Our server costs ~$56 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: 21st November 2017, 11:33

Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
The Mathematical Proof Thread
#51
RE: The Mathematical Proof Thread
Hello everyone.  I was working on a proof yesterday and thought that it was cool enough to share here. Hence, the proposition and my direct proof of it is below.

With that said, in creating this thread, it was my intent for people to be able to post whatever beauty they encountered in mathematics for as long as possible. Thus, I apologize if my post violates community guidelines or causes any inconvenience to the staff and/or AF members. 

Proposition: For every four real numbers a,b,c, and d, sqrt(a^2 + b^2)*sqrt(c^2 + d^2) ≥ ac + bd.

Note the proposition can be re-written in “if-then” form, which yields the following equivalent statement: “if a,b,c, and d are real numbers, then sqrt(a^2 + b^2)*sqrt(c^2 + d^2) ≥ ac + bd.
 
Proof.


 

Notes





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














Reply
#52
RE: The Mathematical Proof Thread
This morning, I completed a proof of the following proposition: there exist four distinct real numbers a,b,c,d such that exactly four of the numbers ab,ac,ad,bc,bd,cd are irrational. (IMO, this proposition is pretty cool).  Note, the proof method to use here is proof by contradiction. Well, happy proof-writing AF members and anyone else.  

Hint 1



Hint 2 (Analysis of Proof/Spolier)



P.S. I'm not a math guru/nerd/expert, so I apologize if my writing/reasoning is inelegant.
"I'm fearful when I see people substituting fear for reason." Klaatu, from The Day The Earth Stood Still (1951)














Reply
#53
RE: The Mathematical Proof Thread
Kernel, I think you're ready for this link: https://artofproblemsolving.com/community

(or have I shown it to you, before?)
Reply
#54
RE: The Mathematical Proof Thread
(21st September 2017, 05:28)pocaracas Wrote: Kernel, I think you're ready for this link: https://artofproblemsolving.com/community

(or have I shown it to you, before?)
You have shown it to me before, and it was in relation to math proofs.  The site looks pretty cool: I've certainly got a lot more to learn and could  always improve.  Out  of curiosity, is the site  being recommended due to poor/incorrect mathematical reasoning/proof-writing in my previous post? Thanks.

P.S. In general, to all AF members, if I've made an error in a proof/mathematical work that I've posted here, then if it's okay with you, could you please let me know, so that I can post a corrected version? Thanks, and I appreciate everyone's participation in this thread.  Live long and prosper.
"I'm fearful when I see people substituting fear for reason." Klaatu, from The Day The Earth Stood Still (1951)














Reply
#55
RE: The Mathematical Proof Thread
(21st September 2017, 10:59)Kernel Sohcahtoa Wrote:
(21st September 2017, 05:28)pocaracas Wrote: Kernel, I think you're ready for this link: https://artofproblemsolving.com/community

(or have I shown it to you, before?)
You have shown it to me before, and it was in relation to math proofs.  The site looks pretty cool: I've certainly got a lot more to learn and could  always improve.  Out  of curiosity, is the site  being recommended due to poor/incorrect mathematical reasoning/proof-writing in my previous post? Thanks.

Actually, I posted this link first and read your proof after.
It's always good to have a bunch of people interested in the same thing discussing these things you're so passionate about... and I think you'll find a much more eager audience there... and, mainly, a lot of teachers.


Personally, I'm not a fan of showing this kind of proposition... my physics background sees no use in them! Tongue




(21st September 2017, 10:59)Kernel Sohcahtoa Wrote: P.S. In general, to all AF members, if I've made an error in a proof/mathematical work that I've posted here, then if it's okay with you, could you please let me know, so that I can post a corrected version? Thanks, and I appreciate everyone's participation in this thread.  Live long and prosper.

Will do, if I ever spot it. Wink
Reply
#56
RE: The Mathematical Proof Thread
Kernel's humility and diplomacy is breathtaking and astounding.

All I've got going for me is my ability to constantly speak my mind. Or perhaps rather: my inability to fail to constantly speak my mind Tongue
We do not change our minds. Our minds change us.
Reply
#57
RE: The Mathematical Proof Thread
(21st September 2017, 13:27)pocaracas Wrote:
(21st September 2017, 10:59)Kernel Sohcahtoa Wrote: You have shown it to me before, and it was in relation to math proofs.  The site looks pretty cool: I've certainly got a lot more to learn and could  always improve.  Out  of curiosity, is the site  being recommended due to poor/incorrect mathematical reasoning/proof-writing in my previous post? Thanks.

Actually, I posted this link first and read your proof after.
It's always good to have a bunch of people interested in the same thing discussing these things you're so passionate about... and I think you'll find a much more eager audience there... and, mainly, a lot of teachers.


Personally, I'm not a fan of showing this kind of proposition... my physics background sees no use in them! Tongue




(21st September 2017, 10:59)Kernel Sohcahtoa Wrote: P.S. In general, to all AF members, if I've made an error in a proof/mathematical work that I've posted here, then if it's okay with you, could you please let me know, so that I can post a corrected version? Thanks, and I appreciate everyone's participation in this thread.  Live long and prosper.

Will do, if I ever spot it. Wink

I appreciate your thoughtful response, pocaracas.  I didn't realize that there were a lot of teachers on that site along with so many people interested in mathematics: it seems like a literal goldmine of wonderful ideas and meaningful exchanges based in curiosity and friendship.

With that said, as to your question contained within the hide box, I would say that always being open to the possibility of how I may be mistaken, is a trait that I value and try to exhibit; it can certainly spare a person some embarrassing moments, especially if that individual has committed an error. 

In your post, you said that you have a physics background. Out of curiosity, are you more interested in applied mathematics or pure mathematics (or do you like both)? Thanks again for your response.
"I'm fearful when I see people substituting fear for reason." Klaatu, from The Day The Earth Stood Still (1951)














Reply
#58
RE: The Mathematical Proof Thread
(21st September 2017, 13:52)Kernel Sohcahtoa Wrote: In your post, you said that you have a physics background. Out of curiosity, are you more interested in applied mathematics or pure mathematics (or do you like both)? Thanks again for your response.

I'd say I like both, but I prefer the applied variety.
Give me 3 dimensions and I can still visualize the thing.... go higher and things become tricky.
Reply
#59
RE: The Mathematical Proof Thread
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.
Reply
#60
RE: The Mathematical Proof Thread
Also, this 'u' is commonly used in arithmetic as the number 1. The neutral set member in multiplication.

Our numeric system with the common + - * / operations forms a division ring.

I prefer an analytical approach to math.
Reply



Possibly Related Threads...
Thread Author Replies Views Last Post
  Million Dollar Prize for Math proof of NP problems emilynghiem 6 2061 22nd February 2015, 00:47
Last Post: vorlon13
  "Gödel's ontological proof" proves existence of God Belac Enrobso 41 8986 9th February 2015, 03:22
Last Post: Alex K
  Mathematical proof.. lifesagift 20 4154 26th September 2014, 17:01
Last Post: lifesagift
Information My proof for de morgans law LogicMaster 17 2699 29th May 2014, 19:55
Last Post: Cyberman
  Mathematician Claims Proof of Connection between Prime Numbers KichigaiNeko 10 4924 26th September 2012, 03:18
Last Post: Categories+Sheaves
  Need a proof (real analysis) CliveStaples 8 4023 2nd August 2012, 22:11
Last Post: CliveStaples
  Mathematical proof of the existence of God JudgeDracoAmunRa 20 9257 30th March 2012, 11:43
Last Post: JudgeDracoAmunRa
  Spot the Mathematical Fallacy Tiberius 16 5960 25th March 2010, 06:57
Last Post: Violet
  Mathematical claims of 'Bible Codes'...is there any truth in the maths? CoxRox 12 5880 9th January 2009, 17:23
Last Post: Tiberius



Users browsing this thread: 1 Guest(s)