How Do Field Axioms Prove Properties of Complex Number Inverses?

[SNOOTCHIEBOOCHEE]

Homework Statement

Using only the axioms for a field, give a formal proof for the following:

a) 1/z₁z₂ = 1/z₁ 1/z₂
b) 1/z₁ + 1/z₂ = z₁ + z₂/z₁z₂

The Attempt at a Solution

I really am having a tough time understanding this problem. I know the axioms of a field

i.e. associativity and commutativity for addition and multiplications (those are the only axioms she cares about) but how do i use these to show the above is true?

 

[HallsofIvy]

Well, obviously in addition to associativity and commutatitivity you will need to use the definition of "mutliplicative inverse" since that is what 1/z is. And, just as obviously each of these should have the provision "z1 and z2 not equal to 0". And you are missing parentheses from (b)- it should be 1/z1+ 1/z2= (z1+ z2)/z1z2, not what you have.

For the first one, the left side is the multiplicative inverse of z1z2. To show that the right side is also, you need to show that (z1z2)(1/z1)(1/z2)= 1.

For the second you want to show that (1/z1+ 1/z2)[(z1+ z2)/z1z2]= 1.