3Blue1Brown

<div id="text-container" class="style-scope ytd-channel-name"><a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw">3Blue1Brown</a></div>

</div>

2.15M subscribers</div>

<div id="subscribe-button" class="style-scope ytd-video-secondary-info-renderer">SUBSCRIBED

</div>

<div id="description" class="style-scope ytd-video-secondary-info-renderer">A new and more circularly proof of a famous Wallis product for pi. Here's a supplemental blog post, expanding some of the rigor of this proofs, along with other interesting tidbits about surrounding topics that we didn't fit into the video: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?q=https%3A%2F%2Fwww.3blue1brown.com%2Fsridhars-corner%2F2018%2F4%2F17%2Fwallis-product-supplement-dominated-convergence&event=video_description&v=8GPy_UMV-08&redir_token=Ej2TnNhohhIfr2y5WZFz-kw7ivd8MTU3MDkzODA3OEAxNTcwODUxNjc4" target="blank" rel="nofollow">https://www.3blue1brown.com/sridhars-...</a> Special thanks to the following Patreon supporters: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?q=http%3A%2F%2F3b1b.co%2Fwallis-thanks&event=video_description&v=8GPy_UMV-08&redir_token=Ej2TnNhohhIfr2y5WZFz-kw7ivd8MTU3MDkzODA3OEAxNTcwODUxNjc4" target="blank" rel="nofollow">http://3b1b.co/wallis-thanks</a> If you also want to support videos like these, visit <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?q=https%3A%2F%2Fwww.patreon.com%2F3blue1brown&event=video_description&v=8GPy_UMV-08&redir_token=Ej2TnNhohhIfr2y5WZFz-kw7ivd8MTU3MDkzODA3OEAxNTcwODUxNjc4" target="blank" rel="nofollow">https://www.patreon.com/3blue1brown</a> Another approach to this product by Johan Wästlund: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?q=http%3A%2F%2Fwww.math.chalmers.se%2F%7Ewastlund%2Fmonthly.pdf&event=video_description&v=8GPy_UMV-08&redir_token=Ej2TnNhohhIfr2y5WZFz-kw7ivd8MTU3MDkzODA3OEAxNTcwODUxNjc4" target="blank" rel="nofollow">http://www.math.chalmers.se/_{wastlund...</a}> With more from Donald Knuth building off this idea: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?q=https%3A%2F%2Fapetresc.wordpress.com%2F2010%2F12%2F28%2Fknuths-why-pi-talk-at-stanford-part-1%2F&event=video_description&v=8GPy_UMV-08&redir_token=Ej2TnNhohhIfr2y5WZFz-kw7ivd8MTU3MDkzODA3OEAxNTcwODUxNjc4" target="blank" rel="nofollow">https://apetresc.wordpress.com/2010/1...</a> Music by Vincent Rubinetti: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?q=https%3A%2F%2Fvincerubinetti.bandcamp.com%2Falbum%2Fthe-music-of-3blue1brown&event=video_description&v=8GPy_UMV-08&redir_token=Ej2TnNhohhIfr2y5WZFz-kw7ivd8MTU3MDkzODA3OEAxNTcwODUxNjc4" target="blank" rel="nofollow">https://vincerubinetti.bandcamp.com/a...</a> ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?q=http%3A%2F%2F3b1b.co%2Frecommended&event=video_description&v=8GPy_UMV-08&redir_token=Ej2TnNhohhIfr2y5WZFz-kw7ivd8MTU3MDkzODA3OEAxNTcwODUxNjc4" target="blank" rel="nofollow">http://3b1b.co/recommended</a> Various social media stuffs: Website: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?q=https%3A%2F%2Fwww.3blue1brown.com&event=video_description&v=8GPy_UMV-08&redir_token=Ej2TnNhohhIfr2y5WZFz-kw7ivd8MTU3MDkzODA3OEAxNTcwODUxNjc4" target="blank" rel="nofollow">https://www.3blue1brown.com</a> Twitter: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?q=https%3A%2F%2Ftwitter.com%2F3blue1brown&event=video_description&v=8GPy_UMV-08&redir_token=Ej2TnNhohhIfr2y5WZFz-kw7ivd8MTU3MDkzODA3OEAxNTcwODUxNjc4" target="blank" rel="nofollow">https://twitter.com/3blue1brown</a> Patreon: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?q=https%3A%2F%2Fpatreon.com%2F3blue1brown&event=video_description&v=8GPy_UMV-08&redir_token=Ej2TnNhohhIfr2y5WZFz-kw7ivd8MTU3MDkzODA3OEAxNTcwODUxNjc4" target="blank" rel="nofollow">https://patreon.com/3blue1brown</a> Facebook: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?q=https%3A%2F%2Fwww.facebook.com%2F3blue1brown&event=video_description&v=8GPy_UMV-08&redir_token=Ej2TnNhohhIfr2y5WZFz-kw7ivd8MTU3MDkzODA3OEAxNTcwODUxNjc4" target="blank" rel="nofollow">https://www.facebook.com/3blue1brown</a> Reddit: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?q=https%3A%2F%2Fwww.reddit.com%2Fr%2F3blue1brown&event=video_description&v=8GPy_UMV-08&redir_token=Ej2TnNhohhIfr2y5WZFz-kw7ivd8MTU3MDkzODA3OEAxNTcwODUxNjc4" target="blank" rel="nofollow">https://www.reddit.com/r/3blue1brown</a></div>

</div>

<h4 id="title" class="style-scope ytd-metadata-row-renderer">Caption author (English)</h4>

<h4 id="title" class="style-scope ytd-metadata-row-renderer">Caption authors (Polish)</h4>

<div id="content" class="style-scope ytd-metadata-row-renderer"><a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/channel/UCQYPNNQExri6AG5NmQjmvNg">KhanAcademyPoPolsku</a><a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/channel/UCLY0sVSIlw5qERRhLSMk0tA">Fundacja Edukacja Przyszłości</a><a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/channel/UCBmbEBj4eybdApFesQCcc2w">Centrum Fizyki Teoretycznej PAN</a></div>

<h4 id="title" class="style-scope ytd-metadata-row-renderer">Caption author (Spanish)</h4>

<h4 id="title" class="style-scope ytd-metadata-row-renderer">Category</h4>

</div>

SHOW LESS</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">right, I thinkyou'regonnalikethis.00</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer active">wanttoshowyou a beautifulresultthatreveals a surprisingconnectionbetween a simple seriesoffractionsandthegeometryofcircles.00</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">unlikesomeotherresultslikethisthatyoumayhaveseenbefore,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">oneinvolvesmultiplyingthingsinsteadofaddingthemup.00</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">thevideoyou'reabouttowatchisparticularlyexcitingforusat3Blue1Brown00</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:26because</div>

<div class="cue style-scope ytd-transcript-body-renderer">itcameabout a littledifferentlyformostofthevideosthatwe'vedone.00</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">youstep back andthinkaboutit, thevalueofanykindofmath00</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:33presentation</div>

<div class="cue style-scope ytd-transcript-body-renderer">comesfrom a combinationoftheunderlyingmathandthen all ofthechoicesthatgointohowtocommunicateit, and00</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">almost all ofthecontentonthischannel, theunderlyingmathissomethingthat'swell-known in thefield.00</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">eitherbasedongeneraltheoryorsomeparticularpaper, andmyhopeisforthenoveltytocomefromthecommunication half, and00</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">thisvideotheresultwe'rediscussing, a veryfamous infinite productfor pi knownasthe Wallis Product, isindeedwellknownmath.However,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">we'llbepresentingis, toourknowledge, a more original proofofthisresult.For01</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:06context</div>

<div class="cue style-scope ytd-transcript-body-renderer">, after watching our video on the Basel problem,</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">01:09Sweether</div>

<div class="cue style-scope ytd-transcript-body-renderer">(?), the new 3b1b member who some of you may remember from the video about color and winding numbers, well,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">spentsome time thinkingabouttheapproachtaken in thatvideo01</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">wellasthinkingabouttheconnectionbetweenthe Basel problemandthe Wallis productandhestumbledinto a newproofofthe01</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:25relationship</div>

<div class="cue style-scope ytd-transcript-body-renderer">betweenthe Wallis productand pi. I01</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">, I'll leave open the possibility that an argument of this style is hidden somewhere in the literature beyond what our searching pulled up,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">I can at least say that it was found01</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:36independently</div>

<div class="cue style-scope ytd-transcript-body-renderer">, and that if it does exist out there, it has done a fantastic job hiding itself from the public view.01</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">withoutfurtherado, let'sdiveintothemath.01</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:48Consider</div>

<div class="cue style-scope ytd-transcript-body-renderer">theproduct 2/1 times 4/3 times 6/5 onandonandon01</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">whatwe'redoingisincluding all theevennumbersasthenumeratorsandoddnumbersasthedenominators.Of02</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:00course</div>

<div class="cue style-scope ytd-transcript-body-renderer">, all the factors here are bigger than 1, so as you go through the series, multiplying each new factor in one by one,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">resultkeepsgettingbiggerandbigger.02</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">fact, itturns out thatiteventuallygetsbiggerthanany finite limit, so in that sense,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">not super interesting.It just blowsuptoinfinity.Andontheotherhand, ifyoushiftthingsoverslightly, lookingat02</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">2/3 times 4/5 times 6/7 on and on,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">ofthosefactorsarelessthan 1, so theresultkeepsgettingsmallerandsmallerandthis time theseriesturns out toapproach02</div>

</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">whatifwemixthetwo? Ifyoulookedat 2/1 times 2/3</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">02:39times</div>

<div class="cue style-scope ytd-transcript-body-renderer">4/3 times 4/5 on and on like this where now02</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">partialproductsalongthewaykeepgoingupandthen down andthenupandthen down thenup a littlebitandthen02</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">a little bit less until all of these jumps and falls are of almost02</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">changeat all so nowitmustbeconvergingtosomekindof positive finite value, butwhatisthatvalue?</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">03:01Believe</div>

<div class="cue style-scope ytd-transcript-body-renderer">itornot,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">discoverthatthisequals pi dividedby 2 andtounderstandtheconnectionbetweenthisproduct,</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">03:09apparently</div>

<div class="cue style-scope ytd-transcript-body-renderer">unrelatedtocirclesand pi, we'regoingtoneedtotake a slightdigressionthrough a fewgeometrictools.03</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">a productive digression though since these are some useful ideas to have in your problem solving tool belt for all kinds of other math.03</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">setuphereinvolves a circlewithmanydifferentpoints03</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:29evenly</div>

<div class="cue style-scope ytd-transcript-body-renderer">spacedarounditandthenone additional specialpoint.Thisissimilartowhatwehad in thevideoonthe Basel problemwhere03</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">picturedtheseevenlyspacedpointsaslighthousesandthoughtofthatspecialpointas an observer.03</div>

</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">, the quantity we cared about involved looking at the distance between the observer and each lighthouse,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">takingthe inverse squareofeachofthosedistancesandaddingthem all up.03</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">iswhywehadthewhole narrative withlighthouses in thefirstplacesincethe inverse squarelawgave a reallynicephysical03</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:59interpretation</div>

<div class="cue style-scope ytd-transcript-body-renderer">tothisquantity: it was the total amountoflightreceivedbythatobserver.04</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">despitethatnicephysicalinterpretation,</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">04:08there's</div>

<div class="cue style-scope ytd-transcript-body-renderer">nothingmagicalaboutadding inverse squareddistancesthat just happenedtobewhat was usefulforthatparticularproblem.04</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">totackleournewproblemof04</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">2/1 times 2/3 times 4/3 times 4/5 and so on, we're going to do something similar,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">different in thedetails.Insteadofusingthe inverse squareddistances,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">lookatthedistancesthemselvesdirectly, andinsteadofaddingthemup04</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">bemultiplyingthem, giving a quantitythatI'llbereferringtoasthedistanceproductfortheobserver.That'llbeimportant.And04</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">thoughthisdistanceproduct, nolongerhas a nicephysicalanalogy04</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">still kind of want to illustrate it with lighthouses and an observer because well04</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">don'tknow.It'sprettyand also it's just morefunthanabstractgeometricpoints04</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">forthisproofofthe Wallis product,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">goingtoneedtwokeyfactsaboutthisdistanceproduct; twolittlelemmas. First, iftheobserverispositioned05</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:01halfway</div>

<div class="cue style-scope ytd-transcript-body-renderer">betweentwolighthousesonthecircle, thisdistanceproduct, thethingthatyougetbymultiplyingtogetherthelengthsof all theselines,</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">05:10works</div>

<div class="cue style-scope ytd-transcript-body-renderer">out to be exactly two.05</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">matter how many lighthouses there are.05</div>

</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:20second</div>

<div class="cue style-scope ytd-transcript-body-renderer">, if you remove one of those lighthouses and put the observer in its place this distance product from all of the remaining lighthouses05</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:28happens</div>

<div class="cue style-scope ytd-transcript-body-renderer">toequalthenumberoflighthousesthatyoustartedwith.05</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:34Again</div>

<div class="cue style-scope ytd-transcript-body-renderer">, no matter how many lighthouses there are.05</div>

</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">thosetwofactsseemcrazy, I agree.05</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">meanit'snotevenobviousthatthedistanceproducthereshouldwork out tobe an integer in eithercaseand05</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">, it seems super tricky to actually compute all of the distances and then multiply them together like this.05</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">itturnsout05</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:59there</div>

<div class="cue style-scope ytd-transcript-body-renderer">is a, well, a tricktothistrickycalculationthatmakesitquite simple.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">mainideaisthatthegeometricpropertyofthesepointsbeingevenlyspacedaround a circle06</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:09Corresponds</div>

<div class="cue style-scope ytd-transcript-body-renderer">to a reallynicealgebraicpropertyifweimaginethistobetheunitcircle in thecomplexplane06</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">eachofthoselighthousesnowsittingonsomespecificcomplexnumber06</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">ofyoumightrecognizetheseastherootsofunity06</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">letmequicklywalkthroughthisidea in caseanyofyouareunfamiliarthinkaboutsquaringoneofthesenumbers06</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">has a magnitudeofone so that'sgoingtostaythesame, butthe angle itmakeswiththe horizontal will double06</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">howsquareandcomplexnumbersworks06</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:39Similarly</div>

<div class="cue style-scope ytd-transcript-body-renderer">cubingthisnumberisgoingtotriplethe angle thatitmakeswiththehorizontal06</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">in general raising it to the nth power06</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:47multiplies</div>

<div class="cue style-scope ytd-transcript-body-renderer">the angle byn06</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">forexampleonscreenrightnow06</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:51there</div>

<div class="cue style-scope ytd-transcript-body-renderer">aresevenevenlyspacedpointsaroundtheunitcirclewhichI'llcalll0l1l2and so onand06</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">rotated in such a waythatl0issittingatthenumber 1 onthatrighthandside07</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">becausethe angle thateachoneofthesemakeswiththe horizontal is an integer multiple of 1/7 of a turn07</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:11Raising</div>

<div class="cue style-scope ytd-transcript-body-renderer">anyoneofthesenumberstotheseventhpower07</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:14Rotates</div>

<div class="cue style-scope ytd-transcript-body-renderer">youaroundtolandingonthenumberonein07</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:18Other</div>

<div class="cue style-scope ytd-transcript-body-renderer">words. These are all solutionstothe polynomial equation X totheseventh minus oneequalszero07</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">ontheotherhandwecouldconstruct a polynomial thathasthesenumbersasroots a totallydifferentwaybytaking X minus l007</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:37Times</div>

<div class="cue style-scope ytd-transcript-body-renderer">X minus l1 on and on and on up to X minus07</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">I mean you plug in any one of these numbers and that product will have to equal 0 and07</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:46Because</div>

<div class="cue style-scope ytd-transcript-body-renderer">thesetwodegreesevenpolynomialshavethesamesevendistinctrootsandthesameleadingterm07</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">just X to the seventh in both cases. They are in fact one and the same07</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">take a momenttoappreciate just what a marvelousfactthatisthisrighthandsidelookslikeitwouldbean08</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:04Absolute</div>

<div class="cue style-scope ytd-transcript-body-renderer">nightmaretoexpandnotonlyarethere a lotofterms08</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">writing down whatexactlyeachofthosecomplexnumbersisisgonnalandus in a wholemessofsinesandcosines08</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">becauseofthesymmetryofthesetupweknowthatwhen all ofthealgebraicdustsettles08</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">goingtosimplify down to just being X totheseventh minus one all oftheotherterms will cancel out and08</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">coursethere'snothingspecialabout 7 hereifyouhave n pointsevenlyspacedaround a circlelikethis08</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">aretheBrutesof X tothe n minus 1 equals 0 and08</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">youmight08</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">whythiswouldgive a nicesimplifyingtrickforcomputingthedistance08</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:42product</div>

<div class="cue style-scope ytd-transcript-body-renderer">thatwedefined a momentagoifyouconsidertheobservertobeanyothercomplexnumbernotnecessarilyonthecircleand08</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">youplug in thatnumberfor X thatrighthandside08</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:53There</div>

<div class="cue style-scope ytd-transcript-body-renderer">isgivingyousomenewcomplexnumberwhosemagnitudeistheproductofthe08</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:59distances</div>

<div class="cue style-scope ytd-transcript-body-renderer">betweentheobserverandeachlighthouse09</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">lookatthatlefthandsideitisa09</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:05dramatically</div>

<div class="cue style-scope ytd-transcript-body-renderer">simpler way to understand what that product is ultimately going to simplify down to09</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:10Surprisingly</div>

<div class="cue style-scope ytd-transcript-body-renderer">thismeansthatifourobserversitsonthesamecircleasthelighthousestheactualnumberoflighthouseswell09</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">won'tbeimportant.It'sonlythefractionofthewaybetweenadjacentlighthousesthatdescribesourobserver, which will comeintoplay09</div>

</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">fractionis f thenobservertothepowern09</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:32lands</div>

<div class="cue style-scope ytd-transcript-body-renderer">F of the way around a full circle so the09</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:36magnitude</div>

<div class="cue style-scope ytd-transcript-body-renderer">ofthecomplexnumberobservertothe N minus 1 isthe09</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:41Distance</div>

<div class="cue style-scope ytd-transcript-body-renderer">betweenthenumberoneand a point F ofthewayaround a fullunitcircle09</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">exampleonscreenrightnowwehavesevenlighthousesandtheobserverissitting09</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:52one-third</div>

<div class="cue style-scope ytd-transcript-body-renderer">ofthewaybetweenthefirstandthesecond so whenyouraisethecomplexnumberassociatedwiththatobservertotheseventhpower10</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">enduponethirdofthewayaroundthefullcircle so themagnitudeofobservertotheseven minus one10</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:07Would</div>

<div class="cue style-scope ytd-transcript-body-renderer">bethelengthofthischordrighthere?</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">10:10Which</div>

<div class="cue style-scope ytd-transcript-body-renderer">foronethirdofthewayaroundthecirclethathappenstobeaboutonepointseventhreeandrememberthisvalueis10</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:17quite</div>

<div class="cue style-scope ytd-transcript-body-renderer">remarkablythesameasthefulldistanceproductthatwecareabout10</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">couldincreaseordecreasethenumberoflighthousesandno matter what so longasthatobserverisonethirdofthewaybetween10</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:29Lighthouses</div>

<div class="cue style-scope ytd-transcript-body-renderer">wewouldalwaysgetthelengthofthissamechordasourdistanceproduct10</div>

</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:37General</div>

<div class="cue style-scope ytd-transcript-body-renderer">let'sdefine a specialfunctionforourselveschordof F. Which will meanforanyfractionF10</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">lengthof a chordcorrespondingtothatfractionof a unitcircle10</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">forexamplewhatwe just saw was chordofone-third10</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:52Actually</div>

<div class="cue style-scope ytd-transcript-body-renderer">, it's not so hard to see that chord of F. Amounts to the same thing as two times the sine of F10</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:58halves</div>

<div class="cue style-scope ytd-transcript-body-renderer">timestwopi11</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:01Which</div>

<div class="cue style-scope ytd-transcript-body-renderer">is 2 timesthesineof f pi, butsometimesit'seasierto just thinkofitaschordoff11</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">theresultwe've just shownisthatfor an observer f ofthewaybetweentwolighthouses11</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">total distance product as complicated as that might seem works out to be exactly11</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:19Chord</div>

<div class="cue style-scope ytd-transcript-body-renderer">of F. No matter, howmanylighthousesthereare11</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">in particular think about chord of 1/2</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">isthedistancebetweentwopointsontheoppositeendsof a unitcirclewhichis 2?</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">weseethatno matter howmanylighthousesthereareequallyspreadaroundtheunitcircle11</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:36putting</div>

<div class="cue style-scope ytd-transcript-body-renderer">an observer exactly halfway along the circle between two of them results in a distance product of precisely 2 and11</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">ourfirstkeyfact, so just tuckthatawayforthenextkeyfactimagineputtingtheobserverrightononeofthelighthouses11</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">, then of course the distance product is 0 the distance 0 lighthouse ends up annihilating all other factors11</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">supposewe just gotridofthatonetroublesomelighthouseandconsideredonlythecontributionsfrom all oftheotherones12</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">wouldthatdistanceproductwork out tobe?</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">nowinsteadofconsideringthe polynomial observertothe N minus 1, whichhas a rootat all ofthese n rootsofunity12</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">lookingatthe polynomial observertothe N minus 1</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">12:19Divided</div>

<div class="cue style-scope ytd-transcript-body-renderer">byobserver minus 1 whichhas a rootat all oftherootsofunityexceptforthenumber 1 itself12</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">a little algebra shows that this fraction is the same thing as 1 plus observer plus observer squared on and on and on12</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">toobservertothe N minus 1 and12</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">ifyouplug in observerequals 1 sincethat'sthenumberhe'ssittingonwhatdoyouget?</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">ofthetermsherebecomeone so itworks out toben12</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:48Which</div>

<div class="cue style-scope ytd-transcript-body-renderer">meansthe total distanceproductforthissetupequalsthenumberof original lighthouses?</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">thisdoesdependonthe total numberoflighthouses, butonly in a very simple way. I meanthinkaboutthis12</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">isincrediblethe total distanceproductthat an observersittingatoneofthelighthousesreceivesfrom all otherlighthousesispreciselyn13</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:07Where</div>

<div class="cue style-scope ytd-transcript-body-renderer">n is the total number of lighthouses including the ignored one that is our second key fact13</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">bythewayprovinggeometricfactswithcomplexpolynomialslikethisisprettystandard in math13</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">ifyouwentuptoyourlocalmathematicianandshowedhimor her thesetwofactsorotherfactslikethese13</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">quicklyrecognizeboththatthesefactsaretruth, andhowtoprovethemusingthemethods13</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">just showed and now so can you so next with both these facts in our back pocket?</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">seehowtousethemtounderstandtheproductthatwe'reinterested in andhowitrelatestopi13</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">thissetupwithandlighthousesevenlyspacedaround a unitcircleand13</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:50Imagine</div>

<div class="cue style-scope ytd-transcript-body-renderer">two separate observerswhatI'llcalledthekeeperandtheSailor13</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">thekeeperdirectlyononeofthelighthousesandputtheSailorhalfwaybetweenthatpointandthenextlighthouse14</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">ideahere will betolookatthedistanceproductforthekeeper14</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:04Divided</div>

<div class="cue style-scope ytd-transcript-body-renderer">bythedistanceproductfortheSailor, andthenwe'regoingtocomputethis ratio in two separate ways14</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">thefirstkeyfact.Weknowthatthe total distanceproductfortheSailoris 2 and14</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">distanceproductforthekeeperwell, it'szerosincehe'sstandingrightontopofone14</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">ifwegotridofthatlighthouse, thenbyoursecondkeyfacttheremainingdistanceproductforthatkeeperisn14</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">ofcoursebygettingridofthatlighthouse14</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">also gotten rid of its contribution to the sailors distance product so that14</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:37denominator</div>

<div class="cue style-scope ytd-transcript-body-renderer">nowhastobedividedbythedistancebetweenthetwoobserversand14</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:42Simplifying</div>

<div class="cue style-scope ytd-transcript-body-renderer">this just a littlebititmeansthatthe ratio betweenthekeepersdistanceproductandthesailorsis14</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">timesthedistancebetweenthetwoobservers all dividedbytwo14</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">wecould also computethis ratio in a differentwaybyconsideringeachlighthouse14</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:58Individually</div>

<div class="cue style-scope ytd-transcript-body-renderer">foreachlighthousethinkaboutitscontributiontothekeepersdistanceproductmeaning just itsdistancetothekeeper15</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:06divided</div>

<div class="cue style-scope ytd-transcript-body-renderer">byitscontributiontothesesailorsdistanceproductitsdistancetothesailorand15</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">wemultiply all ofthesefactorsupovereachlighthousewehavetogetthesame ratio in theendn15</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:18Times</div>

<div class="cue style-scope ytd-transcript-body-renderer">thedistancebetweentheobservers all dividedbytwo15</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">thatmightseemlike a super messycalculation15</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">as n getslargerthisactuallygets simpler foranyparticularlighthouse15</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">examplethinkaboutthefirstlighthouseafterthekeeper in a sense ofcounterclockwisefromhim15</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">is a bitclosertotheSailorthanitistothekeeper15</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:39specifically</div>

<div class="cue style-scope ytd-transcript-body-renderer">the angle fromthislighthousetothekeeperisexactlytwicethe angle fromthislighthousetotheSailorand15</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:47Those</div>

<div class="cue style-scope ytd-transcript-body-renderer">anglesaren'texactly proportional tothesestraight-linedistances15</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">as n getslargerandlarger15</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">correspondencegetsbetterandbetterandfor a verylarge n thedistancefromthelighthousetothekeeperisverynearly16</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:00twice</div>

<div class="cue style-scope ytd-transcript-body-renderer">thedistancefromthatlighthousetotheSailorand16</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">thesamewaylookingatthesecondlighthouseafterthekeeperithas an angle tokeeperdividedby angle tocelular ratio of16</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:12exactly</div>

<div class="cue style-scope ytd-transcript-body-renderer">4/3 which is very nearly the same as the distance to keeper divided by distance to sailor ratio as16</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">getslargeand16</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">thirdlighthouse l-3 isgoingtocontribute a fractionthatgetscloserandclosertosixfifthsas n isapproachinginfinity16</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">forthisproof16</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">goingtowanttoconsider all thelighthousesonthebottomofthecircle a littlebitdifferently16</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:36Which</div>

<div class="cue style-scope ytd-transcript-body-renderer">iswhyI'veenumeratedthem negative 1 negative 2 negative 3 and so onifyoulookatthatfirstlighthousebefore16</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">keeperithas a distancetokeeperoverdistancetosailor ratio thatapproaches16</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">2/3 as n approaches infinity, and then the second lighthouse before it L negative 2 here16</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:55contributes</div>

<div class="cue style-scope ytd-transcript-body-renderer">a ratio that gets closer and closer to16</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">4/5 and the third lighthouse L. Negative 3 contributes a fraction closer and closer to 6/7 and so on17</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:07Combining</div>

<div class="cue style-scope ytd-transcript-body-renderer">thisover all ofthelighthouses17</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">gettheproduct 2 over 1 times 2 over 3 times 4 over 3 times 4 over 5</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">17:16times</div>

<div class="cue style-scope ytd-transcript-body-renderer">6 over 5 times 6 over 7 on and on and on17</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">istheproductthatwe'reinterested in studyingand in thiscontexteachoneofthoseterms17</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:25reflects</div>

<div class="cue style-scope ytd-transcript-body-renderer">whatthecontributionfor a particularlighthouseisas n approachesinfinityand17</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">I say contribution I mean the contribution to this ratio of the keepers distance product to the sailors distance product17</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:37Which</div>

<div class="cue style-scope ytd-transcript-body-renderer">weknowateverystephastoequal n timesthedistancebetweentheobserversdividedby 2</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">whatdoesthatvalueapproachas n approachesinfinity?</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">, the distance between the observers is 1/2 of 1 over n of a full turn around the circle and17</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:55Since</div>

<div class="cue style-scope ytd-transcript-body-renderer">thisis a unitcircleits total circumferenceis 2 pi so thedistancebetweentheobserversapproaches PI dividedby N. And18</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:04therefore</div>

<div class="cue style-scope ytd-transcript-body-renderer">n times this distance divided by 2 approaches PI divided by 2</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">thereyouhaveit18</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">product 2 over 1 times 2 over 3 times 4 over 3 times 4 over 5 onandonandon18</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">approach pi dividedby 2</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">is a trulymarvelousresultandit'sknownasthe Wallis productnamedafter a 17thcentury18</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:26mathematician</div>

<div class="cue style-scope ytd-transcript-body-renderer">John Wallis who first discovered this fact in a way more convoluted way and also a little bit of trivia18</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">isthesameguywhodiscovered, orwellratherinventedtheinfinitysymbol18</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">actuallyifyoulook back atthisargument18</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">pulled a littlebitofsleightofhandtoodeep in formalityhere, whichtheparticularlymathematicallysophisticatedamongyoumighthavecaught18</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">wehavehereis a wholebunchoffactors18</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:55Which</div>

<div class="cue style-scope ytd-transcript-body-renderer">weknewmultipliedtogethertoget n timesthedistancebetweentheobserversdividedbytwo18</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">thenwelookedatthelimitofeachfactor19</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:03individually</div>

<div class="cue style-scope ytd-transcript-body-renderer">as n wenttoinfinityand19</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:05Concluded</div>

<div class="cue style-scope ytd-transcript-body-renderer">thattheproductof all ofthoselimitingterms19</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">toequalwhateverthelimitof n timesthedistancebetweentheobserversdividedbytwois19</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">whatthatassumesisthattheproductoflimitsisequaltothelimitofproductsevenwhenthere'sinfinitelymanyfactorsand19</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">kindofcommutingoflimits in infinite areaarithmeticwell19</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">notalwaystrueitoftenholds, butitsometimesfails19</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">, let me show you a simple example of a case where this kind of commuting of limits doesn't actually work out19</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">we'vegot a gridherewhereeveryrowhas a singlesevenandthen a wholebunchofones19</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">ifyouweretotakethe infinite productofeachrowyou just getsevenforeachoneofthem19</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">sinceeveryoneoftheseproductsis 7 thelimitoftheproductsis also 7</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">lookatwhathappensifyoutakethelimitsfirstifyoulookateachcolumn19</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">limitof a givencolumnisgoingtobeonesinceatsomepointit'snothingbutonce20</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">thenifyou'retakingtheproductofthoselimits20</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">just taking the product of a bunch of ones so you get a different answer namely one20</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:12Luckily20</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:13Mathematicians</div>

<div class="cue style-scope ytd-transcript-body-renderer">havespent a lotof time thinkingaboutthisphenomenon20</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">they'vedevelopedtoolsforquicklyseeingcertainconditionsunderwhichthisexchangingofthelimitsactuallyworks20</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">thiscase a particularstandardresultknownasdominatedconvergencequicklyassuresusthattheargument20</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">just showed will go through in full rigor for those of you20</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">areinterestedsweeterhaswrittenup a supplementalblog post tothisvideo?</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">20:36Which</div>

<div class="cue style-scope ytd-transcript-body-renderer">coversthosedetailsalongwithmanymorethings20</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">I should also say we need to be a little careful about how to interpret a product like this20</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:45Remember</div>

<div class="cue style-scope ytd-transcript-body-renderer">wehavecontributionsfromlighthouses20</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:48Counterclockwise</div>

<div class="cue style-scope ytd-transcript-body-renderer">fromthekeeperaswellaslighthousesclockwisefromthekeeperandwhatwedid was interleavethese in ordertogetourproduct20</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">thelighthousescounterclockwisefromthekeepercontributetwooveroneforoverthreesixoverfiveonandonandtheones21</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:04Clockwise</div>

<div class="cue style-scope ytd-transcript-body-renderer">fromthekeepercontributetwooverthreefouroverfivesixoversevenandlike I saidbeforeifyouplayaroundwiththose21</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:11Individual</div>

<div class="cue style-scope ytd-transcript-body-renderer">seriesyou'llfindthatthefirstonegetslargerandlargerandblowsuptoinfinityandthesecondonegetssmallerandsmaller21</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:19Approaching</div>

<div class="cue style-scope ytd-transcript-body-renderer">zero so it'sactuallyprettydelicatetomake sense out ofthisoverallproduct in termsofcomputingthetwohalves21</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:26Separately</div>

<div class="cue style-scope ytd-transcript-body-renderer">andthencombiningthemand a deed will findthatifyouintermixthesetwohalves21</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:32Differently</div>

<div class="cue style-scope ytd-transcript-body-renderer">forexampletakingtwiceasmanyfactorsfromoneofthemforeachfactorfromtheother21</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">couldget a differentresultfortheoverallproduct21</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">onlywhenyouspecificallycombinethem in thisone-for-onemannerthatyouget a productthatconvergestopi21</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">thisissomethingthat falls out ofthewaythedominatedconvergencejustifiesusandcommutinglimitsthewaywedidand21</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:53again</div>

<div class="cue style-scope ytd-transcript-body-renderer">formoredetailsseetheSupplementalpost21</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:56Still</div>

<div class="cue style-scope ytd-transcript-body-renderer">thoseare just technicalitiestheconceptualgistforwhat'sgoingonhereisexactlywhatwe just showed22</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">in fact after doing all that work22</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">wouldbeashamednottotake a quickmomenttotalkaboutonemoreneatresultthat falls out ofthisargument22</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:15Arguably</div>

<div class="cue style-scope ytd-transcript-body-renderer">, this is the coolest part of the whole proof we see we can generalize this whole discussion think back to when we discovered our22</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:22First</div>

<div class="cue style-scope ytd-transcript-body-renderer">keyfactwherewesawthatyoucouldnotonlyconsiderplacingtheSailorprecisely, halfwaybetweenlighthouses22</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">anyfraction F ofthewaybetweenadjacentlighthouses in thatmoregeneral setting thedistanceproductfortheSailor22</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">necessarilytwo, butit was chordof F. WhereF22</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">thatfractionofthewaybetweenlighthousesand?</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">wegothroughthesamereasoningthatwe just didwiththeSailoratthislocationinsteadandchangednothingelse22</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">we'llfindisthatthe ratio ofthekeeper'sdistanceproducttothesailorsdistanceproductisnown22</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:56Times</div>

<div class="cue style-scope ytd-transcript-body-renderer">thedistancebetweenthemdividedbychordof F. Whichapproaches F times 2 PI23</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:03divided</div>

<div class="cue style-scope ytd-transcript-body-renderer">bychordof F as n getslarger23</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">, in the same way as before23</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">couldalternativelycalculatethisbyconsideringthecontributionsfromeachindividuallighthouseif23</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">takethe time toworkthis out the (k)thlighthouseafterthekeeper will contribute a factorofK23</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:22Divided</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:23Minus</div>

<div class="cue style-scope ytd-transcript-body-renderer">f2this ratio and all thelighthousesbeforethekeepertheycontributethesamething23</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">you're just plugging in negative valuesfor K. Ifyoucombine all thosecontributionsover all nonzerointegersK23</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:36Where</div>

<div class="cue style-scope ytd-transcript-body-renderer">in the same way as before you have to be careful about how you bundle the positive and negative K terms together?</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">you'llgetisthattheproductof K dividedby K minus F over all nonzerointegers K isgoingtoequalF23</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:50times</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:51Chord</div>

<div class="cue style-scope ytd-transcript-body-renderer">of F. Put anotherwaysincechordof F is 2 timesthesineof f pi24</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">productisthesameas f timestwo pi dividedbytwotimessineof f pi24</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:06Which</div>

<div class="cue style-scope ytd-transcript-body-renderer">is f pi. Oversineof f pi?</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">rewritingthis just a littlebitmorewhatyougetis a prettyinterestingfactsineofF24</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:16Times</div>

<div class="cue style-scope ytd-transcript-body-renderer">pi is equal to F24</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">timesthisreallybigproducttheproductof 1 minus F over K over all nonzerointegersK24</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">whatwefoundis a wayto express sineof X as an infinite product, whichisreallycool24</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">youthinkaboutit, so notonlydoesthisproofgiveusthewallaceproduct24</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:37Which</div>

<div class="cue style-scope ytd-transcript-body-renderer">isbeingcredible in itsownrightit also generalizestogiveustheproductformulaforthesine24</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">what'sneataboutthatisthatitconnectstohoweuler?</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">24:46Originally</div>

<div class="cue style-scope ytd-transcript-body-renderer">solvedthebaselproblemthesumthatwesaw in thepreviousvideo24</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">was looking at this very infinite product for sine. I mean connecting these formulas for pi24</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">circlesisonething24</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">connectingthemtoeachotherisanotherthingentirelyand25</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">againifyouwantmoredetailson all ofthischeck out thesupplementaryblogpost25</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">video was supported in partbyUdacity25</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">actuallyrecentlydid a bitofworkwithUdacityhelpingtoincorporatesomeofthe linear algebracontent25</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">doneintotheircoursesandthereason I was happy formyworktobe a verysmallpartofwhattheydois25</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">I agree with Udacity s mission namely they want to democratize higher education25</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">all know that learning computer science and AI can get you a good job25</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:30There's</div>

<div class="cue style-scope ytd-transcript-body-renderer">just way more demand than supply at the moment25</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">for a lotofpeoplegoingthrough a traditionalfour-yearprogramtoget a degree just isn't a possibilityeither25</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:40Financially</div>

<div class="cue style-scope ytd-transcript-body-renderer">orlogistical II andUdacitygoalistogivepeopleaccesstocoursesthat25</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:45Meaningfully</div>

<div class="cue style-scope ytd-transcript-body-renderer">preparethemforin-demandjobs in techandnot just withlecturecontent25</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:50There's</div>

<div class="cue style-scope ytd-transcript-body-renderer">lotsofpractice real projectsgoodfeedback all ofthethingsthatgointo genuine skillbuilding25</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">areveryseriousaboutgettingpeopleemployableskills25</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">ofthenewestofferingsthatI'drecommendyoucheck out istheirschoolofAI26</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:02Whether</div>

<div class="cue style-scope ytd-transcript-body-renderer">you just wanttolearn a littlebitof Python orgodeepintocomputervision26</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">got a lotofdifferentpathsandifyougotoUdacitycomm26</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">3 b-1b that lets them know that you came from this channel26</div>

</div>

<div class="cue-group-start-offset style-scope ytd-transcript-body-renderer">:14Really</div>

<div class="cue style-scope ytd-transcript-body-renderer">, if this is a skill you've been wanting to develop go take a lookEnglish</div>

</div>

</div>

</div>

3Blue1Brown

Attachments