Brain Teaser: ImpossibleMath1

Brain teaser:

Reference: [ImpossibleMath1]
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 basic algebra. Let 'a' and 'b' be two variables that are equal, so that a=b. From these two variables, we can show that 2 is equal to 1:

 1              a = b             Given.
 2            a*a = a*b           Left-multiply both sides by a
 3        a*a-b*b = a*b-b*b       Subtract (b*b) from both sides
 4    (a-b)*(a+b) = (a-b)*b       Factor (a-b) from both sides
 5          (a+b) = b             Cancel (a-b) from both sides
 6          (b+b) = b             Replace a with b (given, see line 1)
 7             2b = b             Simplify (b+b) to 2b
 8              2 = 1             Answer:  2 is equal to 1

What is wrong with this reasoning?

Jack Tan and Emmie Chen / jahk@uiuc.edu

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