How do I prove that

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

Thank you

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- May 23rd 2007, 05:32 AMpapercut06field 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, 06:05 AMJhevon
- May 24th 2007, 05:34 PMThePerfectHacker
If $\displaystyle b,d\not =0$ are elements of a field then,

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

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

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

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