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File: EmergingE ''' <link rel="alternate" type="application/rss+xml" ''' href="/rss.xml" title="RSS Feed"> ********> width="25%" |>> ''' <a title="Subscribe to my feed" ''' rel="alternate" ''' href="https://www.solipsys.co.uk/rss.xml"> ''' <img style="border-width: 0px;" ''' src="https://www.feedburner.com/fb/images/pub/feed-icon32x32.png" ''' align="middle" ''' alt="" />Subscribe!</a> _ ''' <a href="https://twitter.com/ColinTheMathmo"> ''' <img src="https://www.solipsys.co.uk/new/images/TwitterButton.png" ''' title="By: TwitterButtons.net" ''' width="212" height="69" ''' alt="@ColinTheMathmo" ''' /></a> <<| ---- My latest posts can be found here: * ColinsBlog ---- Previous blog posts: * ImpossibleToTranslate * WaitingInVain * NonRepeatingDecimals * RationalRepeats * WhyIsItLovely * CompilingCryptoConnections * ExploringConnectionsBetweenCryptoSystems * ElwynBerlekampHasLeftUs * RootCauseAnalysisAndThePhotocopierQuestion * TheUpDownTides * TheForeAftTide * TheSidewaysTide * WrappingUpWrappingUpTheEarth * TheOtherWrappingTheEarthProblem * WrappingTheEarth * TheRingOfSteel * RoundingUpTheRopes * OtherOtherOtherRopeAroundTheEarth * RopeAroundTheEarthRefined * TheOtherRopeAroundTheEarth * ElementaryEstimates * LatitudeCorrection * JustGiveMeTheAnswer * MoreMusingOnPollardRho * IdleThoughtsAboutPollardRho * WhenOptimisingCodeMeasure * ADogCalledMixture * AnotherPayPalScam * WhyTopPostingHasWon * UnexpectedInteractionOfFeatures * ArchimedesHatBoxTheorem * ConsideringASphere * ToLinkOrNotToLink * GenericAdviceForWritingAThesis * JustTeachMyChildTheMaths * NotASpectatorSport * LeftTruncatablePrime * TheDoctorAndTheLawyer * FourPointsTwoDistancesProof * MeetingRonGraham * NapkinRingVersusSphericalCap * TheFourPointsPuzzle * RadiusOfTheEarthPartTwo * GrepTimingAnomaly * TheRadiusOfTheEarth * ThisWorksToCureMyHiccoughs * PerhapsWeSavedOne * ThinkingAboutMastodon * DisappearingTrainsOnVirgin * TheIndependenceGame * OneOfMyFavouritePuzzles * ThinkingAboutRecursion * MemorisingTheTube * SpikeySpheres * SurprisinglyQuick * AnUnexpectedFraction * YouHaveToAdmireTheirOptimism * RepresentativesMatter * PythagorasByIncircle * APuzzleAboutPuzzles * HowNotToDoTwitter * Calculating52FactorialByHand * SmallThingsMightNotBeSoSmall * NotIfYouHurry * FactoringViaGraphThreeColouring * AnotherProofOfTheDoodleTheorem * WhenObviousIsNotObvious * GraphThreeColouring * TheDoodleTheorem * BeCarefulWhatYouSay * TheMutilatedChessboardRevisited * AMirrorCopied * TheOtherOtherRopeAroundTheEarth * PhotocopyAMirror * ThePointOfTheBanachTarskiTheorem * SieveOfEratosthenesInPython * FastPerrinTest * RussianPeasantMultiplication * FindingPerrinPseudoPrimes_Part2 * FindingPerrinPseudoPrimes_Part1 * TheUnwiseUpdate * MilesPerGallon * TrackingAnItemOnHackerNews * HackerNewsUserAges * PokingTheDustyCorners * ThereIsNoTimeForThis * PublicallySharingLinks * LearningTimesTables * GracefulDegradation * DiagrammingMathsTopics * OnTheRack * SquareRootByLongDivision * BeyondTheBoundary * FillInTheGaps * SoftwareChecklist * NASASpaceCrews * TheBirthdayParadox * TheTrapeziumConundrum * RevisitingTheAnt * TheAntAndTheRubberBand * IrrationalsExist * MultipleChoiceProbabilityPuzzle * RandomEratosthenes * WrappingUpSquareDissection * DissectingASquarePart2 * DissectingACircle * DissectingASquare * AnOddityInTennis * DecisionTreeForTennis * DecisionTreesInGames * AMatterOfConvention * DoYouNourishOrTarnish * BinarySearchReconsidered * TwoEqualsFour * TheLostPropertyOffice * TheForgivingUserInterface * SettingUpRSS * WithdrawingFromHackerNews ---- Additionally, some earlier writings: * RandomWritings. * ColinsBlog2010 * ColinsBlog2009 * ColinsBlog2008 * ColinsBlog2007 * ColinsBlogBefore2007 ******** I've re-read this, and it's really very abbreviated and could do with some nice charts, /etc./ I'm working on it ... [[[>60 ********> | *x* | *c(x)* | | 2 | 0.693147... | | #?# | #1.000000...# | | 3 | 1.098613... | | 4 | 1.386295... | | 5 | 1.609439... | ******** There is a reason why taking the square root halves the rubbish - it's because $(1+\epsilon/2)^2$ is $1+\epsilon+\epsilon^2/4$, and when $\epsilon$ is small enough, $\epsilon^2$ effectively vanishes. So $(1+\epsilon/2)^2\approx 1+\epsilon$, provided $\epsilon$ is small. ********< Hence $\sqrt{1+\epsilon} \approx 1+\epsilon/2$ ]]] Take 10, and square root many times over. After a while you get 1 + rubbish. Taking the square root of that halves the rubbish, so we can deduce that $10^\epsilon = 1+c_{10}\epsilon$. I've put the subscript on the $c$ because if you do this again but start with 5 instead of 10 you get a different constant, $c_5$. So now we have a function, $c(x)$ where you start with $x$, successively square root, compute the constant, and that's the value of the function. If we do this there are lots of questions to ask about the function $c$: * Is it well defined? * Is it continuous? and so on. But we can observe that $c(2)<1$ and $1<c(3)$, and probably $c(2)<c(x)<c(3)$ when $x$ is between 2 and 3, so is there a value of $x$ that gives $c(x)=1$? The answer is yes, the answer is $e$. ---- |>> | |>> <<<< Prev <<<< ---- ImpossibleToTranslate <<| | : | |>> >>>> Next >>>> ---- TheBookIsNotAlwaysRight ... <<| | ---- ********> ''' <a href="https://mathstodon.xyz/@ColinTheMathmo"> ''' <img src="https://www.solipsys.co.uk/images/Mastodon_Mascot.png" ''' width="256" height="280" ''' alt="https://mathstodon.xyz/@ColinTheMathmo" ''' /></a> ******** ''' <a href="https://mathstodon.xyz/@ColinTheMathmo/">You can follow me on Mathstodon.</a> _ _ _ _ [[[> ''' <a href="https://twitter.com/ColinTheMathmo">Of course, you can also<br>follow me on twitter:</a> ''' <a href="https://twitter.com/ColinTheMathmo"> ''' <img src="https://www.solipsys.co.uk/new/images/TwitterButton.png" ''' title="By: TwitterButtons.net" ''' width="212" height="69" ''' alt="@ColinTheMathmo" ''' /></a> ''' <img src="/cgi-bin/CountHits.py?EmergingE" alt="" /> ]]] ********< ---- !! Send us a comment ... 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