What exactly is e? Exploring e in 5 Levels of Complexity

[@DrSeanGroathouse]

To try everything Brilliant has to offer-free-for a full 30 days, visit brilliant.org/DrSean . You’ll also get 20% off an annual premium subscription.

[@MC-ij9dw]

As an engineer I know based on the fundamental theorem of engineering that e = pi = 3.

[@lyingcat9022]

Close enough :)

[@SpeedyMcMichael]

this hurts me

[@Roller11111]

I'm an engineer that deals with higher safety factors than other streams of engineering. For us pi = 5.

[@SpeedyMcMichael]

@@Roller11111 i’m slowly going insane

[@Flesh_Wizard]

As an astronomy nerd, lets round pi to 10 for the sake of ease, as it's going to be a few million light years off anyway

[@mohammadmorshed4684]

This man explains math in such an intuitive way and his videos are rlly high quality, but he only has 15k subs. Actually underrated fr

[@ExtraTrstl]

For real. This is some of the most accessible and coherent explanations. Dude is one of the best teachers I’ve ever had.

[@2ksubswith0vids]

A very underrated math channel for sure

[@Tristanlj-555]

One of the rare times “underrated” is used correctly:)

[@westongunningham7151]

I'd just like to say I followed him before 5k

[@AdityaPutatunda]

Agreed! The same way your comment needs some vowels

[@linuxp00]

My favorite representation is the Taylor's Series, because it relates e with sine, cosine, i, pi, sinh, cosh and hyperreal calculus. Also, as an infinite series you're mind blown when you see that it's derivative is really itself!

[@Bighomie39]

Yeah, I was learning power series derivation in class today, saw the MacLaurin for e^x, and differentiated it for fun, only to finally get a concrete proof for e^x's derivative being itself instead of just being told it is in Calc 1

[@S-payanage]

A letter duh

[@nahuelsotomayor32]

Literally just e, why make it harder

[@the-boy-who-lived]

When someone asks me what e is, I just say some number, just like π.

[@RSLT]

😂😂

[@Deja_flu]

What the sigma

[@stevenvandervlist2958]

Can you prove that?

[@eulerfan2718]

I have a presentation for an important exam in literally 2 days that is exactly about the number e, as well as the exponential function; and a video such as this one truly is appreciated edit : i went crazy with it tysm

[@dw06meow]

bro went so crazy he decided to rename himself into a fan of euler

[@nkacey2000]

Lmaoooo ​@@dw06meow

[@christoffelgoosen4568]

Such a good idea for a video. Normally e is only introduced using calculus, but using compound interest and probability is such a great way to convince students that this number pops up everywhere and is very useful.

[@mastod0n1]

I'm pretty sure I was taught about e before calculus and in pre-calc or even algebra 2. And the compound interest example was the first example taught.

[@SeanRaleigh]

Both levels 4 and 5 are mind-blowing. Well done!

[@kavehtehrani]

I'm a math graduate and I find your videos to be educational even to me! Keep up the good work the quality is top notch my friend!

[@DrSeanGroathouse]

Thanks so much, I'm glad to hear that!

[@RabinSaidÖsteränggymnasietNA1C]

e is the most insane number I have ever seen. I started learning it yesterday and I was shocked when I realized how versatile e is. For example the derivative of e^x is e^x and e^((x^h-1)/h)=x as h approaches zero

[@Simpson17866]

You can approximate e to 18 trillion trillion decimal places using the digits 1-9 once each :D (1 + 9 ^ (-4^ (7*6)) ) ^ (3^2^85)

[@Ny0s]

I enjoyed this video very much. The fact that one can find e in Pascal's triangle really blew my mind. There are so much hidden gems in this mathematical object, I feel like I discover more of them every time, from unexpected grounds. Thank you!

[@Unchained_Alice]

The probability one was always my favourite. I worked it out myself without first knowing a long time ago so it is special to me. Plus probability theory is up there with my favourite fields in maths.

[@DanielC618]

Great job! By far the best explanation I found 👏👏👏let's get that KZbin algorithm going, this channel needs way more exposure!

[@Fire_Axus]

no

[@bemusedindian8571]

Level 5 was mind blowing. Never heard this before.

[@DrSeanGroathouse]

I'm glad you liked it! It's probably my favorite

[@basisTermium]

I think the 1st one is the best. Its all coming back to simple definition. I just recently discover volterra's product integral (1st kind) in which could be see as another way to create e.

[@varoonnone7159]

The first one is the only one I understood

[@marcoottina654]

This is gold! It should be taught in every college!

[@guglielmotranchina249]

McLovin's smart doppelganger

[@karhukivi]

The Black-Scholes option pricing model in financial economics has a component with "e" in it and this was causing my MBA class some grief until I pointed out that it was simply continuous discounting of the interest rate included in the strike price. Like many people, they thought that interest is calculated annually, or possibly monthly, but the concept of continuous compounding was new to them. The textbooks usually have the example shown in part 1 of the video, but who reads them!

[@Iddy267]

Yall wanna know why humanity is stuck at a type 0 civilization level? It is because access to knowledge is often locked behind paywalls just like with "BRILLIANT", and learning is controlled by rigid structures. Imagine if education were free and stress-free, where people could learn at their own pace, focused on what genuinely interests them. This freedom would make learning enjoyable, which would foster curiosity, critical thinking, and a much broader skill set across society. Instead, we live in a system where nearly everything, including education, comes at a cost. In Norway, for example, students pay up to 70% of their income just to keep a roof over their heads-it's absurd. If education were truly worth the hassle, it would be accessible to all, without forcing students to jump through financial and structural hoops. With a society more empowered to learn and think critically, we could make significant strides toward becoming a more advanced civilization. All this because people need their greed to be satisfied instead of thinking big as a freaking race.

[@yplayergames7934]

If all my friends were good to understand english and wanted an advice of someone to teach them very good, I would recommend your channel, you're explanations are very understandable and well conected

[@randyzeitman1354]

Superb. Far away, the best explanation of e.

[@randyzeitman1354]

e is far more important than pi. Pi explains how many straight segments make up a circle. e explains how those circles integrate into reality itself.

[@davidellis1929]

I was taught that e is the limit of (1+1/n)^n as n grows beyond all bounds ("approaches infinity"). The compound interest example is a great way to introduce e. Before learning calculus, limits must be treated informally and intuitively. It took mathematicians two hundred years between the initial development of calculus and the formalization of what a limit is. With hindsight, the epsilon-delta definition can be derived from the requirement that the limit be unique if it exists, by defining the condition that leaves behind all other candidates for the limit.

[@jeremyi4693]

In high school calculus, our teacher taught us a mnemonic device for the approximate value of e. Think of a picture of Andrew Jackson in a square frame with a diagonal line from one corner to the other corner. Andrew Jackson served 2 terms, he was the 7th president, he was first elected in 1828, because he had 2 terms, we use 1828 twice. And the angles in the frame are 45-90-45. So, 2.718281828459045

[@bsbrawl1653]

😮 cool

[@ianbennett2443]

unfortunately, i know more about integrals than i do us history

[@andyghkfilm2287]

Agh but what if I don’t know what Andrew Jackson looks like??

[@jeremyi4693]

​@andyghkfilm2287 think of a square with the name Andrew Jackson written in it.

[@carultch]

@@andyghkfilm2287 He's on the most common printed bank note of US currency. He's Mr $20 Bill.

[@jacobmanning7983]

I think one of the most important properties of e is that ln(e)=1. We could still do calculus with exponentials with a different base, however we would have to divide by the log of the different base

[@crosseyedcat1183]

I study linear systems, geometry, and rotations and to me, the compounding version of e is the best characterization. Understanding how e relates discrete and continuous actions when the actions encoded as products makes understanding what e to the power of a matrix or an imaginary number or quaternion means.

[@jaymethodus3421]

E as I use it: Exact; Equivalent; Expression (energy), e^i for 'computational cost' but the most [E]vil way I use it, as to denote exponential constant values, for scaling of base 3/4 calculation expressions into self-similar real-number ratios of irrational "digits" being operated on logarithmically.

[@AndrewDangerously]

Can you explain this at level 1 and 2?

[@jaymethodus3421]

@@AndrewDangerously It would require an exponential amount of text. Do you describe that 'amount' of that text using units derived from paragraphs? from words? from characters? From sentences? Pixel on/ off rate? The various electrical circuitry quantitues, taking their own exponential functions into account of this unknown value exponent? See, 0,1, and 2, are not real. 3 is where the real value baseline begins, as far as the instructional code for reality. 012 is a *continuum constant* that acts as a function instructing relative operational order of value exchange between real quantities. Dimensions aren't real. Yet trigonometry is extremely correlative to the deep-scaling of that very concept. Idk what to call my theory yet, but seems to be very well supported by every stone I turn over in my expansive search.

[@jaymethodus3421]

@@AndrewDangerously Uhh. I tried... So 1^2 is 1. Terrence Howard really screwed me on this shit ngl lol.... but he's crazy. And I'm both/neither. He is onto something deeply irreducible about the discrepencies of '1', '0', and 2; to the exponent of the discrepancies from using -=X/ as our 4 highest order "math operations". 1 is actually an irreducible scale unit that represents an infinitely irreducible and unique value composed of higher and lower order integral values as they are ALL, mutually calculated. In %base10linear: 1=sqrt(-2)

[@jaymethodus3421]

@@AndrewDangerously How's that for level 1 and 2? Pun intended lol

[@jaymethodus3421]

Terrence has glimpses and he's high EQ, he knows what he saw, and he just runs with it. But he has no idea wtf he's talking about it what it actually means, or when and where to actually appy it without sounding like a snake oil salesman.

[@LambdaBam]

The fact that I understand all of this, means that I owe my entire adult learning life to KZbin.

[@Mikel08ll8]

Your derivation at 5:00 could be done with every number right? With 2^x you would also get that its derivative is 2^x, which is obviously not true

[@fransisigos]

Yeah this was very handwavy. Not good at all

[@Ki0212]

Well, by the way he did it, the derivative of 2^x will be 2^x times its derivative at zero, which is log(2)

[@hamedajab2483]

Quality is absolutely crazy

[@eliteteamkiller319]

I love this channel so much.

[@DrSeanGroathouse]

Thanks so much! I'm glad you like it

[@bredlispythonguy]

The section on calculus is circular though. Defining what an exponential function over the real line means requires defining a power series with the appropriate properties expected from rational exponentiation. Therefore using e^x to define it's power series is circular because e^x is simply defined as a particular power series.

[@barrybence4555]

All this is very true, but there is always that One Big Question about the special numbers and constants such as i, e, pi, etc., and that is they seem to be necessary for our structured reality to exist, but why? Why do all the ratios and integrals and probabilities exist the way they are? Did we invent them, discover them, or did the Cosmic Mathematician design them? Inquiring former math majors would like to know. I'm 81 and would like an answer soonest!

[@Skellborn]

I'm sorry, i dont get the Limit at 5:30: e^x lim((e^h-e^0)/h) is 0/0 for h-> 0. Meaning you have to do l'hospital. But dir this you have to differentiate e^x and you start all over again. How do you know it's 1? By stating it 1min earlier?

[@joelganesh8920]

As stated in the video, the limit is the definition of the derivative of e^x at x=0, which was already assumed to be 1.

[@sachavalette1437]

exp is the reciprocal function of ln so its derivative is 1/f’(f^-1(x)) = 1/(1/exp(x)) = exp(x). This is how to prove it.

[@viz8746]

Perfect. Thank you!

[@DingleTwit]

The derivative part of level 3 made me literally put down my book and go “whoa” when I read it. That’s the version that finally made e click for me.

[@legobuildingsrewiew7538]

Masterful video. All true. Wow.

[@dodgecoates8760]

Great video.

[@aaravgulati2]

11:42 Hey Sean, great video, but there is a small mistake, while taking area you used b*h but you should have used 0.5*b*h as an area.

[@DrSeanGroathouse]

Ah, good catch, thanks!

[@Ascendance1992]

I now understand what English sounds like to foreigners

[@ariuwu1234]

my favourite definition is first defining exp : C -> C and defining e as exp(1)

[@MishaChorniy]

e looks like having 'weightlessness' characteristics among math functions

[@nbooth]

No mention of the eigenfunction of the differential operator?

[@seanknapp1271]

My smart counterpart, thank you.

[@xjgal7702]

Great video! My only question really is, what formula are you using for area of a triangle in Level 5? I understand why the sides are x(1-z) and y(1-z) and why you integrate, but why is the area xy(1-z)^2, the product of the two sides? Wouldn’t that be the area of a rectangle with those dimensions?

[@aaravgulati2]

He should have wrote 1/2b*h as area, but he did a slight mistake. Also, he was able to apply the formula because it is a right angled triangle.

[@NicholasAngelidis1]

another great video!

[@kashemvai5025]

great channel and great video

[@moonwatcher2001]

Excellent, interesting and amene!!!

[@bonquva]

5:40 what happens to the e^h - e^0 / h ???

[@James-l5s7k]

This is a good channel for math. Real math. Not toy math.

[@iamdigory]

The opposite of real math is false math. Any math, no matter how silly or fun is real unless it's false.

[@VirgilDemery]

Nice!

[@SiddhantSharma181]

this video made me follow you!

[@DrSeanGroathouse]

Thanks so much!

[@bonitageorge6410]

5th letter of the alphabet

[@geraltofrivia9424]

Great content

[@SobTim-eu3xu]

Great video, I love it❤

[@varoonnone7159]

Why am I so stupid ? I don't feel stupid but yet, I never could be this good at maths, why ?

[@paulregener7016]

Math requires practice, it is not a skill you read and understand like English you must use English to understand it. You cannot read a book learning basics of English and expect to read, write, and speak proficiently. Language HAS to be taught as you do not learn how to read or write naturally you must create Neural networks to even begin thinking a certain way. Math is the exact same way, you must read it, watch it, write it out and solve problems otherwise you’ll never get used to the grammar of math

[@paulregener7016]

The one thing that makes my brain feel better about the subject is, numbers are limited like us, even if we work at an astronomical scale it can be bounded and quantified. And there are only SO MANY EQUATIONS. And there are only 4 known forces in the Universe we operate in. This makes me feel better because Math feels unlimited but it’s only limitation is imagination. Making it a box and an open space to work with at all times. Weird concept

[@varoonnone7159]

@@paulregener7016 You're being kind but I've been to a merit based elite school in my home country, Mauritius (The Royal College of Port-Louis) I had a higher than average level in maths throughout my schooling, an A in Maths main subject for my Cambridge International Examinations Board organised A levels Yet, many others in my school and equivalent elite schools had a natural inclination to it I still remember a classmate of mine who seemed to immediately know the solution of any maths problem He ultimately won a scholarship after coming out second in the whole country in the Economics/Maths field His brain was clearly wired differently Still today, I miss out totally on the logical aspects of things, as if my brain was wired wrong My sister has a PhD in Chemical Engineering and she's got a natural inclination to maths, I became a lawyer but not being a scientist has always bothered me I feel that I have no logic at all whenever I try a maths problem, it's kinda depressing

[@varoonnone7159]

@@paulregener7016 You're being kind but I've been to a merit based elite school in my home country, Mauritius (The Royal College of Port-Louis) I had a higher than average level in maths throughout my schooling, an A in Maths main subject for my Cambridge International Examinations Board organised A levels Yet, many others in my school and equivalent elite schools had a natural inclination to it I still remember a classmate of mine who seemed to immediately know the solution of any maths problem He ultimately won a scholarship after coming out second in the whole country in the Economics/Maths field His brain was clearly wired differently Still today, I miss out totally on the logical aspects of things, as if my brain was wired wrong

[@emin-x3t3f]

Good morning 🔔🎁

[@baolongpiano]

4:18 And that value is e! (factorial, = 4.26081...)

[@Fractured_Scholar]

Care to do a Level 6 for Rotors?

[@haddib]

This is not the video I expected to watch while drunk on a Saturday night

[@yawninglion]

I was expecting the final level to be some circles in the complex plane.

[@andrealves6545]

The last one took me by surprise ahah

[@francisgrizzlysmit4715]

This guy looks like a teenager or younger, or is it that I am getting so old at -> 62 lol😆🤣😆, anyway very good can't fault you in anything I have seen so far

[@Kilgorebass7]

I see Qbert hopping around Pascal's triangle

[@InfiniteLeleveling]

There are more ways to intuitively think about e. My favorite is the “e is the image of 1 by the exponential function” approach. But for that to really make sense, you would have to really understand what we mean by the exponential function and its many definitions. The exponential function can be defined as the inverse of the natural logarithm, but I find this definition to be superior: the only function whose derivative is equal to itself and is 1 at 0.

[@Neodynium.the_permanent_magnet]

Yeah, baby, yeah!

[@MrPoornakumar]

Brilliant.

[@Blin.gde.moy.stariy.nik.]

4:00 isn't that 0 ? Both of them ?

[@evanl854]

Both of the values of each function at x = 0 are 1. However, the slope of each function at x = 0 is different.

[@keithpage3842]

You have to use a log to differentiate 2^x slope is 2xln(2)

[@ChadwickMann]

Even the first explanation was confusing to me… just a wall of numbers and terms

[@m_c_8656]

Word!

[@eranronen573]

As a physics major, is there a level 0?

[@fredsalter1915]

You left out Level 6

[@AliceObscura]

I basically use the level one explanation verbatim when I introduce e to my math students. Since it directly involves money, it is more interesting to most of them and it's a handy way to explain compounding interest at the same time. Two birds with one stone 😂

[@fungouslobster5123]

its also the map from a Lie algebra to a Lie group, literally can never escape it lol

[@cato451]

Euler’s number is the greatest number

[@josueolivos3783]

great fucking video!! this helped me understand EVERYTHING better lol

[@Garfield_Minecraft]

give me extra e-2!

[@ThalesWell]

Oh, I've heard this one! It's mc^2 right? :-p

[@rikisanity6045]

Engineers: e=pi=3

[@erik9420]

Would've liked to see e^(i*pi)+1=0

[@orologioimpazzito]

Why you look like Sheldon´s brother 😀

[@carultch]

He doesn't look anything like Georgie.

[@orologioimpazzito]

@@carultch 😪

[@RobertoCarlos-tn1iq]

a medical doctor and a mathematician! congrats!

[@kingforgotten9090]

e=3

[@RSLT]

If you multiply 0.999... by 10^n, where n is the number of decimal places( number of 9s), the result is 'wait for it' 1/e.

[@hovnovamjepomymjmenu4710]

The letter "z" is "zeď", no "ziiiii".

[@joedalton77]

E is actually also a meme

[@astro6774]

did we invent e or did we discover it...

[@SLURPZZZ4461]

I say discover

[@aaravgulati2]

Nothing is invented in math

[@miguelvieira4445]

It’s just a number used by people in the oil industry. Get it? Ha ha ha

[@TheGiggleMasterP]

I tell people how great E is and they always think I'm talking about drugs.

[@ElGnomistico]

*E*

[@joey_zhu]

level 6: lie groups

[@delta9990]

e > π calculus > geometry i will die on this hill

[@sebas31415]

What about Calc 3 and 4 which touches on geometry (as in proof of area, surface area, and volume formulae)

[@danmiltenberger5616]

2.718 is not > than 3.14.........

[@unicorn3232]

@@sebas31415 that is barely geometry tbh, and that's actually fun

[@Cow.cool.]

Abstract linear algebra > calculus any day

[@hydropage2855]

@@sebas31415I’ve definitely seen you before somewhere else. Another gd player

[@ianowens1905]

This guy looks like he should have eyeliner and a nose ring

[@assassinraider442]

e

[@carultch]

Integral z^2 dz From 1 to the cube root of 3 Times the cosine Of 3 pi over 9 Is the log of the cube root of e

[@overlordprincekhan]

Call me mad but e=π=3 g=10

[@gaigor]

[@domelessanne6357]

MDMA

[@scottfarcus1667]

He didn't explain what it is though? Just pointed out some places it pops up. No insight offered here...

[@fractodacto]

cool

[@willie333b]

👀

[@Fire_Axus]

why sponsor this video?

[@JJW410]

So he can earn money?

[@deuce2293]

math major would just set e=exp(1)