Proving Conditional Identities :

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I am learning how to prove conditional identities like

(a^2-c^2+b^2+2ab)/(c^2-a^2+b^2+2bc) = (s-c)(s-a)
if a+b+c = 2s
- Derived from Herons formula

I have understood the proof for the above , but i want more problems to work on.
Can anyone suggest some link where i can find similar problems/more explanation on conditional identities?
I searched a lot , but couldnt find anything at all.

It would be great if you could tell me about how to formulate such conditional identities by taking a formula like herons law.
 

Answers and Replies

  • #2
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I've never heard that term "conditional identity" before so I'm confused as to what the condition is exactly. Is it that a,b,c are the sides of a triangle or that a+b+c=2s? If it's the former then it seems like you just want to some geometric identities to prove but if its the latter than a "conditional identity" isn't any different than a regular identity if you substitute the variable in.
 
  • #3
arildno
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Well, I don't see what's wrong with calling it a "conditional identity".

In (a,b,c,s)-space, the condition restricts the validity of the equality to the hyperplane a+b+c=2s.
 
  • #4
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Perhaps what you're thinking about is something like

Condition(S) => Conclusion(S), where S is some sort of problem domain. You could then say something like S = {a, b}, have Condition(S) = "a * b = 0", and have Conclusion(S) = "a = 0 OR b = 0".

I suppose you could look at this proof as a proof of a conditional statement... and I suppose that all proofs, really, could be written in a similar form.

I think what qntty is trying to get across is that you normally take for granted that there are strings attached... all equalities are conditional to some extent (except trivial reflexivities like "a=a").

I think it's just a way of looking at things that you shouldn't let hang you up.
 
  • #5
arildno
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Wherein lies the "triviality" in million-term absolute identities??
 
  • #6
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This is not anything related to geometry.
what AUmathtutor says is what i mean to convey.

Any tips on solving these?
And can you point me to any links which give some help?
I found a lot of stuff for trig , but none at all for algebraic identities. :(
 
  • #7
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Well as for some tips on proving algebraic identities (and inequalities), you should (1) Know the formulas that can be used to prove the identity very well and always look for similarities between the current form of the identity you have and identities that you know (2) be acquainted with techniques needed to change the form of an algebraic expression (factoring, decomposing to partial fractions, using substitutions cleverly, etc) (3) The use constraints will be necessary to solve the problems so try to get rid of the constraints if they don't help and if you can (eg by making a substitution that eliminates the constraint) (4) know the standard tricks for the certain type of identity/inequality that you're trying to prove if there is any. Of course there's no substitute for a lot of practice if you want to get good.
 

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