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The Mathematical Proof Thread
#1
The Mathematical Proof Thread
Via Lara Alcock (How to Study as a Math Major, 2013: oxford university press), math proofs are like an internal combustion engine: we drive just fine without knowing how a car actually works, but if we are ever curious why a car operates the way it does, then we need to understand the internal combustion engine, which is the broader system that makes driving a reality. 

I've created this thread for people to post, share, discuss, or inquire about any proof they want.  It would be really cool if the proof was something beautiful to you. Perhaps we could even form a mini proof library.  

Here's a basic example to get things rolling.

Inquiry: suppose we have the even numbers 2,4,6.  We know that these numbers are even.  We also know that if we square these numbers then the squares are also even.  But, although we intuitively know this, how could we show that this result is true for 2,4,6, or more importantly, for all even integers x?

Definition of an Even Number: An integer n is even if n=2a for some integer a. (Hammack, Book of Proof, 2013, pg 89)


Proposition: If x is an even integer, then x^2 is even. 

Proof (direct). Suppose x is an even integer.  Then x=2a for some integer a via the definition of an even number.  Now substitute x=2a into x^2, which gives x^2=(2a)^2=4a^2=2(2a^2).  Consequently, x^2=2b for some integer b=2a^2.  Thus, x^2 is even by the definition of an even number.
"I'm fearful when I see people substituting fear for reason." Klaatu, from The Day The Earth Stood Still (1951)

"It's not the mediator's job to get the parties to reach settlement, to promote empowerment, to solve a problem, or to foster reconciliation, although a mediation may achieve all of these.  Instead, the mediator seeks to restore or catalyze the parties' ability to cooperate in meeting each other's reasonable needs and hopes, whatever those may be.

What do you need? How can I help you get there?" Jennifer E. Beer & Caroline C. Packard, from The Mediator's Handbook (2012), pg. 9.












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#2
RE: The Mathematical Proof Thread
Don't know much math, but that sounds like WIFOM to me.
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#3
RE: The Mathematical Proof Thread
My kid is studying some stuff in her 8th grade advanced honors algebra class. I'll have to run this by her to see if she understands. Once you start throwing letters into a math equation, I get lost. Letters don't belong with numbers for us dumb folk.
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#4
RE: The Mathematical Proof Thread
I do love me some proofs.

I remember for whatever reason the proof of Borel's Lemma being particularly interesting to me back in Advanced Number Theory but that could just be me remembering the joy of finally getting to any sort of measurable accomplishment - the *entire* semester was spent proving the Cayley-Bacharach Theorem. It's been nearly 5 years since I've done any higher-level mathematics like that with any consistency, so both of those proofs are currently wayyy beyond my ability to explain.

P.S. I'll post some cool proofs that I *do* remember and understand soon!
How will we know, when the morning comes, we are still human? - 2D

Don't worry, my friend.  If this be the end, then so shall it be.
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#5
RE: The Mathematical Proof Thread
(14th September 2016, 00:04)Nymphadora Wrote: My kid is studying some stuff in her 8th grade advanced honors algebra class. I'll have to run this by her to see if she understands. Once you start throwing letters into a math equation, I get lost. Letters don't belong with numbers for us dumb folk.

if she's doing advanced honors algebra, then this is just basic stuff for her.
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#6
RE: The Mathematical Proof Thread
This wasn't presented as an occasion for producing a proof but I found it a fun problem anyway.  I wonder what you think of my argument in favor of the proposition.  (I'll hide it in case you want to have a go at it yourself first.)

Proposition:  The cube of any odd number ≥ 3 decreased by that same odd number will always be divisible by 24


(16th February 2017, 18:16)TheOther JoeFish Wrote: So what you're saying is that I can harass all of the members I want for the next 168 hours, as long as I do so in my signature?
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#7
RE: The Mathematical Proof Thread
Whatevers, cool, after a first reading, it looks watertight to me. As far as I am concerned, you could shorten your second step to
"
n^3-n=(n-1) * n * (n+1)
If n is odd, n-1 and n+1 are even. In fact, one of them is divisible by four because that os always the case for two consec. even numbers.
"
without introducing a. That made it harder to read and I don't see why you needed it.
The fool hath said in his heart, There is a God. They are corrupt, they have done abominable works, there is none that doeth good.
Psalm 14, KJV revised edition

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#8
RE: The Mathematical Proof Thread
I'll try to write an induction proof for it later, the problem seems perfectly suited for that
The fool hath said in his heart, There is a God. They are corrupt, they have done abominable works, there is none that doeth good.
Psalm 14, KJV revised edition

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#9
RE: The Mathematical Proof Thread
(14th September 2016, 01:59)Alex K Wrote: Whatevers, cool, after a first reading, it looks watertight to me. As far as I am concerned, you could shorten your second step to

n^3-n=(n-1) * n * (n+1)
If n is odd, n-1 and n+1 are even

without introducing a. That made it harder to read and I don't see why you needed it.


Glad to hear you say so.  Felt like sticking legs on a snake to me too.  I guess I was trying to make it more math-y.
(16th February 2017, 18:16)TheOther JoeFish Wrote: So what you're saying is that I can harass all of the members I want for the next 168 hours, as long as I do so in my signature?
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#10
RE: The Mathematical Proof Thread
The more I think about it, the more it becomes apparent that an induction proof could only be more complicated than yours because it would basically contain it.
The fool hath said in his heart, There is a God. They are corrupt, they have done abominable works, there is none that doeth good.
Psalm 14, KJV revised edition

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