "divide both sides by ," we have to check that we are not dividing by 0.
Rebuttal: But the statement is true if . Then gives us .
Reply: Yes, the statement is true when . However, the question is for all possible values of , and . In particular, the statement need not be true when .
Rebuttal: But by the rules of algebra, you can divide by
Reply: Yes, but that implies that is not equal to 0. After all, you cannot divide by zero.
Want to make sure you've got this concept down? Try this problem: