Brain Teaser: ImpossibleMath2


Brain teaser:
Reference
[ImpossibleMath2]
Submitter
Jack Tan (jahk@uiuc.edu)
Submission date
14 April 1997

About the ImpossibleMath problems: here are some math problems for various grade levels, from algebra up to calculus. These are strange because the problems give results such as "2 is equal to 1" and "-1 is equal to 1"! Can you find the flaw in the reasoning? (My apologies to those more oriented towards literature and language.)

This problem requires advanced algebra (imaginary numbers). Let the symbol 'Sqrt(x)' stand for "the square root of x," and let the symbol 'i' stand for the unit imaginary number, whose square is equal to -1, and let 'x^y' stand for "x raised to the yth power." We can show that -1 is equal to 1:

 1              i = i                  Trivial
 2            i^2 = i^2                Square both sides
 3            i^2 = i*i                Expand the definition of i^2
 4   (Sqrt(-1))^2 = Sqrt(-1)*Sqrt(-1)  Expand the defintion of i
 5   (Sqrt(-1))^2 = Sqrt(-1 * -1)      Rule:  Sqrt(x)*Sqrt(y) = Sqrt(x*y)
 6             -1 = Sqrt(-1 * -1)      Rule:  Sqrt(x)^2 = x
 7             -1 = Sqrt(1)            The product of -1 and -1 is 1
 8             -1 = 1                  The square root of 1 is 1
What is wrong with this reasoning?


Jack Tan and Emmie Chen / jahk@uiuc.edu

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