Our server costs ~$33 per month to run. Please consider donating or becoming a Patron to help keep the site running. Help us gain new members by following us on Twitter and liking our page on Facebook!
Current time: 17th January 2017, 02:54

Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
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.













Reply
#2
RE: The Mathematical Proof Thread
Don't know much math, but that sounds like WIFOM to me.
Reply
#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.
Please consider becoming a patron to help offset site costs and to get rid of EGO
Click here for more info and to sign up Smile
Avatar compliments of Luckie with some assistance by Losty Heart
Reply
#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!
Don't forget.
Always, somewhere, someone is fighting for you.
As long as you remember her, you are not alone.
Reply
#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.
Reply
#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


Belief .. is the insistence that the truth is what one would .. wish it to be. The believer will open his mind to the truth on the condition that it fits in with his preconceived ideas and wishes. 


Faith, on the other hand, is an unreserved opening of the mind to the truth, whatever it may turn out to be.  


- Alan Watts, The Wisdom of Insecurity
Reply
#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

Reply
#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

Reply
#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.
Belief .. is the insistence that the truth is what one would .. wish it to be. The believer will open his mind to the truth on the condition that it fits in with his preconceived ideas and wishes. 


Faith, on the other hand, is an unreserved opening of the mind to the truth, whatever it may turn out to be.  


- Alan Watts, The Wisdom of Insecurity
Reply
#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

Reply


Possibly Related Threads...
Thread Author Replies Views Last Post
  Million Dollar Prize for Math proof of NP problems emilynghiem 6 1818 22nd February 2015, 00:47
Last Post: vorlon13
  "Gödel's ontological proof" proves existence of God Belac Enrobso 41 8091 9th February 2015, 03:22
Last Post: Alex K
  Mathematical proof.. lifesagift 20 3445 26th September 2014, 17:01
Last Post: lifesagift
Information My proof for de morgans law LogicMaster 17 2315 29th May 2014, 19:55
Last Post: Stimbo
  Mathematician Claims Proof of Connection between Prime Numbers KichigaiNeko 10 4540 26th September 2012, 03:18
Last Post: Categories+Sheaves
  Need a proof (real analysis) CliveStaples 8 3629 2nd August 2012, 22:11
Last Post: CliveStaples
  Mathematical proof of the existence of God JudgeDracoAmunRa 20 8411 30th March 2012, 11:43
Last Post: JudgeDracoAmunRa
  Spot the Mathematical Fallacy Tiberius 16 5616 25th March 2010, 06:57
Last Post: Violet
  Mathematical claims of 'Bible Codes'...is there any truth in the maths? CoxRox 12 5320 9th January 2009, 17:23
Last Post: Tiberius



Users browsing this thread: 1 Guest(s)