# Math Help - Proving ismorphism

1. ## Proving ismorphism

Define on RXR by setting (a,b)(c,d)=(acbd,ad+bc)Show that the function h from system (C,)(RxR,) given by h(a+bi)=(a,b) is a one to one function from the set of complex numbers that is onto RxR and is operation perserving
I dont really have a background in complex numbers so i am kinda having difficulties

2. ## Re: Proving ismorphism

for "operation-preserving" to make sense, you have to say WHICH operation on C, h is preserving. in this particular case, it is clear to me the operation is "complex multiplication".

in particular, since C is a field, the usual laws of arithmetic apply (we can "FOIL"), so:

(a+bi)(c+di) = ac + a(di) + (bi)c + (bi)(di) = ac + (ad)i + (bc)i + (bd)i2.

now in the complex numbers, we have i2 = -1, so:

ac + (ad)i + (bc)i + (bd)i2 = ac - bd + (ad + bc)i.

what you have to show is the following:

h((a+bi)(c+di)) = h(a+bi)⊕h(c+di), that is: