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The Mathematical Proof Thread
#61
RE: The Mathematical Proof Thread
Earlier this month, I did a proof that was pretty cool (IMO).  As a result, I thought I'd share it here.  As always, I'll put my post in hide tags.  In addition, I'd like to clarify  that my writing is not meant to be arrogant: my mathematical writing style is how I've learned to communicate mathematics (I definitely have lots of room for improvement). With that said, here is the following exercise from my mathematical proofs book by Gary Chartrand:

Prove that if a,b, and c are positive real numbers, then (a+b+c)(1/a + 1/b + 1/c) ≥ 9


Hint/Analysis of Proof




Proof (condensed version)




P.S. I've included an analysis of proof in order to illustrate the thought processes that are involved in constructing a proof (please note that I had more scratch work for each of the statements listed in the analysis; thus, these steps were the result of me taking the time to work things out on paper).  For me at least, I gain an understanding of a proof via the analysis of proof and not by the condensed version.  Well, thanks for your time and attention,  and I hope that more people will post cool math stuff in this thread. Also, there was no solution for this exercise in the book, so the work that I have posted here is entirely my own.











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#62
RE: The Mathematical Proof Thread
(September 21, 2017 at 4:07 pm)LastPoet Wrote: When you analyse an algebraic division Ring, a set E with with 2 operations called '+' and '*' with the axioms of the algebraic goup (E.+) previously proven with commutivity a+b = b+a, the exitence of one element we called 0 where for all x in E: x+0 = x, with the set not being empty or singular;

Given the properties of the operation '*':

1: there exists a member we call u where for all set members a, a*u = a
2: for all members of E, a*b = b*a
3: for all a,b,c in E, a*(b+c) = a*b+a*c

With these in mind prove that a*0=0 and while you're at it prove that 0 is not equal this 'u' mentioned on the axioms.




Note: we do not need to know a*b=b*a for this result.
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#63
RE: The Mathematical Proof Thread
(December 17, 2017 at 4:39 pm)Kernel Sohcahtoa Wrote: Earlier this month, I did a proof that was pretty cool (IMO).  As a result, I thought I'd share it here.  As always, I'll put my post in hide tags.  In addition, I'd like to clarify  that my writing is not meant to be arrogant: my mathematical writing style is how I've learned to communicate mathematics (I definitely have lots of room for improvement). With that said, here is the following exercise from my mathematical proofs book by Gary Chartrand:

Prove that if a,b, and c are positive real numbers, then (a+b+c)(1/a + 1/b + 1/c) ≥ 9


Hint/Analysis of Proof




Proof (condensed version)




P.S. I've included an analysis of proof in order to illustrate the thought processes that are involved in constructing a proof (please note that I had more scratch work for each of the statements listed in the analysis; thus, these steps were the result of me taking the time to work things out on paper).  For me at least, I gain an understanding of a proof via the analysis of proof and not by the condensed version.  Well, thanks for your time and attention,  and I hope that more people will post cool math stuff in this thread. Also, there was no solution for this exercise in the book, so the work that I have posted here is entirely my own.


OK, good. Now, suppose that x_1 ,...x_n >0. Show that

(1/x_1 + 1/x_2 +...+1/x_n )(x_1 +x_2 +...x_n )>=n^2.

(you did the n=3 case).
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#64
RE: The Mathematical Proof Thread
Thanks for your last post, Polymath.  I must admit that I kept getting stuck during each of my attempts and decided to let it go for the time being.  However, this evening, I noticed that your post also appeared as an exercise in one of my math books, so I decided to give it another try. 

That said, for anyone interested, I will post the result to be proved and then post the proof in hide tags.  Also, in my proof, I make use of another fact/result, so I will post that fact and my proof of it (in hide tags) above my main proof. That said, should anyone read the proofs below, please feel free to point out any errors/typos that I have made; I'm very appreciative of any feedback that you can offer.  Thanks

Result: for every two positive real numbers a and b, it follows that a/b +b/a ≥ 2. 

Proof





Main Result to Prove: for every n ≥ 1 positive real numbers a_1,a_2,…,a_n, it follows  that (a_1 +…+ a_n)*(1/a_1 +…+ 1/a_n) ≥ n^2

Proof













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#65
RE: The Mathematical Proof Thread
My apologies for this post.  I'm posting a slight clarification in hide tags.













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#66
RE: The Mathematical Proof Thread
Prove Bernoulli’s Identity: For every real number x > -1 and every positive integer n, (1 + x)^n ≥  1 + nx

Proof













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#67
RE: The Mathematical Proof Thread
(July 6, 2018 at 6:40 pm)Kernel Sohcahtoa Wrote: Prove Bernoulli’s Identity: For every real number x > -1 and every positive integer n, (1 + x)^n ≥  1 + nx

Proof



Looks sound enough to me.
It's amazing 'science' always seems to 'find' whatever it is funded for, and never the oppsite. Drich.
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#68
RE: The Mathematical Proof Thread
I was taking 2nd semester calculus (spring of '78) and the prof called on students to go to the board and solve a problem from the homework set. I had done all the homework, and had to do the proof that a particular function was continuously differentiable. Having just finished a refresher on college algebra the semester previous, I whipped that proof right out. I was pleased with myself, for sure. You should be, too, for your proof! Rewarding, isn't it?  Big Grin
I never thought I'd live long enough to become a grumpy old bastard. Here I am, killing it!
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