Exercise 9.  Let's use the symbol * for a new type of multiplication of complex numbers defined by   .  
This exercise shows why this is an unfortunate definition.

9 (b).  Show that, if you use this new multiplication, nonzero complex numbers of the form    have no inverse.  
That is, show that, if  ,  there is no complex number w with the property that  ,  where    is the multiplicative identity you found in part 9 (a). 

Solution 9 (b).

See text and/or instructor's solution manual.

According to the definition of  "*",  for any complex number    we would have  ,  which can't possibly equal  .

This solution is complements of the authors.

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