which logic is being redifined here? i dont understand.
if you take a set of objects and perform certain operations on them' date=' sometimes you will realize that the set isnt closed under that operation.
for example, when the greeks tried to take the square root of 2, they couldnt fathom the existence of a number not part of the rational numbers. do the so called "irrational" numbers lead to a contradiction? i would hope to think not.
so when you've got the idea of the square root of a negative number, the solution falls outside the original set of numbers. so just as math was plunged into irrational numbers to find solutions to square roots of "non-square" numbers, the "imaginary" number was created and the complex number system was used to find solutions to square roots of negative numbers.
"irrational" and "imaginary" arent meant to be taken literally.
they're perfectly valid number systems which allow calculations to be made.
just because you cant have 3[i']i[/i] apples doesnt mean that the number system is a fallacy. i mean you cant have pi apples either (no matter how hard you try).