consider this: [1] x^2-x^2=x^2-x^2 is always true for every real number. [2] Factoring both sides in different ways, it becomes (x-x)(x+x)=x(x-x). [3] Dividing both sides with (x-x) {which by the way is equal to 0 for every real number}, the equation becomes 1(x+x)=x(1) {or, (0/0)(x+x)=x(0/0)}. [4] By convention, 0/0 is indeterminate, which in turn makes the whole equation undefined. but, by the 3rd step, it turns out that 0/0 and 1 are equal because (x-x)/(x-x) can be simplified to 0/0 {since the numerator and denominator are both equal to 0}. [5] Working with (x-x)/(x-x), since the numerator and denominator are equal, it is equal to 1. but if simplified first, this becomes 0/0. by convention {and reality!}, 0/0 is NOT equal to 1. [6] continuing from step 3, that equation becomes x+x=x, which is 2x=x. Since it was stated in step 1 that x can be any real no., if x=1, then the equation becomes 2=1, which we all know {hopefully!} is NEVER true. What is happening here?!
2006-08-12
01:16:52
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9 answers
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fictitiousness ;-)
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Mathematics