e (Euler's Number) is seriously everywhere | The strange times it shows up and why it's so important

Sign up with brilliant and get 20% off your annual subscription: brilliant.org/MajorPrep/
STEMerch Store: stemerch.com/
Support the Channel: / zachstar
PayPal(one time donation): www.paypal.me/ZachStarYT
Instagram: / zachstar
Twitter: / imzachstar
Join Facebook Group: / majorprep
Animations: Brainup Studios (email: [e-mail])
Timestamps/Extra Resources
2:42 - Derangements (en.wikipedia.org/wiki/Derange...)
3:11 - Optimal Stopping (en.wikipedia.org/wiki/Optimal...)
5:25 - Infinite Tetration (en.wikipedia.org/wiki/Tetration)
6:25 - 1958 Putnam exam question (mks.mff.cuni.cz/kalva/putnam/...)
9:33 - Fourier Transform (GIF credit to 3blue1brown, check out his video on the FT here: • But what is the Fourie... )
10:56 - Gamma Function (en.wikipedia.org/wiki/Gamma_f...)
12:53 - Casimir Effect Paper (arxiv.org/pdf/hep-th/9901011.pdf)
13:20 - Higher Dimensional Spheres (en.wikipedia.org/wiki/Volume_...)
►My Setup:
Space Pictures: amzn.to/2CC4Kqj
Magnetic Floating Globe: amzn.to/2VgPdn0
Camera: amzn.to/2RivYu5
Mic: amzn.to/2BLBkEj
Tripod: amzn.to/2RgMTNL
Equilibrium Tube: amzn.to/2SowDrh
►Check out the MajorPrep Amazon Store: www.amazon.com/shop/zachstar

[@ojasbhagavath5484]

Both English professors and Math professors agree that 'e' is everywhere.

[@jimo9555]

😂👍

[@guythatdoesthings4935]

Good one lol

[@ojasbhagavath5484]

@@sthenios7026 eeeeeeeeeeeeeeeee.

[@ltloxa1159]

"You can't write a text problem without using the letter e"

[@binmahin7184]

Comment of the Year

[@gubx42]

Imagine a bank that pays 100% interest per year, after a year you have... $0, because it's a scam.

[@DreadKyller]

Imagine a bank that pays 100% interest per year, but it's really that they pay 1,000,000% interest after 10,000 years.

[@joaoviniciusrm]

It's not a scam, it's called Argentina

[@ShamoKwok]

You don't have any money in the bank

[@herbie_the_hillbillie_goat]

A bank pays 100% interest? Now we're talking imaginary numbers.

[@smanzoli]

@@joaoviniciusrm that would be a bit of interest + a lot of inflation (monetary correction)...

[@zoeymccann124]

e is everywhere, literally. even in the word everywhere four times

[@rsfakqj10rsf-33]

underrated

[@elgordobondiola]

It isn't in this conjunction

[@augusto256]

wich is the number of times where a fraction gets it's closest approximation

[@splenden2235]

@@songe_d.automne the

[@zebran4]

e isn't litteraly everywhere. For instance, it isn't in my couch.

[@ivanfranco458]

ALL I KNOW FROM ENGINEERING SCHOOL IS THAT E = 3

[@takvacs]

I'm almost certain we went to different schools. There's about 2/e chance.

[@Someguy-cs6cj]

Yeah same. e=3=π

[@chrisryan6464]

@@Someguy-cs6cj e=3=pi it.s almost never used, and shouldn.t... that's why pi is always 3.14 when you divide the circumference to the diameter, AND NOT 3... and with e the smaller you go the more it goes towards 2.25something

[@Guztav1337]

@Chris Ryan No, that is not it. In Engineering e=3=π. Same goes for 2+2=5 for really large values of 2. (i.e. Under a uniform distribution 2+2=5 in 12.5% of the times)

[@chrisryan6464]

@@Guztav1337 I have heard of 2+2=5 before but never looked into it... what's the deal?

[@brookej0]

I feel like 'e' is some snippet of our universe's source code that we weren't supposed to discover and the developers must be really pissed off about it 😂

[@paulwolf3302]

It comes from geometric algebra.

[@AirshipToday]

@@paulwolf3302 No. It comes from compound interest

[@carsonllorente481]

@@paulwolf3302 you’re probably thinking of pi, e doesn’t come from geometrical algebra

[@kiwiboy1999]

Definitely does seem trancendant, even with increased dimensions from our own. There's something weird in that.

[@AirshipToday]

@@kiwiboy1999 😂😂

[@koktszfung]

The number 1 is seriously everywhere

[@Guztav1337]

We are number ONE!

[@Cjnw]

…and that happens to be e to ( 2 * pi times the square root of -1 )

[@JustinSulak]

You know what’s amazing is that your comment was posted 1 month ago 😮

[@nT-sp4qd]

11:11

[@nickl6752]

When he was talking about convergence, I was like this is why I start at 1.

[@Tiqerboy]

One of the first applications of 'e' I remember is in the case of radioactive decay because the rate of decay is proportional to the amount that is left.

[@javierduenasjimenez7930]

I also remember that there was an exact formula on the reducement of sound intensity while it "moved" trough the air.

[@pseudoduckk3029]

Yeah radioactivity too, also in growth and decay of current in inductive circuits, simple harmonics, wave motion(strings and electromagnetic), basically e is everywhere but I like only when it is in differentiation lol

[@m19u3l11]

Underrated comment

[@repeater64]

Yeah I think quite a lot of the places it comes up is as solutions to differential equations due to the property of e to the power of x differentiating to itself. That explains the cooling curve, damped harmonic motion, radioactive decay, and probably explains some of the other things in this video.

[@justgame5508]

3:59 Note to self, always go in the 38th place to an interview

[@gabrielonibudo5710]

what if it's 101 people

[@martinclever342]

@@gabrielonibudo5710 actually 101/e~37.2

[@mykekinkster440]

agreed!

[@dyllandonze744]

This only works if they are interviewing 100 people and the employer decides to use this strategy in hiring

[@martinclever342]

@@dyllandonze744 It works for 101 people as well: 99/e < 36.5 < 100/e 101/e < 37.5 < 102/e

[@malou5290]

d/dx(eˣ) = eˣ is mindblowing really

[@zachstar]

Even though I'm so used to it, I still find it very weird.

[@telecorpse1957]

@@zachstar For me, as someone who didn't really get into calculus yet, it seems the least weird thing about it. Isn't that the precise definition of e? Is it less weird that if you divide a circle's circumference by its diameter, you get π?

[@zachstar]

Position will match. Plug in 4.844 into the equation e^t, you'll get the same thing as velocity and acceleration, they're all the same function.

[@CalculatedRiskAK]

@@AndreasNilsson96 The position does match. d/dx(e^x) explains exactly why. Take the derivative of position, you get velocity. Take the derivative of velocity, you get acceleration. Those three values are all the same for e^x.

[@AndreasNilsson96]

@@CalculatedRiskAK I was wrong, you're right. that make a lot of sense, thanks guys.

[@jokerbug1985]

Press e to Pay Respects.

[@sureshn.b.bhatta7116]

E

[@anim8dideas849]

e

[@josephmutuku7678]

e

[@phamnguyenductin]

2.718281828...

[@that_one_guy934]

2.718281828459045235360287471352662497757247093699959574966967627724076630353 547594571382178525166427427466391932003059921817413596629043572900334295260 595630738132328627943490763233829880753195251019011573834187930702154089149 934884167509244761460668082264800168477411853742345442437107539077744992069 551702761838606261331384583000752044933826560297606737113200709328709127443 747047230696977209310141692836819025515108657463772111252389784425056953696 770785449969967946864454905987931636889230098793127736178215424999229576351 482208269895193668033182528869398496465105820939239829488793320362509443117 301238197068416140397019837679320683282376464804295311802328782509819455815 301756717361332069811250996181881593041690351598888519345807273866738589422 879228499892086805825749279610484198444363463244968487560233624827041978623 209002160990235304369941849146314093431738143640546253152096183690888707016 768396424378140592714563549061303107208510383750510115747704171898610687396 965521267154688957035035402123407849819334321068170121005627880235193033224 745015853904730419957777093503660416997329725088687696640355570716226844716 256079882651787134195124665201030592123667719432527867539855894489697096409 754591856956380236370162112047742722836489613422516445078182442352948636372 141740238893441247963574370263755294448337998016125492278509257782562092622 648326277933386566481627725164019105900491644998289315056604725802778631864 155195653244258698294695930801915298721172556347546396447910145904090586298 496791287406870504895858671747985466775757320568128845920541334053922000113 786300945560688166740016984205580403363795376452030402432256613527836951177 883863874439662532249850654995886234281899707733276171783928034946501434558 897071942586398772754710962953741521115136835062752602326484728703920764310 059584116612054529703023647254929666938115137322753645098889031360205724817 658511806303644281231496550704751025446501172721155519486685080036853228183 152196003735625279449515828418829478761085263981395599006737648292244375287 184624578036192981971399147564488262603903381441823262515097482798777996437 308997038886778227138360577297882412561190717663946507063304527954661855096 666185664709711344474016070462621568071748187784437143698821855967095910259 686200235371858874856965220005031173439207321139080329363447972735595527734 907178379342163701205005451326383544000186323991490705479778056697853358048 966906295119432473099587655236812859041383241160722602998330535370876138939 639177957454016137223618789365260538155841587186925538606164779834025435128 439612946035291332594279490433729908573158029095863138268329147711639633709 240031689458636060645845925126994655724839186564209752685082307544254599376 917041977780085362730941710163434907696423722294352366125572508814779223151 974778060569672538017180776360346245927877846585065605078084421152969752189 087401966090665180351650179250461950136658543663271254963990854914420001457 476081930221206602433009641270489439039717719518069908699860663658323227870 937650226014929101151717763594460202324930028040186772391028809786660565118 326004368850881715723866984224220102495055188169480322100251542649463981287 367765892768816359831247788652014117411091360116499507662907794364600585194 199856016264790761532103872755712699251827568798930276176114616254935649590 379804583818232336861201624373656984670378585330527583333793990752166069238 053369887956513728559388349989470741618155012539706464817194670834819721448 889879067650379590366967249499254527903372963616265897603949857674139735944 102374432970935547798262961459144293645142861715858733974679189757121195618 738578364475844842355558105002561149239151889309946342841393608038309166281 881150371528496705974162562823609216807515017772538740256425347087908913729 172282861151591568372524163077225440633787593105982676094420326192428531701 878177296023541306067213604600038966109364709514141718577701418060644363681 546444005331608778314317444081194942297559931401188868331483280270655383300 469329011574414756313999722170380461709289457909627166226074071874997535921 275608441473782330327033016823719364800217328573493594756433412994302485023 573221459784328264142168487872167336701061509424345698440187331281010794512 722373788612605816566805371439612788873252737389039289050686532413806279602 593038772769778379286840932536588073398845721874602100531148335132385004782 716937621800490479559795929059165547050577751430817511269898518840871856402 603530558373783242292418562564425502267215598027401261797192804713960068916 382866527700975276706977703643926022437284184088325184877047263844037953016 690546593746161932384036389313136432713768884102681121989127522305625675625 470172508634976536728860596675274086862740791285657699631378975303466061666 980421826772456053066077389962421834085988207186468262321508028828635974683 965435885668550377313129658797581050121491620765676995065971534476347032085 321560367482860837865680307306265763346977429563464371670939719306087696349 532884683361303882943104080029687386911706666614680001512114344225602387447 432525076938707777519329994213727721125884360871583483562696166198057252661 220679754062106208064988291845439530152998209250300549825704339055357016865 312052649561485724925738620691740369521353373253166634546658859728665945113 644137033139367211856955395210845840724432383558606310680696492485123263269 951460359603729725319836842336390463213671011619282171115028280160448805880 238203198149309636959673583274202498824568494127386056649135252670604623445 054922758115170931492187959271800194096886698683703730220047531433818109270 803001720593553052070070607223399946399057131158709963577735902719628506114 651483752620956534671329002599439766311454590268589897911583709341937044115 512192011716488056694593813118384376562062784631049034629395002945834116482 411496975832601180073169943739350696629571241027323913874175492307186245454 322203955273529524024590380574450289224688628533654221381572213116328811205 214648980518009202471939171055539011394331668151582884368760696110250517100 739276238555338627255353883096067164466237092264680967125406186950214317621 166814009759528149390722260111268115310838731761732323526360583817315103459 573653822353499293582283685100781088463434998351840445170427018938199424341 009057537625776757111809008816418331920196262341628816652137471732547772778 348877436651882875215668571950637193656539038944936642176400312152787022236 646363575550356557694888654950027085392361710550213114741374410613444554419 210133617299628569489919336918472947858072915608851039678195942983318648075 608367955149663644896559294818785178403877332624705194505041984774201418394 773120281588684570729054405751060128525805659470304683634459265255213700806 875200959345360731622611872817392807462309468536782310609792159936001994623 799343421068781349734695924646975250624695861690917857397659519939299399556 754271465491045686070209901260681870498417807917392407194599632306025470790 177452751318680998228473086076653686685551646770291133682756310722334672611 370549079536583453863719623585631261838715677411873852772292259474337378569 553845624680101390572787101651296663676445187246565373040244368414081448873 295784734849000301947788802046032466084287535184836495919508288832320652212 810419044804724794929134228495197002260131043006241071797150279343326340799 596053144605323048852897291765987601666781193793237245385720960758227717848 336161358261289622611812945592746276713779448758675365754486140761193112595 851265575973457301533364263076798544338576171533346232527057200530398828949 903425956623297578248873502925916682589445689465599265845476269452878051650 172067478541788798227680653665064191097343452887833862172615626958265447820 567298775642632532159429441803994321700009054265076309558846589517170914760 743713689331946909098190450129030709956622662030318264936573369841955577696 378762491885286568660760056602560544571133728684020557441603083705231224258 722343885412317948138855007568938112493538631863528708379984569261998179452 336408742959118074745341955142035172618420084550917084568236820089773945584 267921427347756087964427920270831215015640634134161716644806981548376449157 390012121704154787259199894382536495051477137939914720521952907939613762110 723849429061635760459623125350606853765142311534966568371511660422079639446 662116325515772907097847315627827759878813649195125748332879377157145909106 484164267830994972367442017586226940215940792448054125536043131799269673915 754241929660731239376354213923061787675395871143610408940996608947141834069 836299367536262154524729846421375289107988438130609555262272083751862983706 678722443019579379378607210725427728907173285487437435578196651171661833088 112912024520404868220007234403502544820283425418788465360259150644527165770 004452109773558589762265548494162171498953238342160011406295071849042778925 855274303522139683567901807640604213830730877446017084268827226117718084266 433365178000217190344923426426629226145600433738386833555534345300426481847 398921562708609565062934040526494324426144566592129122564889356965500915430 642613425266847259491431423939884543248632746184284665598533231221046625989 014171210344608427161661900125719587079321756969854401339762209674945418540 711844643394699016269835160784892451405894094639526780735457970030705116368 251948770118976400282764841416058720618418529718915401968825328930914966534 575357142731848201638464483249903788606900807270932767312758196656394114896 171683298045513972950668760474091542042842999354102582911350224169076943166 857424252250902693903481485645130306992519959043638402842926741257342244776 558417788617173726546208549829449894678735092958165263207225899236876845701

[@douglasstrother6584]

I'm an old geezer (56), and have a little envy about this and all of the KZbinr's who publish informative and captivating videos on mathematics, physics, etc. The scope of topics and their graphical representations are captivating and enlightening. Keep up the good work: it is influencing many lives for the better.

[@friendlyone2706]

I'm an older geezer and VERY envious. I struggled to find the books I wanted to read in school & public libraries in the various towns we lived in while growing up. In grade school, Compton's Encyclopedia was my first algebra teacher. I was obsessed with how eyes worked, what was light that eyes could even work... A child today can go to the net and indulge in "rabbit hole" style learning to his/her heart's content. Rabbit hole learning leads to more interesting questions and greater insights than can ever happen sitting obediently in class. My only regret: Today's young will never fully appreciate the feast before them because they have never known starvation. I don't want them to experience starvation, but it is good to recognize and appreciate luxury.

[@douglasstrother6584]

@@friendlyone2706 Agreed! "The World Book Encyclopedia" and "Time-Life Science Library" was what I grew-up with.

[@friendlyone2706]

@@douglasstrother6584 One of the schools I attended had World Book, and another favorite were the All About books.

[@theblinkingbrownie4654]

​@@friendlyone2706I am a youngin and I sort of wish to experience this starvation you speak of to increase my desire for learning. Too bad it would waste too much time that could be saved by simple google searches. Oh well, our generations have to encounter its fair share of problems too with corporations getting increasingly more efficient at capturing our attention, getting addicted is easier than ever. How I wish I could obtain the benefits of every era without the drawbacks

[@Bhagya-AP]

You're not old at all! 56 is absolutely young and thriving.

[@Porururidimu]

I don't understand a single thing but sounds cool

[@MrCmon113]

That's better than the illusion of understanding.

[@halonothing1]

Watch enough times and/or enough different videos about e and natural log and you will.

[@--_9623]

@@halonothing1 Did you so?

[@playerscience]

@@stonecat676 the concepts that you mentioned are not at all hard... 🤦😂😂😂

[@ossiehalvorson7702]

@@halonothing1 Ehhhh, not really. Mathematics is ultimately logical. You could copy a logic circuit enough times and maybe understand how the signal travels, but without understanding the gates it's made of, you won't actually understand it. The second someone asks you to design your own variation on the circuit, it falls apart. A structure without a foundation is going to be a very unstable structure. You need to understand the elements that come before first, otherwise it's closer to mimicry than it is to an understanding.

[@peybro]

2:24 when you get caught stealing cookies by your German grandmother

[@rextransformation7418]

😂😂😂😂😂😂😂👍 Das ist smart!!! 😂😂

[@ian7792]

Lol

[@ramimerza644]

I'm falling in love with math

[@wesleymiller8661]

-Ing?

[@kunaldhawan5952]

@@wesleymiller8661 you and me both

[@DavidThomas005]

i like math videos. i hate math class

[@sadatnafis2032]

same

[@liambennett1389]

Oh yeah? How fast?

[@algorithminc.8850]

People don't become mathematicians or engineers because it is easy - quite the opposite. This is a great channel - might help people understand how much fun math and engineering really are. He makes many of these topics approachable by those new to the concepts (the viewer is along for the ride in learning in some cases), while also being really interesting to those who already have backgrounds.

[@MasterofFace]

Math is easy

[@kostas919]

@@MasterofFace then you probably know shit

[@Tom-vu1wr]

@@kostas919 definitely not at uni yet

[@arqamkhan8230]

Have u ever thought of the possibility that somethings would never be fun for someone no matter how much they invest their time in it unless they have nothing else to do which is unlikely and btw before u say something i have my fair share of math

[@lucusekali5767]

Agree

[@bryan.flores]

Im starting to notice that numbers are everywhere in math🤯

[@nikstoun9478]

😑

[@jimo9555]

Snap out of it, you're being irrational

[@adamuhaddadi5332]

@@jimo9555 get real smh ;)

[@jimo9555]

@@adamuhaddadi5332 don't be so negative :p

[@adamuhaddadi5332]

@@jimo9555 couldint find an other joke :( **sed music plays**

[@kennedystapleton2279]

Not a physics or math student, I literally limped my way through school to end up slightly successful in banking, and you make me want to rediscover my passion for learning and make me feel like I could tackle quantitative finance. Thank you genuinely and I hope you continue to inspire others to reach higher.

[@kennedystapleton2279]

Not specific to this video alone you’re incredibly funny and intelligent and we need more people like you.

[@984francis]

"That's what you learned at high school, minimum." I seriously doubt more than 2% of the population knows that.

[@takvacs]

2.71828% to be exact.

[@holdenhill28]

It's in the basic Algebra 2 curriculum

[@takvacs]

@@holdenhill28 Studying is not knowing.

[@jakobwachter5181]

2% of the population knows it because they only learn it once per year. If it was continuously learned then 2.71828% of the population would understand and this video could reach a larger audience. I blame the public school system.

[@SirNyanPanda]

People dumb amirite

[@BangMaster96]

What i don't get is, how can people figure this out? I can do basic calculus, algebra, linear algebra, etc.., but then, i see things like from this video, and it just completely blows my mind. Like, how is an individual able to find out formulas for real world systems? For example, formulas for fluid dynamics, or circuit analysis, etc..What do they do that helps then derive these formulas? Again, look at Maxwell's equations, how did he come up with all those equations? What did he do that led to him deriving those equations? Also with fourier transform, how did he figure out what variables and equations to use? Math just goes out of my mind once i start learning many of the abstract concepts, there truly are some real geniuses on this planet, just the fact that those people can understand what an equation is doing, and how to derive those equations is beyond my scope of understanding.

[@Victor-jd5om]

Ikr

[@Tomahawks360]

It is mostly because these people know what math means. What a derivate, an integral, a sum, etc. means on a practical level. For example, the typical example of a bacteria colony. When studing it, you can measure that, for example, they tend to reduce their pop with the influence of an antibiotic. You get data, the pop of bateria on different period of time, graph the results, and then you can see that they tend to reduce their pop with a certain speed. What speed means on math? Derivate. So you have a derivate of an unknown function. But for that is calculus! You use differentials, integrals and natural log to make an equation that sync with this mysterious function. As you could see, math is not the only thing needed, but also a fuck ton of experimentation and data registred. I mean, Maxwell needed first some data about the electromagnetic phenomena to actually make math that could describe the behaviour he was seeing. And again, the same as before: Ah, electromagnetic field kinda work like vector fields. Ah, I see that the magnetic field always close on itself, I can describe that using the Divergence. Ah, I see the change of magnetic field makes the electric field spin, I could describe that using derivates (that describe change) and curls (that describe rotation). It is like seeing math like a language; as certain objects and actions can be transcribe in words, certain phenomenas can be transcribe in functions and operators, and knowing correctly the basic ones can be enough to write everything you see on math. Now, there exist the really true geniuses that can discover the depht secrets of nature without any experimental data, just using math. I'm amaze and confused about how Newton discover the Lineal Momentum using only math. But again, it seems it's because he knows what math means out of the paper. So, my recommendation? Try to learn what your calculus and algebra means. What are the descriptions and meanings of limits, derivates, integrals, and so on. And not just how to use them. I mean read, words. And then, try to do inverse engineering on all these famous equations, what they say to you just seeing the math. Obviously, that doesn't mean you are going to get as gut as Einstein or Newton just doing that. But at least it could help to take a step forward on your understanding about the world. And if you're an engineer, it could really help. I mean, legends say that Faraday was absolute shit with maths, and that that is the only reason of why today we talk about Maxwell's equations instead of Faraday's equations.

[@BangMaster96]

@@Tomahawks360 I agree, at some point, you need to have a solid foundation of basic mathematics, after which, you can build on top of your foundations. Newton, Einstein, Maxwell, were definitely one of the smartest minds this planet has ever had. Another one is David Hilbert, who was one of the most influential mathematician of 20th century. These people didn't see math as manipulation of number, but rather, a tool to use and explain nature and the universe. I will try to rebuild my foundations, and hope i can at least get a glimpse of what it feel like to be a true mathematician.

[@Tomahawks360]

@@BangMaster96 I wish the best of luck to you. If this is your passion, then it will be one of the most fun and joyful experience you could have.

[@JordanMetroidManiac]

Well, it takes a deeper understanding of reality (everything) to discover and prove such properties of a mere number. The people who do that don’t stop learning about math at basic calculus, algebra, and linear algebra. They go onto learning about more abstract mathematics such as topology, modern algebra (A.K.A. group theory), etc. Such areas of math include concepts that aren’t so applicable to the real world as calculus or linear algebra are, but they are applicable to revealing *deep* relations in the universe. It’s mind blowing, to say the least.

[@rextransformation7418]

10:30 "Or when you take a π out of the oven..." '¬_¬

[@Dr.FeelsGood]

Oh my...an obvious pun masquerading as a joke. Your only way out of this one is to admit you are a dad.

[@javierduenasjimenez7930]

Bruh, I thought that he was reffering to number π until the pie picture appeared on the screen😂

[@doomerman965]

You can also use Euler's formula to solve higher order differential equations with complex roots.

[@rithvikmuthyalapati9754]

The fifth case of high order DE's

[@joshgoddard472]

Ive just found out about your channel and only having watched three of your videos I am already in love with the information you are interested in and the way that you are able to communicate said information and am very excited to binge your content and add to your already large amount of support

[@korelly]

In another video on this subject, they explained this: Take the letters ABC and count all permutations possible (ACB, BAC, BCA...) you will find there are 6 (or 3!) of them. Count the cases where no letter is in its alpbabetical order. You will find 2 cases: BCA and CAB. 6 divided by 2 makes 3, which is slightly higher than 2.71828... Now do the same experience with ABCD. You will get 24 permutations possible, or 4!. And count how many cases have no letter in its alphabetical order. There are 9 of them. 24/9 = 2.66666666.... That is closer to e. If you do that experience with growing numbers of letters, you will get always closer to e.

[@sambhavsongara3087]

What you are talking about is the sub factorial notarion, !n

[@iftahkotlerr7601]

Another great video by majorprep, I actully try to prove stuff from the probability part, I have test on that next week so thanks for the more preparation on that!

[@infernocaptures8739]

You're the first person who made me *actually* understand the gamma function.

[@herkules593]

Cool then explain why Γ(n)=n! holds Getting explained what it does isn't the same as understanding. Understanding in maths is understanding proofs.

[@Fokalopoka]

@@herkules593 gamma of n is actually (n-1)!, the n! function is capital pi of n function, which is same function as gamma of n-1

[@leonthethird7494]

@@herkules593 so pretentious

[@DreadKyller]

@@herkules593 Do I need to understand every detail about how the bread I use is made in order for me to make a sandwich? Do I need to understand exactly how the planks of wood I use are harvested and cut to use them for building? Do I need to understand exactly how the circuitry in the CPU of my computer is constructed in order to write a program for it? People already figured out many times over how the gamma function works and how to calculate it, if I need to use it as part of a larger product/problem I would almost never need to understand exactly why it works, only that it does. Really the only time I'd absolutely need to, is in constructing particularly complex proofs about whatever I find using it. If I'm not dealing with trying to find proofs, there's little reason to need to know, in depth, how the gamma function works, just what it does. It's a bit pretentious to act as if not understanding all the details means you don't understand anything about it.

[@Nylspider]

@@herkules593 extremely rude

[@fizwiz81]

The part about Euler’s formula being all over textbooks is 100% true. I am in college right now and it is all we do in my physics class, and it’s all I do in my differential equations class

[@qfox16789]

Loving your recent run of videos. Very interesting dedicated channel. Keep it up 👍

[@vib80]

And, of course, you posted this on 1/e day! The point of the year that's 1/e through it, so it's the point where if you were doing an optimal stopping algorithm over the year you'd switch from sampling to select.

[@Pining_for_the_fjords]

This video is trippy. Is this what doing an e feels like?

[@davinonnenmacher7272]

Quality content. Thanks!

[@matthewcarrillo4287]

Lovin the support from Brilliant! Great platform supporting a great mind. Congrats.

[@Drarck_]

I found this channel few days ago now I'm hooked in it, how didn't I know you existed? xD keep the good work dude

[@kebabmarley2505]

"E" - markiplier

[@ViratKohli-jj3wj]

Lolololol

[@elliotbaker5416]

Markiplier = Leonhardt Euler

[@chlo_z7566]

*sees Euler's identity* *remembers how hard complex analysis was* *hisses*

[@kerrym7089]

This was my favorite course in undergrad!

[@xMrJanuaryx]

... e made complex analysis easier not harder foo

[@mykekinkster440]

lol, agreed!

[@chlo_z7566]

@@xMrJanuaryx Oh it made the lower math stupid easy, but I meant the class in general. I struggled with key hole contours mainly.

[@allenzhang8261]

take spherical harmonics

[@crazyfly5505]

I understood almost everything you said. You're really good at this! Thanks

[@gat0tsu]

dude this channel is awesome. thanks for the videos

[@Kevin-cy2dr]

Man you are doing a great job. Trust me, you deserve better.

[@lizapiashko9105]

3:03 the 5 of diamonds and queen of clubs remained in the same location

[@lunam7249]

Your awesome!!you made “e” happy!!

[@jlpsinde]

First video from you I've seen, really cool! I loved it! Hug from Portugal.

[@punditgi]

Fascinating explanation as always. Go, Zach!

[@jpkellerman7056]

I am a chemistry major and use e all the time but never understood where it came from or why to use it. It was just something I accepted as remember formula and use calculator but the first bit about (1+1/999999)^999999 just somehow made me grasp it alot more. would love to see more on this but maby focused more on why and how. thank you alot by the way

[@nagarmalsharma9992]

You're a chemistry major and you didn't study limits in your highschool? Kinda weird

[@gspaulsson]

as an ex-programmer, the job interview example reminds me of a tournament-tree sorting algorithm. It also works for evolution by random mutation and natural selection. And e is also the most frequent letter in the English alphabet, so of course it turns up everywhere.

[@RobertBrownbrowncone]

I Really enjoyed this video, you explained this topic in a very clear and understandable manner. Thank you for making e and imaginary numbers a fun topic to watch.

[@Kamel419]

as always, your videos are outstanding! (apparently didn't see the notification when this was first published lol)

[@redpandarepresent6380]

We love u man...... thank you for all of this

[@zachstar]

If you want an explanation regarding the even dimensional spheres I made a follow up video! kzbin.info/www/bejne/o4nTYomde7yVgcU But several other examples here don't really come with an explanation since I was trying to go for range rather than depth (or else the video would've been very long). But I put several resources in the description for anyone who wants further reading.

[@natevanderw]

Great video. I wish my students rose to your level. I jump up down in class trying to explain how cool the number e is. Next time I make them watch this video and give them a quiz!

[@LYNXzTwist]

@@sifatullahalk e^x has many applications in APD and ODEs, things you'll learn in engineering, as the differential of e^x is e^x it is crucial for solving ordered differentials

[@GaryTugan]

flippin awesome vid! I love all things "e", and so your vid caught my eye. I was impressed. So MUCH meat here, I think I'll have to watch this a few times. Seriously excellent.

[@antonbashkin6706]

This is really clear! Great explanation.

[@IzzetRight]

Loved the video. I'm in algebra 2 but I enjoying learning about higher math concepts as I absolutely love math. Every now and then I'll spend some time (1h +) trying to find new math patterns or explain something to myself. I've been wondering for a while how fractional factorials were calculated. Now I know. Mind = Blown.

[@HollywoodF1]

There are a few things that you come across where there is actually a more general explanation for something than the one you first learned. Turns out that the explanation for factorial that people are taught is a special case for integers. Just like a circle is a special case of an ellipse and Newtonian Mechanics is a special case of Relativistic Mechanics. It's an important lesson for the science-minded: Beware of the possibility of a more general explanation. Some of humankind's greatest discoveries resulted from this kind of thinking.

[@IzzetRight]

@@HollywoodF1 That is really cool.

[@mathematicsfanatic832]

This channel is love

[@fizixx]

There is a king's ransom of golden nuggets of information and fascinating concepts here. Great, greatvid, thanks!

[@emilyesnyman]

Absolutely LOVED this video! Thanks Zach :)

[@kvartlapp9724]

Back in high school I had a very serious math teacher. He never smiled or showed any other emotions. Until one day, after being done with both trig. functions and imaginary numbers. You could see the excitement behind his eyes and he said "I really have to show you something! It's not really in the curriculum, but you really need to see this!". He then proceeded to derive Euler's formula on the blackboard, and he had tears in his eyes when he was done. He almost started crying while stating that "this is the most beautiful thing in the universe".

[@razmuzen1090]

Wtf :D

[@yat_ii]

i made a proof for eulers formula and i love it but im not quite that mad about it

[@tyler89557]

"When would e show up in the real world?" *Calculates composite interest* *Calculates things like Gibbs Free Energy and K in chemistry*

[@csebastian3]

This is a FANTASTIC video! Thank you.

[@shantanushimpi1694]

Excellent video showing detailed study on e. Thanks

[@robertschlesinger1342]

Absolutely superb mathematics video that I recommend for most all HS and college students, as well as those that have gone beyond a college degree.

[@jakez082]

"Or just look at an electromagnetism text book" I swore I'd never open that thing after E and M again.

[@lurdesgoldenwolf4356]

Awesome explanation of e! Love the channel! Thank you!!! :)

[@dylanalexander5163]

so thankful for this video

[@crazyfly5505]

This would be a great series. You should do all the constants! Please 🙏

[@crazyfly5505]

Ok I just looked at Wikipedia to see what I just asked you to do. Supergolden ratio??? Mathematicians, SMDH

[@crazyfly5505]

Ok I just looked at Wikipedia to see what I just asked you to do. Supergolden ratio??? Mathematicians, SMDH

[@farhanislam8463]

Thanks a lot. I'm learning statistics, and I always wondered why they just assume euler's number. Probability makes so much more sense now.

[@dijonstreak]

SO thankful. !! and timely ..been studying Laplace Transform + other Integrals and always just what the heck did " e " stand for. ??!! thanks to YOU now i KNOW. Whoo-Hoo. !!

[@emmagraceburfoot6188]

Thank you, great video 🙏

[@evccrtt1480]

I wish I had money to pay brilliant

[@randomdude9135]

Or some Noble soul would pirate it for greater good

[@mridulbarman4114]

What a Channel !!! KZbin algorithm sometimes surprises you.

[@yepyaesoneoo5266]

Thank you. That's very nice. Well, I am studying about underwater explosion. No wonder the shock wave profile (due to explosion) follows the exponential decay function!!! Also e (Euler's number) solves half of my problem since I am trying to find solutions for my differential equations. I just can't help feeling amazed by the wonderfulness of this number. This is a really deep, fundamental and beautiful number indeed!!!! Again, thank you so much for your explanation :)

[@markpointer2967]

This is simply awesome, thank u!

[@robertschlesinger1342]

There are several great books on e at varying levels of difficulty. Amazingly, Euler had vision problems for a significant part of his life and was blind during his later years, yet he continued to produce important mathematics. He endured a cataract operation, and during those years it must have been pure torture. The operation was not successful.

[@johnbillings5260]

It makes you wonder about all of the things that have yet to be discovered to have math tied to e.

[@CarlosLopez-kj1bv]

The Euler-Mascheroni constant as well. Many kind of numbers or coefficients can offer a complete representation of that constant. One article connects the Jensen coefficients, Turán moments and the even Taylor coefficients of the Riemann Xi function with the Euler-Mascheroni constant. It is a new topic that after the revision of an article will open the eyes regarding the unexpected consequences of the Riemann hypothesis through the Riemann Xi function.

[@rparl]

My first differential equation text was a revision which covered the Heavyside transform and had a short section showing how similar the Laplace transform was. A later text discussed the Laplace transform but completely omitted the now obsolete Heavyside transform. When I got to UC Berkeley, I looked up the origin of the Laplace transform and found that it was an offshoot of his tome on celestial mechanics. At one point he needed a technique and interrupted the tome to dash off a slim volume on an obscure element of math. He then used that in his tome. The slim volume was mostly ignored by those studying Laplace's work and it was well after WWII that the Heavyside transform was discarded as an unsupported "seat of your pants" technique.

[@jakobwachter5181]

If you've taken calc you know why it shows up so often. It's got to do with that pretty little mention that the derivative of e^x is just itself. (For non-calc students, the derivative of something is what the slope of that thing is if you made a straight line out of a really tiny segment of it). It was mentioned in passing in this video but it's really a powerful assumption. It allows things like differential equations, which relate a function to their own derivative, to be solvable. Sticking a mass on a spring in a fluid is a prime example of a differential equation. As follows: The sum of all forces = mass times acceleration; acceleration being the second derivative of position with respect to time. F = ma. The force on a spring, as described by Hooke's Law, is F1 = -kx. There's also a dampening force caused by the fluid, which is proportional to the velocity (a derivative of position); F2 = -bv. F1 + F2 = ma, which can then all be put in terms of X (You may use the ' [prime] symbol as a way to recognize a derivative); -kx - bx' = mx'', which then leads to mx'' + bx' + kx = 0. You can thus easily see how physical systems take into account their position and its relation to everything else. In fact, this is more common than you think: dynamic systems, circuits, electric fields, classical mechanics and plenty of other fields take into account this relation of position and its derivatives to the motion of a body. You really don't want to have to use more than one equation to explain this behavior because you'd like to keep things as simple as possible. Therefore e really is a big player in a whole bunch of different fields. It's worth checking to see whether or not a function you've had to memorize was brought about by a differential equation because quite often it is. NOTE: Really the only situation in which the above function works is if your function x is either 0 or an exponential function. By proxy, a combination of sines and cosines will also represent the value, but this is only in specific circumstances (AKA whenever imaginary numbers get involved). Any other type and it would spiral out of control! Go ahead: try and solve this function with anything that isn't in the form e^(cx), where c is any value. I guarantee nothing will line up quite right.

[@lewissjj1417]

I honestly have a lot of respect for Matematicians, Physicists and scientists, etc. It's just amazing how they can explain and understand unexplainable phenomena. I'm a 10th grader and i just know how to use sin, cos and tan (pretty lame yeah). So you can imagine my reaction throughout this video seing "gamma function" and "quantum mecanics" stuff. It's not that I'm not interested at all, It's just really overwhelming.

[@BirdTurdMemes]

How is this now 19 year old doing?

[@devangshekhawat]

Love your videos Zach! :)

[@bustacap503]

Wow mind Blown!!!!! Incredible and amazing!

[@Iguield]

These videos are awesome and when you film yourself instead of just commenting over images the video gets much more interesting!

[@zachstar]

Thanks! Yeah been trying to find the right balance of that, think the in person aspect at least makes it more personal.

[@folzbox487]

Why ?

[@XetXetable]

I'm surprised you didn't even mention the basic calculus of e, since I'd say that's where all these applications come from, ultimately. e, or more accurately the exponential function exp(x) = e^x, is the unique function such that exp(0) = 1 and the derivative of exp(x) is exp(x). This means that the rate of growth of exp(x) is exactly the size of exp(x). This is why the exponential function (and really never e on its own) shows up; it's THE way we have to model situations where size directly determines rate of growth. It's also worth noting that the trig functions come from almost the same place; in the case of cos(x), it's the unique function such that the second derivative of cos(x) is -cos(x), while cos(0) = 1 and cos'(0) = 0. Sine is almost the same as cos, but sin(0) = 0 and sin'(0) = 1. In other words, when acceleration of growth is negatively proportional to size, sin and cos will show up in much the same way exp(x) does.

[@MatthewLuigamma032]

You're right, but I think the focus in this video was on the applications of e. At 7:00, the reason the velocity is equal to the position is because velocity is the derivative of the position function. As the derivative of e^x is e^x, they're the same! Same deal with acceleration, which is the derivative of velocity.

[@philipphoehn3883]

He literally went over it in the video but used velocity to show differentiation and area to show integration

[@sonic5d]

This is amazing^^

[@doctorwilly]

This is amazing!

[@PM-gt9mh]

Why didn‘t you say a thing about Leonard Euler himself? He was such a genius, even besides the Euler number his work is just everywhere. The man really is the father of mathematics (I read that description of him in a book about great mathematicians). Would‘ve loved to learn more about him! Otherwise very cool video, you really put some effort in finding all these examples!

[@zachstar]

There’s always more I want to talk about in these videos. But after 15 minutes I really just have to choose. Euler himself could definitely be a great video on its own regarding just some of the things he did.

[@PM-gt9mh]

MajorPrep Yeah of course, you definitely have to narrow things down a lot... If you make a L. Euler Video, I would hit that like button so damn hard!

[@L0j1k]

Because everybody already knows Euler was dope. Also isn't it really a video about Euler anyway, that the video is about this number, one of the most fundamental constants in all of math and science, which is named after Euler?

[@mykekinkster440]

i never knew his first name was leo.... thnks for that

[@erik-ic3tp]

Why Euler? Why not Euclid or Pythagoras? :)

[@mr.chaoticgood1469]

Hey , can you please do a comparison video between chemical and materials engineering?

[@janbergen]

oh nice. i can now calculate umbrella picking probabilities. Thanks Euler. needed that.

[@MitchBurns]

The first half of this video I hadn’t heard of any of it before, but it was interesting and made sense. The second half was all stuff I learned in college as an EE. They really drill it into you, and don’t explain it half as well as they really should.

[@LimLux]

Fun fact: Euler wasn't the first one to discover this number, but he was the one that made it famous, named it after him and did unimaginable things with it.

[@ravitejakakarala7858]

Who discovered it ?

[@zertico_kawaii670]

​@@ravitejakakarala7858 me trust me fr

[@Amine-gz7gq]

@@ravitejakakarala7858 bernoulli

[@thegabrielchannel818]

The golden ratio is the most mysterious, in my opinion.

[@saleplains]

sometimes i watch too much of your other channel and i forget you do this type of stuff

[@BOSS-bk2jx]

very intersecting and informative.... thank you so much

[@franciscobolzan9134]

2:15 Ahhhh yes. The engineering theorem is everywhere

[@mackk123]

this video taught me that banks are ripping us off.

[@gabrielbarrantes6946]

Lol, that is true but not for that reason

[@christianescobar-gonzalez890]

I'm starting school as a math major, love your videos!

[@tylershepard4269]

d/dx[ln(E(e^sx))] is called a moment generating function and it gives the moments (mean, variance, skewness, etc) of a distribution. Very amazing stuff.

[@tyler1107]

9:30 I though that looked familiar and looked at my E&M textbook and what do you know, it’s the same one. I guess every professor assigns this book :) (For anyone wondering, it’s Introduction to Electrodynamics 4e by David J. Griffiths)

[@mizuhonova]

3:33 was a weird example. If you have N candidates and you first interview N/X of them, then your probability of having the best candidate in that pool is gonna be 1/X no matter what X is. I'm not sure why e is special or relevant in this example.

[@ananmaysahu4563]

wwell... so ur saying i interview all of them and pick none, which would be the same as picking n/x with x being one, then the chance of picking the best candidate would be 1/x, therefore 1/1 as in i would deefinitely pick the best one? wtf bro/sis/whatever ur genders nickname is. also i think u understood the example wrong so maybe watch it again? interviewing n/e people (ill call them the first category)and then picking the next one who is better than all the ones in the first category gives u a one by e chance of getting the best candidate.

[@mizuhonova]

@@ananmaysahu4563 You should watch the video again if you're unsure how the math works. In the video, the values are N=100 and X=e. The formula is exactly the same. I just abstract it to N and X to show that the case X=e is not special.

[@mizuhonova]

For example, if you did N=100 X=2 You interview the first N/2=50 people Then your probability of picking the best candidate is 1/X = 1/2 = 50% It's pretty obvious that interviewing 50 people out of 100 will net you 50% chance of finding the best one. That's why I'm saying there's nothing special about using X=e here. It's an obvious conclusion that interviewing N/X will net 1/X chance of getting the best person whether X=e or X was some other number.

[@ananmaysahu4563]

@@mizuhonova did... Did u not read my comment?

[@ananmaysahu4563]

Abstract the value to 1. X=1 Just try it.

[@JosephElfassi]

Great stuff!

[@tupaicindjeke275]

Dude you are a BOSS. Good luck, you will soon have a million subscribers...

[@tristanlj3409]

Euler: I will eat your pie if you'll be my plus 1. + 1: OK then we're quit

[@andysmith5997]

1/e is not enough, I want a better chance, I’m moving to a universe where e is bigger,oops,sorry,smaller

[@realbignoob1886]

Lmao

[@AlcyonEldara]

For the Casimir effect, if this is the "old" paper explaining it using an old version of pertubation theory, you get a divergent series, but after a renormalizzation using the zeta function (which is linked to the gamma function), you get the observed result.

[@TheFarmacySeedsNetwork]

What an awesome walk through of math and physics! Thanks!