Exercise 9.  Use your knowledge of the principal square root function to explain the fallacy in the following logic:

                        .  

Solution 9.

See text and/or instructor's solution manual.

Solution.  The fallacy lies in the assumption implicit in the second equality that    for all complex numbers  .  

Assuming the principal square root is used, then  .  

This will equal    precisely when     (Explain!)   

The latter equality is plainly false when  .  (Again, explain.)

To give a very thorough answer to this problem, you should state precisely when the last equality is true, and prove your assertion.

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell