Let's use the symbol ∗ for a new type of multiplication of complex numbers defined by z1∗z2=(x1x2,y1y2). This exercise shows why this is a bad definition.
(a) Use the definition given in property (P7) and state what the multiplicative identity ζ would have to be for this new multiplication.
(b) Show that, if you use this new multiplication, nonzero complex numbers of the form (0,a) have no inverse. That is, show that, if z=(0,a), there is no complex number w with the property that z∗w=ζ, where ζ is the multiplicative identity you found in part (a).