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 (a).  Use the definition given in property (P7) and state what the multiplicative identity    would have to be for this new multiplication.

Solution 9 (a).

See text and/or instructor's solution manual.

We would want to find a number    such that for any    we have  .

Obviously, if  ,  then according to the definition of  "*" we would have

        .  

Thus, the multiplicative identity in this case would have to be   .

This solution is complements of the authors.

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