June 1st, 2014, 06:25 PM  #11 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,654 Thanks: 2632 Math Focus: Mainly analysis and algebra  No. We use notation like $\lim f(x) = \infty$ to denote divergence, but it doesn't mean any equality in the usual sense of the word. The fact that your continued fraction diverges means that it doesn't converge to any value, so it tells you nothing with the possible exception that you can't make $i$ by adding and dividing real numbers. Last edited by skipjack; June 7th, 2014 at 07:16 PM. 
June 2nd, 2014, 01:34 AM  #12 
Member Joined: May 2014 From: India Posts: 87 Thanks: 5 Math Focus: Abstract maths!  I use only two hypotheses: 1) Division is closed for real numbers. 2) Subtraction is closed for real numbers. I think the first one is wrong because of division by zero. P.S. I suggest you read through my work. It has left senior mathematicians puzzled!!! 
June 2nd, 2014, 06:03 AM  #13 
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 
You're clearly using more than that because you're manipulating a(n infinite) continued fraction which brings in all kinds of continuity and convergence issues. You may wish to read about regularization before continuing further: Euler summation  Wikipedia, the free encyclopedia Abel's theorem  Wikipedia, the free encyclopedia Borel summation  Wikipedia, the free encyclopedia CesÃ ro summation  Wikipedia, the free encyclopedia Ramanujan summation  Wikipedia, the free encyclopedia MittagLeffler summation  Wikipedia, the free encyclopedia 1 + 2 + 3 + 4 + â‹¯  Wikipedia, the free encyclopedia 1 + 1 + 1 + 1 + â‹¯  Wikipedia, the free encyclopedia etc. 
June 2nd, 2014, 07:51 AM  #14  
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  Quote:
Maybe just one more thing about convergence/divergence. If a sequence of real numbers $a_1$, $a_2$, ... (which your cf is, it's even a sequence of rationals) diverges, it doesn't converge, so there's no a â‚¬ R such that for any $\epsilon$>0 there's $n_0$ â‚¬ N such that for any n > $n_0$ the absolute value of $a_n$  a_n_0 <$\epsilon$. There are 2 types of divergence. We can get a sequence like 1, 2,...for which we say it diverges to infinity, it will exceed any value. Or 1, 2, ... for which we say it diverges to  infinity, it will become smaller than any value. The other type is a bounded sequence, like 1, 0, 1, 0, ... Last edited by skipjack; June 7th, 2014 at 07:19 PM.  
June 2nd, 2014, 08:38 AM  #15 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 
DELETED Disregard , i must have laid a brain fart. It is interesting to see the Continued Fraction truncated at regular intervals $$x + 1 \ = \ \frac{1}{11} \ = \ \frac{1}{0} \ \ $$ undefined $$x + 1 \ = \ \frac{1}{1 \frac{1}{2  1}} \ = \ \frac{1}{11} \ = \ \frac{1}{0} \\ $$ undefined again And the pattern continues , you get undefined at every truncation. Nice trick BTW , I upvoted this thread to 5 bars. Last edited by agentredlum; June 2nd, 2014 at 09:23 AM. Reason: More clear explanation + erased brain fart. :) 
June 2nd, 2014, 08:28 PM  #16 
Member Joined: May 2014 From: India Posts: 87 Thanks: 5 Math Focus: Abstract maths!  Sorry!
I just realized that I am a big unknowing fool standing between great mathematical geniuses.!!! Forgive me for this stupid thread. 
June 2nd, 2014, 08:31 PM  #17 
Member Joined: May 2014 From: India Posts: 87 Thanks: 5 Math Focus: Abstract maths! 
There may be a positive aspect though. Firstly, I learnt many new things. Secondly, I may write a book someday about faulty lawdefying proofs and include this one.(among others which I have discovered such as the proof that 1=2, 1+2+4+8+16+... = 1, etc.) LOL

June 2nd, 2014, 08:34 PM  #18 
Member Joined: May 2014 From: India Posts: 87 Thanks: 5 Math Focus: Abstract maths! 
Here's another proof that i is real. I know the mistake, you also try to find it. (âˆš(1))^2 = âˆš(1) * âˆš(1); = âˆš(1*1); = âˆš1; = 1; =>âˆš(1) = âˆš(1) = 1 
June 2nd, 2014, 08:36 PM  #19  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,654 Thanks: 2632 Math Focus: Mainly analysis and algebra  Quote:
All I would say is that at 14 you are unlikely to shake the foundations of any part of mathematics. If you think you have found something that does, you have probably made a mistake somewhere. So perhaps asking what is wrong would be better than claiming a new discovery.  
June 3rd, 2014, 05:24 AM  #20  
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:
Anyone else have similar admissions to make?  

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