May 30th, 2014, 04:35 AM  #1 
Member Joined: May 2014 From: India Posts: 87 Thanks: 5 Math Focus: Abstract maths!  A discovery which could shake the foundation of complex numbers!
I am a 14 year old (please continue reading despite my age) Math lover and explorer. I just discovered something I earnestly want to share with experienced Mathematicians. It is for this reason I have joined this site. What I discovered is that i(root of 1) can be expressed as a continued fraction. The proof is fairly simple but the discovery is important. I want to safely share this discovery because I believe that this discovery has the potential to prove that i is real(though I couldn't prove it myself since I could not find a good definition for real numbers.) I want it safe because even though I don't want to gain popularity, I don't want others to copy my proof without credit. I tried arXive but it requires you to be endorsed and I am only a 14year old schoolboy. Is there another way to publish it "safely"? Please help me as the excitement is difficult to contain. Last edited by skipjack; June 7th, 2014 at 07:09 PM. 
May 30th, 2014, 05:34 AM  #2 
Senior Member Joined: Apr 2014 From: zagreb, croatia Posts: 234 Thanks: 33 Math Focus: philosophy/found of math, metamath, logic, set/category/order/number theory, algebra, topology 
It would be unbelievable if i was real, because it simply isn't! A real number is a member of the set of real numbers, which is defined as a set that satisfies certain 16 axioms. Most common way to prove that such a set exists is by socalled Dedekind cutsintuitively you define a real as a convergent sequence of rationals (fractions). Well, that's not exactly it, but let's say it is. https://en.wikipedia.org/wiki/Real_number https://en.wikipedia.org/wiki/Dedekind_cut Complex numbers are defined as the set of all ordered pairs of reals, with addition "componentwise", and multiplication as follows: (a, b) (c, d) = (acbd, bc+ad) So i^2 = (0, 1) (0, 1) = (1, 0). The reason for such a definition of multiplication is the fact that C is then an algebraically closed fieldeach polynomial in C has a zero. Then again, I'm not saying you're wrong, stranger things have happened. Last edited by raul21; May 30th, 2014 at 06:01 AM. 
May 30th, 2014, 06:15 AM  #3  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Quote:
Quote:
i = [i; ] but you could write it in alternate ways like i = [i1; 1] if you like. i is not a member of $\mathbb{R},$ the real numbers. Quote:
An alternate method is axiomatic: we list certain properties that the real numbers have which are jointly enough to define them. Both approaches are discussed, for example, here: Construction of the real numbers  Wikipedia, the free encyclopedia Edit: Actually, Dedekind cuts (as Raul mentioned) are also quite popular. All three methods work, just pick the one you like best. Quote:
(It's arXiv, by the way.) I'm glad you find math so exciting! I hope you're able to keep that feeling. Last edited by skipjack; June 7th, 2014 at 07:12 PM.  
May 30th, 2014, 06:42 AM  #4 
Senior Member Joined: May 2014 From: Allentown PA USA Posts: 111 Thanks: 6 Math Focus: dynamical systen theory  To Rishabh
Don't be afraid, Rishabh. Most of the My Math Forum community are good people. I myself am somewhat foggy on some rather important concepts, given that I am middleaged. You have probably have heard of Srinivasa Ramanujan, Sir Chandrasekhara Venkata Raman, and Subrehmanyan Chandrasekhar. There is a friend of mine, whom I haven't seen in many years, who has his master's degree in math from the University of Madras. Good luck! Sincerely yours, Carl J. Mesaros 
May 31st, 2014, 02:15 AM  #5 
Member Joined: May 2014 From: India Posts: 87 Thanks: 5 Math Focus: Abstract maths! 
Thanks to all who have replied. I myself was very unsure that such a thing could happen, so I wanted to discuss it. But before that, I have already fallen in love with this site.! Here's the original discovery. To prove: âˆš(1) = 1/(11/(21/(11/(2...))))  1; Proof: Code: Let x = 1/(11/(21/(11/(2...))))  1; â‡’x + 1 = 1/(11/(2(x+1))); â‡’x + 1 = (1  x)/(x); â‡’x^2 + x  x +1 = 0; â‡’x^2 + 1 = 0; â‡’x = âˆš(1); Hence proved. 1/(11/(21/(11/(2...))))  1 is real. (Not very sure of this proof.) Proof: Axioms/theorems used in this proof: â€¢Division is closed under real numbers. â€¢Subtraction is closed under real numbers. Statement: Since this continued fraction involves only division and subtraction, I don't see why it shouldn't be real. Afterword: I don't forcefully assert this proof. I want to know what exactly is wrong with the proof. I will be very thankful to those who help. 
May 31st, 2014, 03:05 AM  #6 
Senior Member Joined: Apr 2014 From: zagreb, croatia Posts: 234 Thanks: 33 Math Focus: philosophy/found of math, metamath, logic, set/category/order/number theory, algebra, topology 
Maybe the error lies in the fact you assumed in advance the continued fraction converges.

May 31st, 2014, 12:05 PM  #7 
Senior Member Joined: May 2014 From: Allentown PA USA Posts: 111 Thanks: 6 Math Focus: dynamical systen theory 
Rishabh, Whether you realize it or not, complex variables are very useful in some areas, such as electrical engineering. You're one up on me, kiddo! I personally deal with complex numbers very rarely. Good luck and may you find mathematics very good! Sincerely yours, Carl J. Mesaros 
June 1st, 2014, 01:40 AM  #8  
Member Joined: May 2014 From: India Posts: 87 Thanks: 5 Math Focus: Abstract maths!  Quote:
This however still poses a very important question. If it diverges then it will be equal to infinity. But i is not equal to infinity!!!  
June 1st, 2014, 01:43 AM  #9 
Member Joined: May 2014 From: India Posts: 87 Thanks: 5 Math Focus: Abstract maths!  
June 1st, 2014, 04:27 PM  #10  
Math Team Joined: May 2013 From: The Astral plane Posts: 2,265 Thanks: 932 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
Dan  

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