Researchers thought this was a bug (Borwein integrals) 

imagine copying lol

Games have been doing this for _decades._

Lol the universe is like mojang

The developers of Real Life™ just left a bug in production and hoped no one would notice.

Math and physics seem to contain some undocumented easter eggs.

Good point! If anyone out there has a deeper knowledge of computer algebra systems, I'm all ears.

@@3blue1brown Ask Matt Parker himself to do so 😂

The answer is π times 692235940415362523136988414491285998468620532382124599554066975879968202372479018941687306133557125141812009840009662733497578477395741589958741155007862285485649171111258286647871898412035813448185128487166238219335182872053769745063205146240398270221977832380760762866554366743397019522289256347615462644913261775369992728315584923236659323759817418582764754173499371387884058167010542953584434449476393697721676981883264752309900228411652423246081739021978704316749310333533596904537502580519003591630854375995694511316758712127072335981655643021189629703319518996608891858801563606731511756259150271536904664925444915995745598487882850973342179949112232261107451564475708164124601869338680457040736426834176357325238700023154772340405663484960868000544476177063934327405358840986142240740495891233632352852053087368646776262360895352822595554176491656178820976720387079767602962842304015276653872951276656719564661860009852322150747843167248021400524688931060413853949705429841350499311344844142812690878735649021359350878799892991941300536391836009746220081646980020619328507232729433224792490941993693654589654207336860144043824383616426523896328586666972201974975363869745131277430423497779482704923699635266814730743056122797451467295167944104959148848306171227734538923653674351260090426832081683750824578884795592847739029407231100114031692028834847718052811109661505435074338037197807509927683710026016782011198945921757041861903371723076024299552942686154078275262558274383125240246903963660244565495743790100779385689120612914314126748249032328644943967606168810945505133744051503793356677613465767506133403785838880077428117672171491305463631982985278470240463605873903007823368419732452411249428087806823995726037033029954428007284645945501678886834962638266386697203029172069359055069598160085557611071250819586513883262334808499877279404265739246453854314818930473784012514484630065265839504131463613716847280435096127167440453437234925013899740472517629852858007350702055473349166597916007035221009345839579731778913437555757452869569584725521765148238483120629334049015846611258643709781106104555540382601644817619693116271703781814763254333627569201647746337166728676209480537183667033744980348862985594703362234073685730010342405696049810927652018855284359782308568790335680407039194097771231043231125155243319999767116121609430970384357269423481232518264366416525210424503728896257581964154317685655227495297650147999531562443287526368243680227419154845562643905990854032891584723971919362819173221100539566110801807612543052603376782159573210538409554672405396295376724610561 / 466192705877572353389835231940601222951670101070134561239973049203355736797679164397624010041403280350403890172469013022072991611009891420460184365109844228257723354893359318552165960075193563963432495384489969443453032619043283947700320752061908507268402779707752507800584175761024396184554311926902174604278990716347817088215671830701294237737416084870577308225709433802172004179436883634224430186712907011255416169182951793876196020581124124871790191131026190817826295390668469484921153061628957182315532723627541561158527229962601975323545963581536503234088278520594697564268665056763039200953239157471488828427155355325822506814512931380692989201738194683671358027936731158164868428465160293810220914942856829006898677125278247051066719835366903281173005006414793603140852470513412338483125238870107400749975506910479167617492429188364161622090075380794841274064530078229588882401307373157838975174030799988659511414398333740739995186397781300193196193202757302833310640180721583180401458815210422678535674359307400703540019934939449079991546032209790021358416751689023180874723355242538833041051750335417189450271841435425842357705548951520412541807195011790313725815748442413653722842379292155259290874590674053279873682477462608416130078159361725049569136018863847769692711434304284648779994670136133745537730928927221014707549796285465899515205737181699779348683309919359212078797505708653203063359044435493651560670169020403706390659381796376355011534232215252329581727323206390983456393841689668653574596495767581620929318398055395068946540290994175198141078405809664168143560635261493106792900807310948239120108625189411525965104535724046516823181955565824022550341414576665251862252517262971555570824456334217133885793566352181827821799733460800562650876183474302696913558747118873383788058316498534666549582395993896633701984857946104957599858399846326694339088056360491291651580401291235916846039420291797011958951903525756596083400499833765015610160614682572190562581996175482897349716398200856207446583669457688398879025589194587622790093420738922616627264007078524712707438700538087126382407147484598708878421359007944754077319874892451013305146739264196759332202156279663574573700221915659282202010270420525069461223193977455730690186787162810767959480118208312050676001116374540365682589708346162179047550107133580021937933978889261168248964550472603798305071959551200598606625251939526329942481081494978482611041645840498095348015739875843098411344378730289107163926051684865403472243934262440308576997235702810712358963888128180945434447661046236283823959416606140230636235659383407828994007185098833311796358669296501389331388587322635430609807372093200683593750000000000000000000000000000000000000000000000000000 which has 137 decimals equal to 0.

@@user-gc8cy5tx1w christ wow

@@user-gc8cy5tx1w holy mother of Gauss

I remember turning my homework paper to 'landscape' to solve Fourier transforms 'by hand' in order to fit them on one line.

That's the most EE thing I've heard in a while, and I work as a plant electrician....

I thought exactly the same thing, I studied undergraduate electrical engineering 30 years ago which was very heavy on laplace and fourier transforms, and convolutions. This video would have helped me understand them infinity better back then!

Could you please explain how Fourier transform comes into picture in your case ?

@@HemantKumar-xn8mn The radius I was machining decremented by .01" from 9.500" - 8.500". When the control got down to , for example, 9.130, 9.130-.01=9.12. Not so. the variable read 9.119999999... . Then when the control got down to say, 8.729999999..., instead it read 8.7299999999...8. This happened on every piece I machined. No one could explain why.

@@SUNRA131 this is probably actually due to floating point imprecision and not due to the problem featured in this video. basically, theres only a finite amount of floating point values that can be represented with a certain number of bits, and since some numbers dont perfectly translate to a corresponding floating point value, itll choose the nearest one instead. most of the time this works fine, but sometimes it doesnt. a good example of this is if you try doing 0.1 + 0.2 in many programming languages, itll compute to 0.30000000000000004. entering 9.13-0.01 into python returns 9.120000000000001

@@EinsteinsBarber Makes sense. Thanks.

@@SUNRA131 Going into it a little bit more: Computers represent everything as binary, including floating point numbers. The way floating point works is you divide your 32 bits (or 64, or 128... you get the picture) into 2 distinct parts. The exponent, and the mantissa. This is analogous to scientific notation in decimal: 1.2*10^5 has 2 parts: the exponent (10^5) and the mantissa (1.2). What ends up happening is that since we have a limited number of binary bits for both the exponent and mantissa, we end up with gaps where certain numbers cannot be represented exactly (without using more bits). In addition to that, it seems that nobody uses the error handling defined in the relevant standard to detect when a number is not representable. This can lead to compounding errors when an inaccurate representation happens in the middle of a multi-stage computation. If you're really curious, look up IEEE 754 on wikipedia :)

yes

If you want to guide someone to a destination, show them the whole map before giving individual instructions. That way if they make a wrong turn, they can have some sense that they’re going the wrong direction. Landmarks and reviewing the map partway through are important for humans learning how to get somewhere.

I don't think that's a spoiler, rather that's a hook. Like movies doing "you must be wondering how I got here" type. Hooks are really important in story telling as that builds the interest in the subject matter. The actual Spoiler in this case is the relationship between the two graphs via Fourier Transform.

Don't worry. It is just two ways of making presentations. While Grant does claim his is superior (in some of his other videos), not everyone agrees. I suspect I would enjoy your storytelling style.

even in storytelling: foreshadowing or even straight up giving answers ahead of time to give a sense of dramatic irony is a useful tool for creating hitchcock-esque suspense in a situation where surprise is not sufficient for making the story good. It's one thing to know *what* happens, another to see *how* it happens, and sometimes knowing what happens makes you wonder how

hi

E

Hi, Destin.

i hate averages

Laminar transform

Congratulations on the Bachelors, that's outstanding! And thanks for such kind words, it means a lot to me.

I think your space bar might be broken

@@TheOneAndOnlyZelenkaGuru utala pi toki ala (impossible, 100% fail)

@@TheOneAndOnlyZelenkaGuru it's probably just a moving averages problem 🤷🏽‍♂️ 🤣😂🤣

@@TheOneAndOnlyZelenkaGuru i think it might come from using a foreign language keyboard on an iphone. when i was learning Mandarin, if i typed an English sentence the space bar would add extra large spaces for some words.

Born to be a sommelier.

And as just a German, you think to yourself why they're putting boron in that wine.

​@@deaconmaldonado7947BORn to be a sommelier

Arnold Blackback approves.

@@mortenbund1219All the other elements argon

What a time to be alive! Thank you. Also what a good timing! The last episode of Veritasium was also about the fast Fourier transform and there, Derek mentioned you! :-)

Bless your soul! Your videos are the only thing that bring me sanity

I like the hint about multiplying large numbers being related to convolution. It took me until well after grad school to realize that the long multiplication I was taught in second grade, was actually a convolution.

@@stevenspencer306 Really? Seems interesting!

@@Math4e Are you holding on to your papers?

Yep! This is the foundation of all signal processing! Takes me back to my analog systems and signals class!

😒

Im also an electrical engineer student and we see this next semester. afterso many classes I realize the entire world can be broken down into vectors and sin() cos().

@@sebagomez4647 It's even cooler than that. Using the trigonometric functions is convenient because you're familiar with them already, and they're easy to generate with analog circuits. The Laplace transform and Z transform generalize this further to also take complex arguments (instead of a real number x). And in digital signal processing, all hell breaks loose -- Why not transform any function using a rectangular wave? Why not transform them using quantized waves? Look up leaflet transforms [correction: WAVElet transform].

Systems and signals is the class that makes you appreciate Fourier and Laplace Transforms, and math in frequency domain / complex numbers as litteral magic. The trick is that litterally any real world function, and many "mathland" function can always have their Fourier transform taken and be expressed as an infinite sum of sinusoids or complex exponentials (which are easier to work with), and then you just do regular multiplication and perform the inverse transform and you have the answer. One of our jokes is that "laplace is god" because its just that much easier for solving differential equations. (And most high level physics equations are differential equations in their most generalized form)

But we would have to multiply by 2cosx

How can we check this if it is true or not

@@kaanetsu1623 I could probably do it rn but I'm busy; but just use a calculator/desmos no? If not desmos use a graphing engine and input the function

That's not the same as the series from before, because the numbers were all decreasing by -1/2, your suggesting to do 1/2^n

@@kaanetsu1623 one could study the convergence of the series towards PI

it's not weird. it's nice to find out that there are things you don't understand that will work themselves out in a very elegant way.

Not at all.. it's pretty much the story of my life :D Downside is I really have to put in some discipline to not be binging on interesting content too much :P (or maybe it is, but in that case I love to be weird)

I find these so calming and beautiful, despite never really being good at maths. There’s such a sense of elegance and awe to these big concepts, and they always make me feel like I’m experiencing something beautiful.

Not at all! Looking "under the hood" & getting an explanation of How Stuff Works is fun for the Curious, whether they're going into math/manufacturing/car repair/etc or not

yes

Anything specific you're hoping to learn? Or any specific contexts where you saw them and were confused?

Me too! I was always confused how convolutions seems to be meaning different things at once, like folding and multiplying functions and doing f(g(x)) ..

@@3blue1brown As a chemical engineer, the only context I've learned them in is just for how to use them to take inverse integral transforms (basically just using the definition of convolution). I'd love to see more about the motivation and intuition behind that definition

@@ollerich32 f(g(x)) is composition, not convolution.

@@3blue1brown I find this topic in my statistics classes..i would like if u cover this in context of convolution of probability distributions.

The very best hour of my undergraduate was a day where Peter Borwein, 10 minutes into our scheduled hour long analysis class on a hot summer day, chatting about anything but the course material, said "I didn't want to teach today anyways", and we spend an hour just talking about mathematics and science. I would pay good money for a recording of that hour.

Woah nice! Was it analysis in R^n and general metric spaces or more like measure theory and functional analysis?

Had no idea they were professors at SFU! I've just started my first year at SFU as an undergraduate majoring in data science.

Very cool that they are teaching at SFU. SFU really deserves more credits than it gets. Despite all the trash talk from UBC, SFU seems to be quite strong in several departments.

@@MaximBod123 I am doing DS at Michigan. Seems like the SFU program has quite a bit of business focus that is absent in our program. Goes to show how underdefined the discipline is ig

I was thinking the same thing 😅 Szeretnél velem inni egy pohár bort?

Yes and Grant has made what, about 5-10 hours total of videos in this manner for his channel? While in your math classes, you get 40 hours or so of content for every course. RU-vid will always win out for ‘most interesting’ content. A good in-person educator will take the best of what is online though and bake that into the daily teaching.

Maths pronouns: they,them lol

Once he showed that moving average it made click in my head and all the lost knowledge about fourier and convolutions from my university came back to me.

@@Herdatec I flashed back instantly to second year college and getting a B- in Signals and Systems. Heard him say sinc(x) and everything repressed came back

Yeah, god bless learning filter design many years ago...

Yeah must be, but that's not me lol

It's also not really nice when you don't know or better yet understand anything in the video from start to finish. That's me.

Well, as one science communicator on youtube puts it: "Physics is everything"...

Only that there is absolutely no relation whatsoever between telomeres and this. It's an artificial relation that only exists in your head, sorry.

Thats not how telomeres work. They aren’t “used up” they are just buffers.

this person seems to know what they are doing, I know this is 3 months old, but hoe do you think they work

If this didn't immediately trigger you Fourier transform reflex as an electrical engineer you would have grounds to sue whatever school gave you the degree. That sort of negligence would be unheard of.

He uses a Python library called Manim to make them.

@@seneca983 He created Manim! :)

I get you. I study molecular immunology, far from the math lands too, but these videos help me grasp the wonderful elegance of mathmatical problemsolving. Fascinating stuff!

For what it is worth, what you are doing is essentially the Laplace transform. You just first note that the integral is even, so you only worry about the positive half of the axis.

It's not too crazy once you realize you're just trying to eliminate the pesky x in the denominator

Guessing exp(-ax) is not crazy necessarily. It is done often in physics, because 1/x diverges when integrated, so one uses a strongly decaying function like exp(-ax) to "help" make it converge faster, then you remove the "help" at the end. A similar trick is used in quantum electrodynamics where the Coulomb force has a potential V(x) ~ 1/x. The exp(-ax) factor corresponds to if the photon actually had mass a, and then at the end of the calculation we set a = 0 because photons are actually massless.

Yep, you're definitely describing the laplace transform and it is actually a generalization of the fourier transform

I'm only just beginning to actually intuitively grasp the Fourier transform over the last year or so of some really excellent videos coming out, but now I got the Laplace Transform to try and figure out! lol

And he's so cultured that he didn't use the word niche, he said "esoteric".

They aren’t that niche and complex but yeah! Master work. ⚙🕰

@@05degrees they absolutely are lol. Most people never even go past solving a triangle. For most people even basic differential calculus is completely foreign

These topics are certainly ... complex

Fourier transform and convolutions are not exactly niche ... It is bread an butter in electronics, to name a few: control theory, communications, signal processing.

I agree. Having to learn something for a test takes the fun out of the thing that I'm trying to learn.

Please elaborate on the formula of the egg 🎉

@@caseyj1144 If I remember correctly, the formula didn't exist. There were close approximations. The book as a whole explored lots of mathematical mysteries both solved an unsolved.

You gave me so much nostalgia with that comment! I loved trying hopelessly to understand math and physics problems as a kid but still getting exposed to some interesting ideas.

We use them in dealing with the waves produced by an earthquake in geotechnical engineering.

Electrical engineering is all about abusing complex numbers to cheat math and save time. The Fourier transform is probably my favorite function in all of math because of how powerful it is. (Along with the rest of the time->frequency domain transforms, we had a joke that laplace is god in college because it made differential equations so much easier, not to mention how it sinplifies every linear circuit into a simple transfer function multiplication and convert back to time domain, not that it was always easy to do so.)

I've had my own experience like that. I'm not any college student but yes sometimes some topics which I study really pop out on Grant's channel after some days which is a cool coincidence.