This is part of a series on common misconceptions
True or False?
If and then
Why some people say it's true: Well, it just makes sense! How could it not be true?
Why some people say it's false: You shouldn’t just trust your intuition, it can’t be that simple!
The statement is .
Short answer: proving with one counterexample
Let and . This satisfies the conditions that and but since the hypothesis is false.
Multiple things aren’t taken into account if you go with your intuition. Were it just it would be fine, but multiplication is more complex. The most obvious thing your intuition overlooks is the signs of numbers. In this case, there are 16 possibilities for the signs of and
However, we can divide them into 5 cases:
Case 1: Exactly one of or is 0 and exactly one of or is negative and the other is positive is negative Then
Case 2: Exactly one of or is 0 and exactly one of or is 0 Then
Case 3: At least one of or is 0 and both and are negative is positive Then
Case 4: Both and are positive is positive is sometimes true in this case, but not always. Since negative times negative equals positive, it is possible that However, even if and are both negative, this is not always the case — in that case, it is possible that or just that — and if they are not both negative, is always true.
Case 5: and are all negative and are both positive Then
We see that in cases 2, 3, 5, and sometimes 4, our hypothesis is false. There are more counterexamples, as these cases assume that and are all real.
Rebuttal: Why should negative times negative equal positive?
Reply: It is intuitively confusing but is always defined that way in math.
Rebuttal: If and are all negative, isn’t
Reply:No. Let and be real positive integers where Then multiply both sides by -1. We have to switch the sign around, which does make a bit of sense intuitively when you think about it.
Rebuttal: I can’t follow all those cases!
Reply: The exceptions to this rule if and are all real numbers are caused because negative times negative equals positive and negative times positive equals negative. These rules complicate the multiplication. This extra complication is why we must check so many cases if we want a general idea of this rule's truth value if and are all real numbers If you don’t want to go through the whole process, let and This satisfies the conditions that and but and so Simply proving one counterexample is enough for this.
Rebuttal: This is counter-intuitive!
Reply: Keep in mind that if and are all real numbers; the exceptions to this rule are caused by completely counter-intuitive equations (and imaginary numbers are counter-intuitive as they are), so you shouldn’t expect an intuitive answer.