#Help with this algebraic proof
unborn fractalBOT
wispy cradle
Can you explain how this works ?
You said that it is sufficient to show sqrt(a)+sqrt(b)=2
To establish the equality
(why not)
But then you used the conclusion ? And substituted in the hypothesis
jade fiber
Yes you are right it is not clear at all, I should have assumed √a+√b=2 and then showed that both equation are same are just transformation of each other. Still I'm not sure if it is enough to conclude the proof. Sorry for the haphazard proof. I just quickly need to write it down for you all to see.
Hopeful someone will show us a rigorous proof for this.
jade fiber
Hi I finally proved it rigorously, I will forward the solution here.
wispy cradle
I have a solution
If you want
Hope my handwriting is clear
I first solve the system, then I show that sqrt(a)+sqrt(b) is equal to 2
wispy cradle
They are but I found x and y 😁
jade fiber
Not required as per the question.
wispy cradle
Yeah so what
Our approaches are not exactly the same
You added the two equations and noticed that it is the expansion of (x+y)^3
I solved the system first
And then checked that the solutions satisfy x+y=2
jade fiber
Oh you are right
wispy cradle
But I could also find any other relationship between x and y such as x - 7y
wispy cradle
My proof is perfectly rigorous
jade fiber
No extra unnecessary stuff.
wispy cradle
You just did exactly the question
I did something more precise
How is that less clean
Your "proof" is strictly included in mine 😂
jade fiber
But my proof is easier to read. Full stop.
Let's take opinion from the crowd instead of arguing
wispy cradle
Your proof is more basic yes
Mine requires a bit of reasoning
Both are valid
Why so salty
jade fiber
Extra reasoning were things can be solved using basic stuff is unnecessary.
wispy cradle
That's not really how real mathematics work
wispy cradle
Maybe for an exam yes
You just need to write a short answer
But here we are exploring a problem
jade fiber
Have you taken class on mathematical proofs?
jade fiber
Then you would understand why.
wispy cradle
😂
jade fiber
OK then you do understand that being concise in proofs is must.
wispy cradle
This has nothing to do with being concise
jade fiber
Okay. Are you working on any cool problems? Or proofs?
wispy cradle
Being concise is great when you write a proof
And we both are very concise
In the above proof
Its just that I did something to go further
Doesnt mean its not a valid approach
Nor a bad proof
jade fiber
Ok
wispy cradle
I didnt include unnecessary arguments either
jade fiber
But you should remember that main reason for writing Proofs is to convince your reader with easy to read proofs without unnecessary symbols and it is important to keep your proofs short. Also for God's sake add a tombstone symbol at the end of your proofs (for the beauty obviously)
wispy cradle
If you focus on the ideas, you would realise my method is far superior than yours
Why ?
Because if any other question was asked about a and b your method would not hold
So yeah your method is "quick" but that doesnt mean its better
Perhaps whats even more problematic is not keeping an open mind
To other approaches
jade fiber
Your method might be superior than mine but it will be stretch to say that it is a good example of beautiful mathematical proof which in my case is.
Also if I need to find individual x and y, I can use Cardano's formula for the cubic equation for the closed form or Newton's method for the numerical result or some other method.
Also I guess we have criticised each other enough good night. 👍
Nevertheless I would like to know what the audience will think about our methods.