Ahouxs.130
net.math
utzoo!decvax!harpo!ihnss!houxi!houxv!houxs!hansen
Tue May  4 12:47:42 1982
An oldie but goodie
Here's an old paradox I remember from my school days. Find which steps in
this proof are wrong, and what rules they violate.

                                       Enjoy,
                                       Tony Hansen
                                       houxs!hansen
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Prove: -1 is equal to 1.
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                ----             ----
1.              v -1     =     v -1            identity axiom, sqrt(a)=sqrt(a)


*****************************************************************************

2.               ------
               / -1              ----
               /  ---    =     v -1            property of 1
             v    1

*****************************************************************************

                ------   ------
3.              / -1           /  1
               /  ---    =    /  ---           property of negative numbers
             v    1          v   -1

*****************************************************************************

                ----           ----
               v -1           v  1

4.             ________  =  ________           sqrt(a/b) = sqrt(a)/sqrt(b)

                ----           ----
               v  1           v -1

*****************************************************************************

         ----      ----      ----     ----
5.       v -1  *  v -1   =  v  1  *  v  1      cross multiply

*****************************************************************************

6.              -1       =      1              Q.E.D. (?!? Surprise ?!?)

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