What's So Natural About e?

Join us on a journey where we explore a visual approach towards e, uncovering the intuition behind some of its common definitions and features.
This Wikipedia page has rigorous proofs of the facts presented in the video:
en.wikipedia.org/wiki/Charact...
Timestamps
00:00 Intro
00:41 Tally's Growth Rule
04:30 Infinite Series Formula
13:04 Limit Formula
17:39 What about e^x?
21:13 Derivative of e^x
22:45 Outro

[@LinesThatConnect]

I've never seen that process of splitting up the growth into colors before. That's a super cool way to show that the two "e" definitions are equal!

[@anon.9303]

holy crap its the man of recursive formulas

[@asheep7797]

@@anon.9303 the points that connect!

[@-originalLemon-]

You used the letter e fifteen times.

[@Omni_ro]

The fact that you somehow used minecraft and machinarium soundtracks in a video about the number e will never cease to amaze me.

[@jursamaj]

I thought it sounded familiar.

[@itscky2007]

ikr lol

[@anstropleuton]

I was about to comment that LOL

[@WizardOfDocs]

okay, for your next trick, how about working out whether anything in Minecraft depends on e?

[@Starwort]

Is that what the second track (playing at 0:59) was? I knew it was familiar and ended up guessing that it was animal crossing lol

[@eitantal726]

In Summary, "e" is the quantity you'll end up with, when you start with "1 unit of stuff", that _continuously_ doubles itself every 1 unit of time. (As opposed to _discretely_ doubles itself. Then you simply get 2)

[@guyedwards22]

I absolutely love this man. The "aha" moment behind why the Taylor Series for e^x is what it is had my mouth in a wide open smile. It's a sum of iteratively integrated polynomials, with the seed of "1" representing a sort of "principle value" of the stuff growing exponentially.

[@forthrightgambitia1032]

One of the things I have discovered is that what seems 'elementary maths' has really deep beauty embedded in it that a purely calculatory way of using it will only hint at.

[@-originalLemon-]

What are polynomials?

[@-originalLemon-]

S-seed of 1? The only seed I have is the one I'm going to be putting in my girlfriend ass.

[@ram6o172]

​@@-originalLemon-a quick google search should give you the definition. I would just explain but i cannot be bothered trying to represent math notation with plain text

[@PC_Simo]

@@ram6o172 I have seen these guys, before. Next, he’ll ask you: ”What are notations?”, or: ”What is math?”, or something; trust me.

[@justinwatson1510]

I love math so much. It is genuinely humanity's greatest achievement, and I don't understand people who say it is boring or useless.

[@diplomaticfish]

I agree :D

[@henryzhang3961]

math is useless until it isn't

[@runo4155]

While I don't like math (I'm better at pure memorization and math is more doing than studying), I agree on its status as humanity's greatest achievement We literally made something up and applied it to the real world to the point where everything makes sense a lot more because some lad decided to assign numbers to stuff way back when

[@faland0069]

I think math is more like the language of the universe, rather than a human-made thing and us humans are just understanding it. But yeah math is great

[@dakota5530]

​@faland0069 Can people stop saying this, math is literally invented. Math literally didn't exist for billions of years and the universe was existing.

[@syllabusgames2681]

I don’t think I have seen a visualization of continuously compounded interest before. It’s usually explained purely numerically in terms of interest, likely because this is usually taught before calculus. I think yours is a more clear approach with less "just trust the equation" needed. I realize the Y scale needs to go up to 3 to contain the full height at the end, but in the section where you are going over integrals (from about 5:00 to 12:00) things would be much clearer if the X and Y axis scales were the same. I understand the integral shows the area under the curve, but I still paused to stare at the yellow line and rectangles thinking “those don’t match” until I noticed the scales were different. This is the second video this year where I have recognized the soundtrack. Nice choice of music. Over all, I liked your video. Thanks for making it.

[@diplomaticfish]

Thanks for the feedback! I agree in retrospect the scaling is a bit off-putting, and it's not necessary to have the y-axis go all the way to 3 for most of the calculations anyways. Glad you liked the music :)

[@schizoframia4874]

@@diplomaticfish the music reminded me of minecraft

[@schizoframia4874]

@@abdullahenaya thanks

[@nutsi3]

@@schizoframia4874that’s because it’s minecraft music

[@schizoframia4874]

@@nutsi3 oh 😂

[@rosskrt]

I love this endless stream of #some2 videos I find myself in. They all turn out to be very interesting, and they're mostly from small content creators like you. We benefit from the great content and you benefit from our views. Win-win!

[@Blyfh]

Exactly! I can't get enough of these. I wish SoME2 would never end.

[@redstocat5455]

Wait, I didn't see this # What does it mean ?

[@evank3718]

Never seen a math video more perfectly synced with Minecraft music

[@BEN-ys6gu]

0.59% dislike ratio, 6.5% like to view ratio and 27% of people who liked also subscribed (myself included). Also 20k views, 1.3k likes, 360 subs from one video in less than a month. These are really really good numbers, if you keep this quality you'll pass the million subs in no time and you deserve it

[@diplomaticfish]

Thanks for the support!

[@GabeLily]

I bet this somehow relates to e

[@BEN-ys6gu]

@@GabeLily 🤣🤣🤣 Nice one, thanks for the laugh

[@ikilledaman]

This video is amazing. You’ve managed to explain e in a way that I, a 13 year old with only some intuition and rough idea of calculus, can understand. Very cool

[@samuelallan7452]

Unwarranted advice incoming.. If I can recommend something I would highly suggest you break into the calculus books. I did that at a similar age and ever since in college a lot of stuff is very easy - the more math you do at your age the more of an understanding you'll be able to have in later life if you go that route. Don't be scared if you don't understand everything - that comes with time but the more you can tackle now the stronger your mental machine later. Try calc, linear algebra, geometry, group theory - as much as you can reasonably tackle without interfering with your life (obviously only if you like the stuff ofc)

[@awaken6094]

​@@samuelallan7452can u suggest some good calculus books?

[@aryatripathi2486]

That's so cool you're interested in such a young age, I was like you too! It's really wonderful:)

[@johnchessant3012]

Very interesting visualization! Intuitively, this makes d/dx (e^x) = e^x almost immediate from the setup, and then derives the series and limit definitions. Great video

[@cedv37]

Intuition and rigor both are essential, and you need both in balance to accelerate your learning. I remember when I was learning about this topic, I was obsessed with rigor and then it made me realize that how much harder to forge intuition and understanding solely by digesting formal proofs and derivations, but with time the more you understand formerly formal things kind of expanding/distilling into your intuition. Great video, I wish I had seen it back then.

[@diplomaticfish]

I've had the exact same thought process. I used to be much more obsessed with rigor, but you definitely do need both. I think the most fun kind of problem is one where it's extremely intuitively clear, but rigorously proving it is challenging.

[@mskiptr]

9:15 For me, the crucial part of figuring out what is going on was noticing that yellow mass is represented using _length_ while the area under orange graph is, well _an area._ That's why they can be equal, even though the scale seems totally different

[@walkastray007]

I watched this video months ago and came back because I remembered it being the most clear way of explaining e to mathematicians/non-mathematicians. Seriously. This was an exceptionally well done video and the visualizations using different colours was more than phenomenal and made it click. Cant wait to see what other things you have in store!

[@sonofmenoetius]

I didn't understand a word brooo 😭😭😭 Too many variables smh

[@Nusret15220]

This video found me after my [kinda painful to understand at first] research about the very nature of the number e. I'm really sure that this video would be SUPER helpful back then, but it's never too late to learn and see new insight about math topics. Great content, keep it up! 👍👍👍

[@mkj1887]

The universe is governed by 2 things: thresholds and reciprocity. Regarding reciprocity, it follows that the base of the natural logarithm is 'unnatural'.

[@Dubile]

I have watched several videos attempting to explain e, because I find the number so fascinating. But I must say, this one related it to real-life the best. I wish this video was around years ago when I just couldn’t wrap my head around e. This video made it so obvious.

[@averagegamer9513]

Excellent video! It’s amazing to see that with all the good videos about e out there, people like you can still manage to come up with a unique take on the visualization of the process to finding it!

[@pascal_pauli]

Nice Video. I first learned about this connection in terms of finance, but the growth of a creature is also fun approach. The second explanation is analogues to a formulation where you invest 1$ in a banking deposit with 100% interest over 1 year. So you get 2$ after a year. But you can also spilt the interest over half a year and get 1.50$ at the half year mark. You reinvest this and get 2.25$ at the end of the year. Split it into three and you get 2.37$, and so on. If you split the year into infinitely many tiny intervals you have exactly e Dollar in you bank account.

[@monodragon]

i checked your channel and when i realized you had one video i was suprised! this looks very professional and super good for a first video

[@pegasus567]

I love this video. You've managed to visually quantify a maclaurin series, continuously compounding growth, and the reason for e's derivative, which no calculus class up to E&M ever managed to do for me. Thank you.

[@qamarat8366]

I've ignored this for a while because of its thumbnail, but wow this is a gem. I understood what e was, but this is another intuitive way of learning that. Good job! I bequeath uponst you a follower.

[@Aris-lq7xe]

Enjoyable and concise. Wow this shouldn't be free

[@jocabulous]

Nice background music

[@craftydoeseverything9718]

This is a really well-made video! The editing and animations are perfect, especially considering you have only one video and 951 subscribers! I am really excited to see what your future videos are like!

[@vedantpatel5985]

A HUGE fucking thanks to you (and Tally). For my whole high school, I have had many sleepless nights (no joking, I am a big maths nerd) wondering "What's so fucking natural about e and log(base"e"). Why is it not 2 ?" I can't thank you enough for your video. It just makes me feel like I know all the ins and outs of "e" and how it works, it's origins and all. Again, thanks a LOT.

[@diplomaticfish]

You're like me then! I'm glad it made sense :)

[@MathVisualProofs]

This is very cool. Nice work. I know this took a lot of work to animate and describe. I enjoyed it.

[@EPMTUNES]

This video, I feel, has a more pleasing explanation than 3b1b’s video on e in his calculus playlist. Great work

[@frostbird3431]

This is soooo good! How are you only at 96 subscribers? So happy I watched this!

[@warpaintish1]

great job guys. really appreciate the time and effort you have put into the video.

[@mnm1273]

Subbed, 300 is coming very soon, it'll keep growing, love the explanations.

[@hydropage2855]

5:38 the instant I saw the yellow line I screamed “THE TAYLOR SERIES” in my head

[@jakesteampson7043]

0:44 Wait, is that..? It can't be..... Oh hell yeah, it's The End from Machinarium OST! Ah, my childhood... Edit: And now there's Mr. Handagote at 17:44... Truly a wonderful choice of music

[@diplomaticfish]

Ah I think you're the first person to recognize the non-minecraft music we threw in there :)

[@victordeandrade7099]

That was truly amazing! For the first time I realized why the Taylor series for e^x actually makes total sense. Greetings from Brazil :)

[@schizoframia4874]

It seems like tally’s height would shoot to infinity. However thats like saying adding infinitly many things will sum to infinity, which is mot nessecarily true. Anyways, very interesting way of thinking about e^x.

[@shikharkothari2608]

This is one of the best videos I have ever come across

[@Matt-qi5ff]

Very helpful, intuitive, clear visualization. Best video I've seen yet

[@Sycamo]

Everything was incredibly intuitive -- great job!

[@michaels.841]

Great video! I was really impressed with your ability to explain the integral of y=x in an intuitive geometric way, without using calculus.

[@SteveMcGreen]

!! best explanation and visualisation ever !!

[@Hahalol663]

This is amazing, such a clear and intuitive explanation!

[@higgsbosonberg4316]

One of the best explanations of e I've seen on youtube.

[@rodjaknenad6984]

One of the better visualizations of e that I've seen, great video!

[@noone-ez6on]

This video somehow helped me anchor both my understanding of integrals but also e itself. I do hope there are more videos with helpful graphics to come!

[@nireusjr]

Thanks for the video, hope to see more from you two :)

[@trelosyiaellinika]

That was beautiful boys! Bravo!

[@umbertocostabitencourt8417]

This video was outstanding! Looking forward to seeing more videos like that.

[@abstrack97]

Thanks a lot for sharing! Finally i have an intuition for this constant. Intuition is as important as knowing the formulas imo ;)

[@przemysawkwiatkowski2674]

Awesome video! (Especially the Taylor series appearing instantly!)

[@hamza9829]

best explanation of e i've ever had, really intuitised the concept for me

[@jackychen7769]

The way I learned of e back in high school was via compounded interest. Basically, if you deposit $1.00 into an account that has a 100% interest rate compounded x times per year, how much would you have after a year? If it's compounded 1 time, then the interest rate applies at the end of the year, so your $1.00 rises by 100%, or $2.00. What about compounding it 2 times per year? In this case, you apply the 100% interest in 2 parts, so 50% after half a year and another 50% at the end of the year. $1.00 + (50% of $1.00) = $1.50 after half a year. $1.50 + (50% of $1.50) = $2.25 at the end of the year. What about compounding it 3 times a year? How about every month (i.e. 12 times a year)? How about every day? How about every second? How about compounding it an infinite number of times in that 1 year? Compounding more often increases how much your $1.00 has become by the end of the year, but that increase doesn't mean you'd get infinite money for infinitely compounding in 1 year. The end result is your $1.00 becoming closer and closer to $2.714... which is the value of e. This was the most intuitive way of learning it for me.

[@granieiprogramowanie2235]

It was hard for me to understand diplomatic fish's explanation, butn this one is better

[@cesarmoreno987y]

Great visuals! Lovely explanations

[@v0ltzyt613]

Hey man I'm a math major and I just wanted to say I love this video so much thank you.

[@GrimblyGoo]

This helped a lot. I hope your channel does well, this is cool

[@martinmonath9541]

Beauuuutiful! Really nice intution!

[@henbotb1178]

It made me unreasonably happy to hear the machinarium soundtrack in a math video, amazing vid automatically (on top of the fact the video was entertaining and informative)

[@maximofernandez196]

I loved the video and the explanations! I hope you'll upload another video

[@bgmarshall]

This was an amazing video. I think especially because I know just a couple basic concepts of calculus (if you have me a textbook calculus problem I probably couldn't solve it) but I do know how to do a few derivatives and that an integral is an "anti derivative," so I could just barely see things like "that looks like the integral of the previous function" and then you saying that in the video was very nice

[@yashovardhanarya7251]

You deserve much more subs bro... Keep up the good work

[@The_NSeven]

This was great :) Loved the minecraft soundtrack in the background too

[@aronadhirathanadhirathan5221]

Wow 🤩 very nice work.

[@WhattheHectogon]

Brilliant work guys!

[@Blade_of_Tomoe]

Excellent video.

[@barrettkepler7618]

This is really good. Thank you so much!

[@robinbruce7838]

Y’all are amazing ❤

[@Mark16v15]

Another way to look at e naturally is that given a choice between having your money compounded at a periodic rate or a continuous rate, you would "naturally" choose a continuous rate because you earn more money that way. To determine that amount of money at the continuous rate requires the calculation of e.

[@user-qm5wv9ef3f]

im a first year petroleum engineer student at Renn`s university, and I haven't studied math for 5 years cuz I didnt like it, but this video is so simple and nice that it helped me with almost all of my questions on the topic, i love it!

[@iamdino0]

Incredible video

[@martian9035]

My mind exploded once I realized that the graphs were just individual taylor series terms

[@thatisjake]

After watching this video, I thought this channel would have at least 10k subs. Great video.

[@abstrack97]

This deserves way more attention btw!

[@kupa121]

That might actually be the best video about e I've seen so far, wow!

[@diplomaticfish]

glad you enjoyed!

[@JohnMushitu]

This was a very intuitive video. Subscribed

[@erawanpencil]

Excellent video! You know, people confidently act like e/natural logarithms are trivial and just a computational convenience one can glaze over, but honestly I don't think modern math or physics really has a grasp as to how deep its implications go and how profound it really is. It's not at all obvious that 'continuous growth' wouldn't be infinite, or that all of the fundamental mathematical operations can be reduced to mere sliding circles (i.e. slide rulers). Change isn't sequence; it's not a mere linear or recursive successor function.... change or time somehow self-grows in a way we don't understand.

[@diplomaticfish]

Great point! People usually try to explain continuous growth as a limit of discrete processes, but in this video we tried to describe the growth as inherently continuous, which is definitely tricky to wrap one's head around.

[@itscky2007]

Really nice explaination, its easy to understand!

[@luxaley]

from the minecraft music to the Riemann sums to the actual calculus, this is beautiful. Thanks. Especially thanks because I didn’t have to pay anything for it.

[@ryantrusty8933]

a lot of this flew over my head in how we ended up at E but the derivite of e to the x part was a nice resolution to a fact that we were just told "that's the way it is" in class for

[@xniyana9956]

Very nice explanation.

[@hkayakh]

You know, now that I’ve watched to around 6 minutes, it reminds me of antimatter dimensions

[@trenchmarian]

hey this is very high quality , well done !

[@lucasf.v.n.4197]

I finally understand "e"!! thank u ❤

[@scotty3739]

i watched this before i took calc 1, and i kinda took the function for tally's height for granted (and the summation of factorials). now that i've taken calc 1 though, i see a really neat relationship between integrals of the function f(x)=x, and the factorials. very cool!

[@void-fp9hq]

Awesome video, thanks!

[@kaionayon8678]

This is genius, and well presented!

[@pepperoniboy57]

amazing video, really well made, very enjoyable

[@titussharman4938]

Awesome visual for what e means!

[@pruusnhanna4422]

I learned something today. Thanks.

[@speedg]

wow pretty good quality for a small channel keep it up!

[@lenardlontoc3585]

I love your imaginative ways to describe or ascribe the theory into visuals. What I didnt take away from this is how to get an understanding of how to d etermine growth of Tally at day 1 in a formula because e^x by itself was explained as a rate of change that is the same. Shouldn't there be a constant in the exponent if growth was to be depicted differently as each new increment abounds throughout the day? Im caught between thinking I am asking something pertinent and wondering if I am just not visualizing this correctly. Maybe I just need further examples of its applications to more things. Thank you for your dedication to such intricate subject matter and its theory and editing. You worked hard on this and I am grateful to you for this.

[@diplomaticfish]

We started by establishing Tally's growth rule: every bit of mass doubles itself in a 1 day period. This is how a day became the relevant unit of time, and why we don't need a constant in the exponent of e^x. We then defined e to be Tally's height after 1 day. We then showed that, based on how her growth works, her height at time x (where x is the fraction of the day that has passed) must be e^x. Of course, we could have rescaled things so that each bit of mass doubles in an hour, and then her height after 1 hour would be e. Let me know if that answers your question.

[@stevenfallinge7149]

Find interesting that the 1/n! denominator can be thought of as "the fraction of permutations that are time-ordered." 1^3/3! would be the portion of a cube with "red-growth yellow-growth green-growth" happening in the order given.

[@chrishtmusic]

excellent motivation of the taylor polynomial of e^x

[@gustavgille9323]

Great video! Currently taking a calculus class, this really helped me understand e!

[@rubensf7780]

Also, the natural log is simply for when you know that something grows like this and you know the end result, so you want to know for how long it was growing (or at least that’s it’s most intuitive use)

[@gane5wc817]

This is literally perfect

[@TheOriginalJphyper]

In school, I developed an intense hatred of the number _e._ It's used to calculate compound interest, which I've never been able to wrap my head around.

[@quantumgaming9180]

I only saw the first 3 minutes of the video and got hooked.

[@thawerz]

I thought minecraft was open in the background, but then I thought I was playing the minecraft soundtrack on a different tab, then I paused the video and I realized it was the video

[@nordmu]

this is so good, thanks you alot!

[@Asterism_Desmos]

Great video! And I was just taught the word “Pedagogical” so that’s going to be used in my vocab more now.

[@michaellyga4726]

The Minecraft music tho

[@punditgi]

Brilliant! 😃