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

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Discussion Overview

The discussion revolves around the mathematical expression and implications of the equation (ab = ac) = (b = c) for all values of a. Participants explore the conditions under which this equivalence holds, particularly focusing on the case when a equals zero.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about manipulating the expression a/(b/c) and seeks assistance.
  • Another participant provides a step-by-step manipulation of the expression, ultimately simplifying it to (ac)/b, while noting the importance of avoiding division by zero.
  • A later reply emphasizes that the equivalence ab = ac implies b = c is not universally true, particularly when a is zero, as this leads to an undefined situation.
  • Participants acknowledge the need to be cautious with the implications of the equation when dealing with zero values.

Areas of Agreement / Disagreement

Participants generally agree on the manipulation of the expression but disagree on the implications of the equation ab = ac, particularly regarding the case when a equals zero.

Contextual Notes

There is an unresolved discussion regarding the implications of division by zero and how it affects the validity of the equivalence ab = ac = b = c.

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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