Is (ab = ac) = (b = c) for all values a?

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The discussion centers on the mathematical expression (ab = ac) = (b = c) and clarifies that this equivalence does not hold for all values of a, particularly when a equals 0. The user demonstrates the manipulation of the expression a/(b/c) to arrive at (ac)/b, emphasizing that division by zero is undefined. The conclusion is that while the equivalence may seem valid, it fails under specific conditions, particularly when a is zero.

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CuriousBanker
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I know that by definition, a/(b/c)= a x 1(b/c)...but from there I am lost. Help please!
 
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Well, I'm terrible at formal proofs, but consider:

a/(b/c)

multiply numerator and denominator by c, so that we get b by itself as the denominator (nobody likes pesky fractions in their denominator):

(c/c)*[a*(b/c)]

You can do this because c/c = 1.

now you have:

(c*a) / [c*(b/c)]

which equals:

(c*a) / b
 
Nevermind I got it!
 
Ah, king you posted that just as I got it. Thanks though!
 
a/(b/c) = ac/b

Just a note that you have to be careful. The above is only true when both b and c are not 0. The reason is that division by 0 is undefined. In the above proof (c/c) is used which is 0/0 when c=0. 0/0 is undefined.

I just got shut out of physics thread by a Mentor poster who believed that:

ab = ac implies b = c for all values a. That is not correct when a is 0.
 

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