# field axioms proof

• May 23rd 2007, 06:32 AM
papercut06
field axioms proof
How do I prove that

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

Thank you
• May 23rd 2007, 07:05 AM
Jhevon
Quote:

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.
• May 24th 2007, 06:34 PM
ThePerfectHacker
Quote:

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