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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.
"I'm fearful when I see people substituting fear for reason." Klaatu, from The Day The Earth Stood Still (1951)














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#62
RE: The Mathematical Proof Thread
(21st September 2017, 16:07)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
(17th December 2017, 16:39)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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