# Math Help - field axioms proof

1. ## field axioms proof

How do I prove that

(a/b) - (c/d) = (ad - bc)/(bd) if b =/ 0 and d =/0

Thank you

2. Originally Posted by papercut06
How do I prove that

(a/b) - (c/d) = (ad - bc)/(bd) if b =/ 0 and d =/0

Thank you
you can literally combine the two fractions on the left and you will get the ones on the right. there are dozens of ways to do this, use your favorite method of subtracting fractions.

3. Originally Posted by papercut06
How do I prove that

(a/b) - (c/d) = (ad - bc)/(bd) if b =/ 0 and d =/0

Thank you
If $b,d\not =0$ are elements of a field then,

$a/b = ab^{-1} \mbox{ and } c/d = cd^{-1}$ be definition.

Then, $ad/bd = (ad)(db)^{-1} = ad^{-1}db^{-1} = a/b$

Similarly, $bc/bd = c/d$.

Hence, $a/b - c/d = ad/bd - bc/bd = (ad)(bd)^{-1} - (bc)(bd)^{-1} = [ad-bc](bd)^{-1} = (ad-bc)/bd$