Marilyn vos Savant, who claims to be in the Guinness Hall of Fame for “World’s Highest IQ.”, has written the book The World’s Most Famous Math Problem, and here is a direct quote from the book, page 61: The square root of +1 is a real number because +1 × +1 = +1; however, the square root of -1 is imaginary because -1 times -1 would also equal +1, instead of -1. This appears to be a contradiction. Yet it is accepted, and imaginary numbers are used routinely. But how can we justify using them to prove a contradiction? Marilyn's book was reviewed by American Mathematical Monthly 102 (1995) 470-473, and the above quote was given as an example of author's misunderstanding and mangling the notion of proof by contradiction. Wikipedia writes that Marilyn is said to misunderstand imaginary numbers. It seems to me that she is not taken seriously because she is not a mathematician but rather a layperson. I see also the same contradiction, why should i^2 = -1 if a negative number times a negative numbers is equal to a positive number? Are the mathematicians themselves making a mistake and a contradiction? Or perhaps it is alright to break the rules if you can do it so cleverly that no-one, especially the laymen, notices it. The best way is to define new rules that apply to the contradiction and call the new numbers imaginary numbers which obey their own rules. That way a negative number times a negative number can be made equal to a negative number, so -1 times -1 would be equal to -1 according to the new rules, because these are just the imaginary numbers obeying the new rules. Maybe it does not matter that these rules may violate the old rules, who cares?