Proving Conditional Identities :

In summary, the conversation is about a person's understanding of proving conditional identities, specifically one derived from Heron's formula. They are looking for more problems to work on and resources for understanding how to formulate such identities. There is also a discussion about the concept of "conditional identities" and tips for solving them. The main takeaways are to be familiar with relevant formulas and techniques, eliminate constraints if possible, and practice regularly.
  • #1
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 couldn't 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.
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  • #2
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
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
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
Wherein lies the "triviality" in million-term absolute identities??
  • #6
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
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.

1. What are conditional identities?

Conditional identities, also known as conditional equations, are mathematical statements that are only true under certain conditions or for certain values of the variables involved. They often involve the use of logical operators such as "if" and "then".

2. How do you prove a conditional identity?

To prove a conditional identity, you must show that the statement is true for all values of the variables that satisfy the given conditions. This can be done through various methods such as algebraic manipulation, substitution, or truth tables.

3. What is the difference between a conditional identity and a conditional equation?

The terms "conditional identity" and "conditional equation" are often used interchangeably, but some people make a distinction between the two. A conditional identity is a mathematical statement that is always true under specific conditions, while a conditional equation is a statement that can be true or false depending on the values of the variables involved.

4. Are there any tips for proving conditional identities?

One helpful tip for proving conditional identities is to work backwards from the desired result. Start with the statement that you want to prove and use logical reasoning to determine what conditions or values of the variables would make it true. You can then use algebraic manipulation or other methods to show that the statement is indeed true for those conditions or values.

5. Can conditional identities be used in real-life scenarios?

Yes, conditional identities have practical applications in various fields such as science, engineering, and computer programming. They are often used to describe relationships between different variables and can help solve problems or make predictions. For example, the laws of thermodynamics can be expressed as conditional identities.

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