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 Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion

 September 15th, 2019, 06:46 AM #1 Senior Member   Joined: Dec 2015 From: somewhere Posts: 734 Thanks: 98 Easy proof of 22/7>pi Let $\displaystyle x=22/7-\pi\;$ and suppose $\displaystyle 22/7<\pi$ or $\displaystyle x<0$. Since x is negative then $\displaystyle sgn(x)=-1+2H(x)<0$. $\displaystyle sgn(x)=-1+2\lim_{s\rightarrow \infty} \frac{1}{1+e^{-sx}}=-1+2\lim_{t_s \rightarrow \infty} \frac{1}{1+e^{t_s x}}$. Since the limit of $\displaystyle 1+e^{-sx}$ converges then : $\displaystyle sgn(x)=-1+2=1$ which is a contradiction of statement :$\displaystyle x<0 \: \equiv \: sgn(x)<0$ . $\displaystyle sgn(x)>0$ or $\displaystyle x>0$ proves $\displaystyle \frac{22}{7}>\pi$. Last edited by idontknow; September 15th, 2019 at 06:56 AM. September 15th, 2019, 07:20 AM #2 Global Moderator   Joined: Dec 2006 Posts: 21,035 Thanks: 2271 You incorrectly assumed that $\displaystyle \lim_{s\rightarrow \infty} \frac{1}{1+e^{-sx}} = \lim_{t_s \rightarrow \infty} \frac{1}{1+e^{t_s x}}$. Thanks from topsquark and idontknow September 16th, 2019, 05:19 PM #3 Senior Member   Joined: Jan 2014 From: The backwoods of Northern Ontario Posts: 393 Thanks: 71 I tend to use simple arithmetic whenever possible for my "proofs." I happen to have memorized the first eight digits of pi: 3.1415926 Then I took my trusty calculator to find the approximate value of 22/7 I got 3.142857143 September 17th, 2019, 12:29 AM   #4
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Quote:
 Originally Posted by idontknow Let $\displaystyle x=22/7-\pi\;$ and suppose $\displaystyle 22/7<\pi$ or $\displaystyle x<0$. Since x is negative then $\displaystyle sgn(x)=-1+2H(x)<0$. $\displaystyle sgn(x)=-1+2\lim_{s\rightarrow \infty} \frac{1}{1+e^{-sx}}=-1+2\lim_{t_s \rightarrow \infty} \frac{1}{1+e^{t_s x}}$. Since the limit of $\displaystyle 1+e^{-sx}$ converges then : $\displaystyle sgn(x)=-1+2=1$ which is a contradiction of statement :$\displaystyle x<0 \: \equiv \: sgn(x)<0$ . $\displaystyle sgn(x)>0$ or $\displaystyle x>0$ proves $\displaystyle \frac{22}{7}>\pi$.
Trying to solve the problem hard way is not the best way. Try to solve problems by an easier, simpler way following basic deductions. September 30th, 2019, 02:33 AM   #5
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Quote:
 Originally Posted by skipjack You incorrectly assumed that $\displaystyle \lim_{s\rightarrow \infty} \frac{1}{1+e^{-sx}} = \lim_{t_s \rightarrow \infty} \frac{1}{1+e^{t_s x}}$.
We are supposing that $\displaystyle x<0$ , then $\displaystyle e^{-sx}$ becomes $\displaystyle e^{tx}\;$, $\displaystyle t>0$. Tags 22 or 7>pi, 22 or 7>pi, easy, proof Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post do like math Calculus 3 August 18th, 2012 12:51 AM HairOnABiscuit Real Analysis 6 October 12th, 2009 10:33 AM HairOnABiscuit Real Analysis 3 October 10th, 2009 10:09 AM Tritive Algebra 3 December 4th, 2008 06:22 PM peri123 Algebra 2 August 5th, 2008 05:16 AM

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