This is a video I've been meaning to do for a long time. It's my best shot at really explaining/proving all the most important/famous facts about e in the simplest way possible. It ties together and complements quite a few of my other videos. A lot of people have asked me how they can support Mathologer. If you'd like to help consider contributing subtitles in your native language to these videos. Having said that Danil Dmitriev has contributed a whole stack of subtitles in Russian and is now the official Mathologer translator for that language and so please don't worry about preparing any subtitles for Russian.
Your presentation style is exactly what I want from math videos. Rather than just spew high level information, you take the time to show why AND how, which really helps me get a deeper understanding than I was able to in school. Thanks for making videos and sharing this knowledge for free.
by and large, appraisal for Mr Mathologer whose demonstration are always very well built without ever resorting to shortcuts ... ( very important not to....) or confusing and incomplete line of thought. Thank you.....
Burkard, you asked specifically for feedback on how successfully you made these explanations accessible. I am an electronics engineer who has always been fascinated by mathematics, but struggled with the higher, more abstract elements of the subject. My way of learning is like constructing a building: laying foundations and fully understanding foundational material before laying the next layer of bricks. And I always felt uncomfortable just learning calculus identities without understanding their derivation. I think that's one of the big things that held me back. Your videos follow a logical progression, building layers of bricks on top of understanding I already have, and at last I can trust in my gut that these identities are correct because you have shown me, even if I cannot repeat the derivation myself. A few times in this video I had to stop, rewind and check some simplification or rearrangement you did, because it wasn't immediately clear to me, but I got there. Thank you for making this fascinating subject accessible in a way I have never seen before. You have rekindled my love of higher math. By the way, I much prefer the taurean version of Euler's identity: e to the i tau equals one, and I think your beautiful graphical demonstration could easily be adapted to demonstrate this form. I'd love to see your taurean conversion of other identities, which will be similar but I think lead to a far greater intuitive resonance than using pi.
It's more of a disciplined classroom material in contrast with more fun material on other math channels. I really loved it and I'm sure people learn more stuff from such approach. Thanks man. I'll watch whatever you post.
9:30 you want a million and one zeroes since the digit after the last zero can still affect the digit before: if for example take the number 5230. Lets say this is a measurement with an error margin of +-1. We can with confidence say what the first 2 digits are, but not the third, since that +-1 also affects the 3.
One drawback of this nice video is the use of ml series to demonstrate the derivative of e(x). Ml series of e(x) uses this property. In its own definition.
That was wonderful! I actually followed all of it (although I'm not sure I could repeat it). I majored in Math, but I've forgotten so much in the 50 years since graduating (I'm 70). But I know this explanation was far easier to understand than any of the ways I learned it back then.
I wish I have seen your videos when i was a middle-school student. an explanation like this one could save me PLENTY of time and nerves. in particular, this part: "and now we come full circle to..." is fatally missing from the standard curriculum. this part really makes all the sense in maths.
After 30 years since I studied calculus this has been the most BRILLIANT explanation of natural logarithms EVER ! FANTASTIC I WILL NEVER FORGET THIS 👍🏻GORGEOUS VIDEO
I have an undergrad in Electrical Engineering (1992) and I must say that I really did not grasp the coefficient expansion/substitution in the beginning of the video but I tracked right along with the calculus examples (but it took some thought) and finally I was right with you and perhaps slightly ahead of you, because I saw where you were going, with the explanation of Euler's Formula. It was fun, thanks for sharing!
I like the following consideration for the derivative of e^x using power identities more than the one from its expansion: Assume e^x with a real number e to have the derivative e^x. Then the differential quotient (e^(x+dx)-e^x)/dx = e^x*(e^dx - 1)/dx has to be e^x. (we have used e^(a+b) = e^a*e^b) => e^x*(e^dx - 1)/dx = e^x Dividing by e^x gives (e^dx - 1)/dx = 1 Straightforward solving for e gives e = (dx + 1)^(1/dx) = lim_n->inf (1/n + 1)^n and we are done (the limit dx->0 being implicitely included to all expressions) This I find pretty amazing, because it is so few lines and not using any expansion, just the differential quotient and the power identity e^(a+b) = e^a*e^b. Of course this "assumption" like consideration has to be formaly rearranged in order to be a proof for e^x being its own derivative by not imposing this and instead showing that (e^dx - 1)/dx is 1 only if e is lim_n->inf (1/n + 1)^n. Which would be basically the same algebraic steps. But I find it more natural in the way it is written above.
Hi Mr. O'Loger, what I think you really need is a video on i^i. These explanations of e^ipi are all similar, and it's easy to get so you can completely internalize how to picture e^ipi without, I think, really really understanding it at all. i^i exposes this weakness. What is REALLY going on here?
Definitely the best Mathologer video yet :) At school they explained limits and binomials to us but never bothered with e. It was really awesome to see all this knowledge piece together into some cool math. Thanks!
The 19/7 thing took me a few minutes to understand. I had to realize: 1. The result has to be an integer over 7! 2. The result can't be 0, because we know the subtracted term is not equal to e.
also he didn't choose the denominator to be 7 because the approximation ended in a 7!. He chose the approximation to end in 7! because he chose the fraction to end in 7. It's was a little confusing because he started with 7! and then chose 19/7 but it's supposed to be the opposite
Yeah the fact that it’s not 0 because it’s equal to a positive remainder is critical. He did gloss over this - but clearly we were both paying attention!
Mr BURKARD , IM 47 NOW AND I THINK THIS EXPANATION OF YOURS FOR e AND ALL ITS FORMULAS IS THE SIMPLEST AND MOST UNDERSTANDABLE OF ALL IVE SEEN SO FAR (1BROWN3BLUE INCLUDED). THANK YOU VERY MUCH. PLEASE KEEPON DOING WHAT YOU DO. BRENIKOU , GREECE.
Standard approach to define e in calculus is to start to find out what is the derivative of a^x. By definition d(a^x)/dx = lim h->0 (a^(x+h) - a^x)/h factor out a^x and you get d(a^x)/dx = a^x(lim h->0 (a^h - 1)/h. Then call lim h->0 (a^h - 1)/h = M(a) as a mystery number. Then define e as the one number that gives M(e) = 1. By examining the function 2^kx one can show that there must be a k that satisfies the equation e = 2^k, so e exists. No-one at this point knows it's value but that's not the point as we just know that d(e^x)/dx = e^x. One can the derive the logarithm ln(x) as inverse function to e^x eg. if y = e^x then x = ln(y). Derivative of the logarithm function d(ln(x)) = 1/x furhermore gives you that m(a) = ln(a) and with that one then get's an expresion to calculate e. When dealing with Taylor series (and as special case McLauren series) one gets the famous infinite sum as expression of e^x.
I prefer the other proof of cis(x)=cos(x)+i sin(x)=e^ix Let f(x)=cis(x), note that f(0)=1 f'(x) = -sin(x)+i cos(x) = if(x) i = f'(x)/f(x) Integrating both sides and using standard u-substition, We get, ix + c = log(f(x)) => f(x) = e^(ix+c) and since we know that f(0) = 1, we get c = 0. Therefore, e^ix = cis(x)
9:20 I'm not sure you need to aim for 1,000,001 zeroes in 1/N! 1/N! is so much larger than the actual error that it underestimates the number of zeroes in the actual error. Error < 1/N! * (2/N+1) < 1/N! (Let me know if staring at the screen at 8:24 for several seconds doesn't make that inequality obvious.) Let's look at a more a more manageable degree of precision, 5 decimal places. e, to 8 decimal places, is 2.71828182. 1 + 1/1! + ... + 1/9! = 2.71828152 to 8 decimal places. The actual error, to 8 decimal places is .00000030. 1/9! is .00000276 to 8 places. It has only 5 leading zeroes. But 1/9! * 2/10 = 0.00000055, rounded to 8 places. The wiggle room allowed by 2/N+1 gets bigger as N increases. I suspect the actual error associated with N for which 1/N! has 1,000,000 zeroes has 6 superfluous zeroes.
I've never seen the infinite polynomial for e^x derived using the binomial theorem; I've only seen it done as the Maclaurin series with e^x being infinitely differentiable as itself. I really appreciate that most derivations you show in your videos are actually easier to understand than the more popular derivations. Thank you for this explanation of various properties of e; I have found it very informative and enjoyable.
Hi. Once again a great video, but i have a question that really bothers me and i would be very, very happy if somebody answers. At 11:27 , how can we know that the question mark stands for a positive number? It doesn't seem very obvious to me. Can anybody help, please?
Yea, this bothered me too. In this case the question mark is actually -20, obviously |-20/7!| is still outside the error range of |1/7!| but this part could definitely use some retuning and more explanations. ( www.wolframalpha.com/input/?i=19%2F7-(1%2B1%2F1!%2B1%2F2!%2B1%2F3!%2B1%2F4!%2B1%2F5!%2B1%2F6!%2B1%2F7!) )
The difference is going to be equal to 1/8! + 1/9! + 1/10! + ... This is a positive number. Therefore, if it is equal to ?/7!, ? has to also be positive.
Yea absolute value brackets would fix the equation, but I still feel the explanation as to why the absolute value of the question mark can't be 0 is missing.
It is obvious that the error must always be positive no matter where you truncate the infinite sum (since there are always more positive terms left out). The main point here is that we are ASSUMING that a certain fraction is equal to e. There are a couple of different ways to see that this assumption leads to a statement that is false. For example, observing the difference to be negative is such a false statement. Any of those false statements implies that our assumption is false to start with. However, only some of these false statements generalise and then allow us to exclude all fractions :)
I enjoyed the explanation....it's very clear without looking complicated....I recall the quote from Einstein that if you can't explain it simply, you don't understand it well enough....you did it...thank you...
Burkard, many thanks for your very informative videos on mathematics. I am a retired professional engineer and it is a pleasure to look back on the areas of mathematics I had to learn for my engineering degree, and you make it so much more interesting and enjoyable as a subject. Kind Regards Tony
I have a question. If you choose 1/7 rather than 19/7 (even when you can clearly see 1/7 is nowhere near e), you can observe how the error gives a negative number, which is clearly smaller than 1/7!. Thus, you can find plenty of a/b solutions that are smaller than 1/b! I assume that the trick is that you have to take its absolute value given that it is an error estimation?
Then, the numerator must have an integer value, since every number adding or substracting is an integer. And the only integer which has an absolute value smaller than 1 is 0. But it can't be zero because there would be no error, which is impossible. Is that right?
The part he skipped over is that the series has only positive terms, so the error is always positive. So the numerator has to be greater than 0, but he proved it is also less than 1 and an integer, so its impossible.
The proof that e is irrational is so short and elegant I had the "aha!" Moment. I love your videos because you give us the tools to know what you are about to say before you say it. For example when you used the irrational proof with denominator 7, I could see how this would be possible with any integer denominator. I love watching your videos and getting the "aha!" moments constantly. Just enough suspense and the pleasure of working it out on our own.
Starting at 14:04. During any basic calculus courses, the fact that e^x is its own derivative is simply shoved down our throats. But after seeing the true explanation for it, I can now consider my mind thoroughly blown, and I can rest in peace.
Your videos are, generally, of high value and readily understood. Mathematics demands continuous practice. Your viewers might draw even more benefit from your teaching if you encourage them to, perhaps even insist, that they go and DO the Maths. You do already have things for people to play with and do, but to let your wonderful presentations sink in and stay there, it must be worked at by the student, or it will all fade and leave one with an empty sense of confidence in one's true grasp of the subject. Understanding the explanation is not the same as understanding what was explained.
Writing and saying log x for the natural logarithm is quite common, for example in schools here in Australia. I actually had a discussion about ln(x) vs. log(x) for this video with my colleague Marty Ross who proofwatched it. He convinced me that log(x) is the way to go. In general, when I write for mathematicians I use ln(x) and I also say infinite series instead of infinite sums, don't talk about the meaning of 3!, etc. On the other hand in these videos I try to use terms that make sense even to someone who is not terribly familiar with formal math :)
As far as I can tell, mathematicians are starting to say log instead of ln since log base 10 really isn't that widely applicable compared to natural log, so it makes sense to simplify the natural log to just "log" since you'll definitely be using that more.
About the million plus one zeros - if we only go to a million, the first non-zero digit of the error and the corresponding digit of our approximation might add up to generate a carry, and then the millionth digit would not be quite right... :)
Those trig functions as infinite series come from the Taylor Series Expansion. You can learn about it on Wikipedia, and it even shows Sin x as an example. en.wikipedia.org/wiki/Taylor_series
It's actually really similar to the derivative of e. As sin takes all the odd powers/factorials and cos all the evens and the derivative of a summand is the summand left to it, it's quite obvious than sin turns into cos and cos into sin. Also as the signs are alternating, sin turns into cos, cos into -sin, -sin into -cos and -cos into sin. Hard to put into words why that way around and not sin turning into -cose, but if you look at it, you should be able to figure it out :)
Robin Powell Yes. Proof x^(1/x)=e^(ln x/x) .The derivative of ln x/x is , by the product rule of ln and 1/x, (ln x/x)'=( ln)'*1/x+(1/x)'*ln(x)=1/x^2-(ln(x)/x^2)=(1-ln(x))/x^2.For x
I've watched quite a few of the Mathologer series and they are consistently well done, ... and without a doubt one of the best at explaining complex mathematical concepts. Wish that these had been around when I was in school.
This crossed my mind too, but ... e is equal to the *infinite* sum given, and every term in that sum is *positive* (e.g., 1/100! >0). This means that if you truncate the infinite sum, such as at the 8th term as he did in the video, you MUST get something *smaller* than e, and therefore there is a POSITIVE error -- not zero.
others have touched on it, but error can never be zero. That's what irrational means. If error was zero, you'd be saying you found a discrete value for e and this violates the definition of irrational. If you did this calculation and found your error to be zero, you probably ought to recheck your work and then take your best suit or dress to the cleaners for a fresh pressing because a Nobel Prize in mathematics is headed your way and you'll want to look sharp for the cameras
@Hortonius Imperialus: What you say is true about irrationals, but at this point in the video, Mathologer had not yet established that e is irrational; that's exactly what he was trying to prove!
true, i suppose. He had yet to test for it but it seems to me that a quantity defined by an infinite sum is by its nature irrational.. what I have an issue with and need to go back to a much earlier math text than i might expect, perhaps as far back as algebra to see by what identity he pulls off what he does at about the 5:00 mark..
I actually proved this to myself earlier this month. I even went a few steps further and found that b^ix = cos(ln(b)*x)+sin(ln(b)*x)*i. AND using this identity: b^(n+m) = b^n * b^m, found that b^(x+yi) = b^x(cos(ln(b)*y)+sin(ln(b)*y)*i) Using that formula one can raise any real number to any complex power!
This explanation was extremely cool! I knew nearly all of the facts presented, but the way they were put forward in a straightforward way from the start of the video (defining e) to the end (progressing from infinite sums to derivatives and integrals to get back to where you started) put all my knowledge of the concept of into one succinct packet! Great video!
Yes, it took me like 10 seconds to realize it, but it's not really obvious. You should have show that e-(the partial sum) is always (obviously) positive, and than your proof would be much easier to understand.
One of my teachers told us a story about how he convinced his girlfriend that factorials are to be pronounced as "boy, oh boy", with an excited tone :)
9:15 You might want to go for 1,000,001 zeros at the beginning of the error estimate because if you just had a million zeros, and the next digit was something large (like a 9 for example) then it would be very possible that your last digit was off by 1. Simple Example: 2.720 +/- 0.009 (three zeros and only two correct digits of e)
+Jay Doherty If you needed to shoot for 1,000,001 zeros in 1/N! to ensure 1,000,000 accurate digits, that would be the reason. But you don't need 1,000,001 zeros in 1/N!, because 1/N! is so much larger than the actual error, you'll have zeros to spare.
It may have been a tiny bit useful to remind viewers that 19/7 - (stuff) is positive because (stuff + positive error) = e. Also at 12:52 you accidentally forgot an "a/b - " at the beginning of your expression
When my kids ge to this stage in school math, I'll make sure they watch these videos; so much better and clearer explanations than anything I've found in the text books... BIG THANKS. again.
I always love your videos, but I just want to say I'm so glad you went into the "why" for the Taylor Remainder Theorem (essentially, even though you didn't quote it outright). I wish I'd have had you as a calculus professor, I never really (really) learned this stuff until much later.
Thank you for the video. I think your videos are always improving and that this was your best one yet. Very clear and to the point. I would love to hear you explain what transcendental numbers are and why e is one, so I look forward to your next vid. I would also like to see you explain why 1 and 0 are unusual, exceptional numbers that are often treated separately from all other numbers. One thing I don't ever hear mentioned is that e raised to the power phi (the Golden Ratio), that is, 2.71828...^1.61803..., is approximately 5. This might not seem important, but 5 is an important number, not least because it is the number of digits on human hands. The number 5 is also important in other contexts. Anyway, thanks for your videos. I will keep watching!
So... finally, multiple years after highschool someone explained to me why e^x is the derivative of itself in a way that I could understand. Thank you!
The irony is that really smart people fail to be able to explain things these things to the layman. It's cute when they think they can though. Nice try mathologer, but you threw e^ipi too fast at me.
I am afraid these videos do require a little bit of fluency in basic algebra and for the latter part a bit of calculus. Otherwise there is not much hope :) Also, if you have problems the first time you watch a video like this you can always watch it a second time, right?
Currently studying for an AP calculus exam and seeing an algebraic proof of the Maclaurin series for e^x is very, very helpful. Great video as always. Thanks!
I had seen e^(i*pi) = -1 back before I properly understood complex numbers, but upon thinking about them for several months straight while employed in a sometimes undemanding manual labour job (potato processing), I accidentally visualized the meaning of imaginary powers (rotations on a logarithmic tube of angle vs. magnitude) and realized that imaginary powers of e were radians of rotation, and then the identity made perfect sense. Taylor series, on the other hand, have mostly just been my bane, because of situations where I've been silently expected to use them on assignments repeatedly showing up and passing me by without realizing that this was at all the intended solution (partly because of my very literal mind with regards to the use of the "=" sign in assignment questions). I don't find them especially difficult to understand, just aggravating to follow when theoretical physics teachers use them without mention. Taylor series and other converging limits never felt as intuitive an explanation with regards to this identity, anyhow.
I have been a subscriber of your channer for 2 years and i can really confirm, this last 3 videos were best and each better than the last one. i really enjoy watching your videos i totally understood everything ! i have great fun doing what i love, math
You started at the right place for me: I have thought of the exponential function as growing out of (the limiting expression) for 'compound interest' -- that is, the rate of increase is proportional to the value of the function. You gave a nice proof that e is an irrational number -- it is "too close" to any rational number (19/7 was a bonus).
Loved the proof of e being irrational. That in itself (followed by some simple steps and of course some assumptions) is proof enough for me of "why" Pi is transcendental. Love your videos, thank you so much for making these. Binge watching these :)
Well if that isn't Mr. "making maths entertaining again!" :-) The way you explain the math of e feels like eating the spinich of the famous Popey the Sea-Man! You create the desire to also want to go deep down into other (hopefully still hidden) secrets of the quenn of sciences! It feels very motivating because you empower/reactivate the inborn intuition of people, where Einstein told us: "Thge only thing that really matters is intuition!" You make math feel come alive again and making it usefull for millions of people! Thank you for that valuable contribution to society! I'm actually a colleague of yours but not so much into software. If you tell me which software to use, then, once I become successfull with it, I'll mention you as the man who gave me the oportunity and connect with you, so we can both help even more people to inspire their lives with maths! :)
thanks for this great video, studying higher mathematics can be very abstract and complicated, watching videos like these renew my love for the subject.
This was a VERY engaging video, rich with interesting facts about e. The explanation of the e-irrational proof was quite easy to follow, and I never knew that estimating the error of approximations of e could be so simple! Also, I have never seen that particular proof that e^x is its own derivative. This video solidifies my choice of e as my favorite number!
Absolutely awesome explanation. Everything that we just took for granted regarding e has been explained so very well here. Thanks for this lovely video.
I think I'm going to start back at the beginning of your playlist and watch everything again but this time with a pencil and paper to follow along. In fact, you might encourage viewers to do that- break out a pencil and paper and follow along. I never really understood e until now. I mean, I kinda got it but no one ever really could break down the nuts and bolts. In high school, he said you'll have to wait till you get to calculus. When I got to calculus, the guy rambled for 15 minutes about natural functions, irrational numbers, scribbled a few lines on the board that really lead nowhere before finally giving up and just rewriting the identity at the bottom of the board.. it was kinda frustrating.. It makes much more sense now.
Your level of confidence in Homer's capacity to even pay attention for that much time, let alone understand you, is just amazing.
Mmmm dooooooonnuuuut *gargle noises*
This is a video I've been meaning to do for a long time. It's my best shot at really explaining/proving all the most important/famous facts about e in the simplest way possible. It ties together and complements quite a few of my other videos.
A lot of people have asked me how they can support Mathologer. If you'd like to help consider contributing subtitles in your native language to these videos.
Having said that Danil Dmitriev has contributed a whole stack of subtitles in Russian and is now the official Mathologer translator for that language and so please don't worry about preparing any subtitles for Russian.
Mathologer great video!
Thanks, you're of a very great help to both skilled and unskilled mathematicians.
Mathologer Explaination was awsome. Really did enjoy the video!
I'd love trying to translate your videos to Brazilian Portuguese! I'm just not sure how to add subtitles to a video.
+Pedro Arthur That would be great. Have a look at this page for the "How to" support.google.com/youtube/answer/6054623?hl=en
Your presentation style is exactly what I want from math videos. Rather than just spew high level information, you take the time to show why AND how, which really helps me get a deeper understanding than I was able to in school. Thanks for making videos and sharing this knowledge for free.
by and large, appraisal for Mr Mathologer whose demonstration are always very well built without ever resorting to shortcuts ... ( very important not to....) or confusing and incomplete line of thought. Thank you.....
Burkard, you asked specifically for feedback on how successfully you made these explanations accessible.
I am an electronics engineer who has always been fascinated by mathematics, but struggled with the higher, more abstract elements of the subject. My way of learning is like constructing a building: laying foundations and fully understanding foundational material before laying the next layer of bricks. And I always felt uncomfortable just learning calculus identities without understanding their derivation. I think that's one of the big things that held me back.
Your videos follow a logical progression, building layers of bricks on top of understanding I already have, and at last I can trust in my gut that these identities are correct because you have shown me, even if I cannot repeat the derivation myself.
A few times in this video I had to stop, rewind and check some simplification or rearrangement you did, because it wasn't immediately clear to me, but I got there.
Thank you for making this fascinating subject accessible in a way I have never seen before. You have rekindled my love of higher math.
By the way, I much prefer the taurean version of Euler's identity: e to the i tau equals one, and I think your beautiful graphical demonstration could easily be adapted to demonstrate this form. I'd love to see your taurean conversion of other identities, which will be similar but I think lead to a far greater intuitive resonance than using pi.
It's more of a disciplined classroom material in contrast with more fun material on other math channels. I really loved it and I'm sure people learn more stuff from such approach. Thanks man. I'll watch whatever you post.
9:30 you want a million and one zeroes since the digit after the last zero can still affect the digit before: if for example take the number 5230. Lets say this is a measurement with an error margin of +-1. We can with confidence say what the first 2 digits are, but not the third, since that +-1 also affects the 3.
One drawback of this nice video is the use of ml series to demonstrate the derivative of e(x). Ml series of e(x) uses this property. In its own definition.
This is a profoundly important channel.
This is as beautiful as a symphony and as brilliant as a Shakespeare play.
That was wonderful! I actually followed all of it (although I'm not sure I could repeat it). I majored in Math, but I've forgotten so much in the 50 years since graduating (I'm 70). But I know this explanation was far easier to understand than any of the ways I learned it back then.
What did you end up doibg after graduation?
I wish I have seen your videos when i was a middle-school student.
an explanation like this one could save me PLENTY of time and nerves.
in particular, this part: "and now we come full circle to..." is fatally missing from the standard curriculum. this part really makes all the sense in maths.
Glad you liked the video and, yes, I agree that this is one thing that is really missing from the standard school curriculum :)
Love how you explained the error of the series. You rock
After 30 years since I studied calculus this has been the most BRILLIANT explanation of natural logarithms EVER ! FANTASTIC I WILL NEVER FORGET THIS 👍🏻GORGEOUS VIDEO
Never knew it would be this easy to prove that (log(x))' = 1/x. I'm definitly gonna use it, when teaching it :)
You are a teacher?!
@@maheshpatel7691 what's the point of asking it's conspicuous
It doesn't make sense that way, they may ask d(eⁿ)/dn=?eⁿ and you can't use series expansion, it's defined bu using derivative
I have an undergrad in Electrical Engineering (1992) and I must say that I really did not grasp the coefficient expansion/substitution in the beginning of the video but I tracked right along with the calculus examples (but it took some thought) and finally I was right with you and perhaps slightly ahead of you, because I saw where you were going, with the explanation of Euler's Formula. It was fun, thanks for sharing!
As a 17 years old school student I've understood everything in this video. It was very well explained and interesting to listen to :)
I like the following consideration for the derivative of e^x using power identities more than the one from its expansion:
Assume e^x with a real number e to have the derivative e^x.
Then the differential quotient (e^(x+dx)-e^x)/dx = e^x*(e^dx - 1)/dx has to be e^x.
(we have used e^(a+b) = e^a*e^b)
=> e^x*(e^dx - 1)/dx = e^x
Dividing by e^x gives
(e^dx - 1)/dx = 1
Straightforward solving for e gives
e = (dx + 1)^(1/dx) = lim_n->inf (1/n + 1)^n
and we are done
(the limit dx->0 being implicitely included to all expressions)
This I find pretty amazing, because it is so few lines and not using any expansion, just the differential quotient and the power identity e^(a+b) = e^a*e^b.
Of course this "assumption" like consideration has to be formaly rearranged in order to be a proof for e^x being its own derivative by not imposing this and instead showing that (e^dx - 1)/dx is 1 only if e is lim_n->inf (1/n + 1)^n. Which would be basically the same algebraic steps. But I find it more natural in the way it is written above.
Best video I've ever watched. As long as I watched, as long as I want to watch it more.
That's great, mission accomplished then :)
Hi Mr. O'Loger, what I think you really need is a video on i^i. These explanations of e^ipi are all similar, and it's easy to get so you can completely internalize how to picture e^ipi without, I think, really really understanding it at all. i^i exposes this weakness. What is REALLY going on here?
Жаль, что я не нашёл этот канал раньше. Это просто потрясающе! Жаль, что у нас в универе нет таких классных учителей по матеше))
Definitely the best Mathologer video yet :)
At school they explained limits and binomials to us but never bothered with e. It was really awesome to see all this knowledge piece together into some cool math. Thanks!
I would consider this an e-ncredible number
Simon Marklund hope that Sanchez stays..
harikrishnan menon Me too, otherwise I'd have to change my profile picture... :(
Simon Marklund sad...haha...
Alexis Sanchez
But I would say that √-1 is more i-ncredible
The 19/7 thing took me a few minutes to understand. I had to realize:
1. The result has to be an integer over 7!
2. The result can't be 0, because we know the subtracted term is not equal to e.
also he didn't choose the denominator to be 7 because the approximation ended in a 7!. He chose the approximation to end in 7! because he chose the fraction to end in 7. It's was a little confusing because he started with 7! and then chose 19/7 but it's supposed to be the opposite
Yeah the fact that it’s not 0 because it’s equal to a positive remainder is critical. He did gloss over this - but clearly we were both paying attention!
last sentence was wrong you don't go full circle with pi you go full circle with tau! ;)
:)
I prefer going full circle with pi because 2 pies are tastier than 1 tau.
der Heider
Where my tau army at? 😂
tau > pi
It's a fact.
why is it mathematicians are obsessed with half circles?
Was curious about 19/7 and was surprised to find that it was a rational approximation of e that's accurate to 2 decimal places (like 22/7 for pi).
At 12:01 I laughed so hard... Never in my life had I heard a less excited "tadah!"
You should familiarize yourself with German sense of humor! ;)
I watched the e^iπ and didn't get anything. I took calculus this year and it finally makes complete sense!!! Math is the best!
Mr BURKARD , IM 47 NOW AND I THINK THIS EXPANATION OF YOURS FOR e AND ALL ITS FORMULAS IS THE SIMPLEST AND MOST UNDERSTANDABLE OF ALL IVE SEEN SO FAR (1BROWN3BLUE INCLUDED). THANK YOU VERY MUCH. PLEASE KEEPON DOING WHAT YOU DO. BRENIKOU , GREECE.
Glad you like these explanations and thank you very much for saying so:)
Standard approach to define e in calculus is to start to find out what is the derivative of a^x. By definition d(a^x)/dx = lim h->0 (a^(x+h) - a^x)/h factor out a^x and you get d(a^x)/dx = a^x(lim h->0 (a^h - 1)/h. Then call lim h->0 (a^h - 1)/h = M(a) as a mystery number. Then define e as the one number that gives M(e) = 1. By examining the function 2^kx one can show that there must be a k that satisfies the equation e = 2^k, so e exists. No-one at this point knows it's value but that's not the point as we just know that d(e^x)/dx = e^x. One can the derive the logarithm ln(x) as inverse function to e^x eg. if y = e^x then x = ln(y). Derivative of the logarithm function d(ln(x)) = 1/x furhermore gives you that m(a) = ln(a) and with that one then get's an expresion to calculate e. When dealing with Taylor series (and as special case McLauren series) one gets the famous infinite sum as expression of e^x.
Now I just want to know how to solve the integral that was on his shirt :S
Tanner Boos maybe try long division?
lol
Looks nasty. Long division seems to be the way to go. Can't imagine how much time it would take to do it though.
Actually it's not that bad. en.wikipedia.org/wiki/Proof_that_22/7_exceeds_π
try geometrically,because the value is the error between the ancient approximation found by a polygon and the real surface of circle,
I prefer the other proof of cis(x)=cos(x)+i sin(x)=e^ix
Let f(x)=cis(x), note that f(0)=1
f'(x) = -sin(x)+i cos(x) = if(x)
i = f'(x)/f(x)
Integrating both sides and using standard u-substition,
We get,
ix + c = log(f(x)) => f(x) = e^(ix+c) and since we know that f(0) = 1, we get c = 0. Therefore, e^ix = cis(x)
9:20 I'm not sure you need to aim for 1,000,001 zeroes in 1/N! 1/N! is so much larger than the actual error that it underestimates the number of zeroes in the actual error.
Error < 1/N! * (2/N+1) < 1/N! (Let me know if staring at the screen at 8:24 for several seconds doesn't make that inequality obvious.)
Let's look at a more a more manageable degree of precision, 5 decimal places.
e, to 8 decimal places, is 2.71828182.
1 + 1/1! + ... + 1/9! = 2.71828152 to 8 decimal places.
The actual error, to 8 decimal places is .00000030.
1/9! is .00000276 to 8 places. It has only 5 leading zeroes. But 1/9! * 2/10 = 0.00000055, rounded to 8 places. The wiggle room allowed by 2/N+1 gets bigger as N increases. I suspect the actual error associated with N for which 1/N! has 1,000,000 zeroes has 6 superfluous zeroes.
I've never seen the infinite polynomial for e^x derived using the binomial theorem; I've only seen it done as the Maclaurin series with e^x being infinitely differentiable as itself. I really appreciate that most derivations you show in your videos are actually easier to understand than the more popular derivations. Thank you for this explanation of various properties of e; I have found it very informative and enjoyable.
That's great :)
Hi. Once again a great video, but i have a question that really bothers me and i would be very, very happy if somebody answers. At 11:27 , how can we know that the question mark stands for a positive number? It doesn't seem very obvious to me. Can anybody help, please?
Yea, this bothered me too. In this case the question mark is actually -20, obviously |-20/7!| is still outside the error range of |1/7!| but this part could definitely use some retuning and more explanations. ( www.wolframalpha.com/input/?i=19%2F7-(1%2B1%2F1!%2B1%2F2!%2B1%2F3!%2B1%2F4!%2B1%2F5!%2B1%2F6!%2B1%2F7!) )
I think the proof should have absolute value brackets like |19/7 - (1+1/1!+1/2!+...+1/7!)|
The difference is going to be equal to 1/8! + 1/9! + 1/10! + ...
This is a positive number. Therefore, if it is equal to ?/7!, ? has to also be positive.
Yea absolute value brackets would fix the equation, but I still feel the explanation as to why the absolute value of the question mark can't be 0 is missing.
It is obvious that the error must always be positive no matter where you truncate the infinite sum (since there are always more positive terms left out). The main point here is that we are ASSUMING that a certain fraction is equal to e. There are a couple of different ways to see that this assumption leads to a statement that is false. For example, observing the difference to be negative is such a false statement. Any of those false statements implies that our assumption is false to start with. However, only some of these false statements generalise and then allow us to exclude all fractions :)
I enjoyed the explanation....it's very clear without looking complicated....I recall the quote from Einstein that if you can't explain it simply, you don't understand it well enough....you did it...thank you...
Burkard, many thanks for your very informative videos on mathematics. I am a retired professional engineer and it is a pleasure to look back on the areas of mathematics I had to learn for my engineering degree, and you make it so much more interesting and enjoyable as a subject. Kind Regards Tony
Not only this is super neatly explained, it also includes The Simpsons and Katy Perry. Best video ever!
I have a question. If you choose 1/7 rather than 19/7 (even when you can clearly see 1/7 is nowhere near e), you can observe how the error gives a negative number, which is clearly smaller than 1/7!. Thus, you can find plenty of a/b solutions that are smaller than 1/b! I assume that the trick is that you have to take its absolute value given that it is an error estimation?
jmiquelmb yeah
Then, the numerator must have an integer value, since every number adding or substracting is an integer. And the only integer which has an absolute value smaller than 1 is 0. But it can't be zero because there would be no error, which is impossible. Is that right?
thanks, this is the simplest answer yet :)
The part he skipped over is that the series has only positive terms, so the error is always positive. So the numerator has to be greater than 0, but he proved it is also less than 1 and an integer, so its impossible.
The left side, the error, is equal to 1/8! + 1/9! + ... so it's obviously positive.
therefore when it's equal to ?/7!, ? has to be positive.
The proof that e is irrational is so short and elegant I had the "aha!" Moment. I love your videos because you give us the tools to know what you are about to say before you say it. For example when you used the irrational proof with denominator 7, I could see how this would be possible with any integer denominator. I love watching your videos and getting the "aha!" moments constantly. Just enough suspense and the pleasure of working it out on our own.
what's the minute for the proof?
The area under a hyperbola between 1 and e is 1. Cool!
Starting at 14:04. During any basic calculus courses, the fact that e^x is its own derivative is simply shoved down our throats. But after seeing the true explanation for it, I can now consider my mind thoroughly blown, and I can rest in peace.
Your videos are, generally, of high value and readily understood. Mathematics demands continuous practice. Your viewers might draw even more benefit from your teaching if you encourage them to, perhaps even insist, that they go and DO the Maths. You do already have things for people to play with and do, but to let your wonderful presentations sink in and stay there, it must be worked at by the student, or it will all fade and leave one with an empty sense of confidence in one's true grasp of the subject. Understanding the explanation is not the same as understanding what was explained.
I honestly think this is one of the clearest videos of yours! Super clear, understandable and informative.
Shouldn't it be ln(x) rather than log(x) since log(x) usually refers to base 10 and not base e?
yeah but i think you can have log x have any base and he chose base e and you are right that would be ln x i hope someone else can confirm
Writing and saying log x for the natural logarithm is quite common, for example in schools here in Australia. I actually had a discussion about ln(x) vs. log(x) for this video with my colleague Marty Ross who proofwatched it. He convinced me that log(x) is the way to go. In general, when I write for mathematicians I use ln(x) and I also say infinite series instead of infinite sums, don't talk about the meaning of 3!, etc. On the other hand in these videos I try to use terms that make sense even to someone who is not terribly familiar with formal math :)
As far as I can tell, mathematicians are starting to say log instead of ln since log base 10 really isn't that widely applicable compared to natural log, so it makes sense to simplify the natural log to just "log" since you'll definitely be using that more.
In Russia we use ln(x) for base e, lg(x) for base 10 and log{n}(x) for base n
"{n}" is a subscript "n"
Mathologer I've seen both in Australia. I was taught to write it as ln(x) in high school but one of my university lecturers writes it as log(x).
amazingly well explained. never too slow to feel dumbed down, not too fast to feel like we're not getting the whole explaination. great job😊😊
About the million plus one zeros - if we only go to a million, the first non-zero digit of the error and the corresponding digit of our approximation might add up to generate a carry, and then the millionth digit would not be quite right... :)
Exactly, but of course a million will definitely do since the estimate I went for was actually a bit wasteful (easier to explain though :)
Your channel is one of my favorite on all of TH-cam, keep doing what you're doing Mathologer!
Can anyone give me the derivations of those trigonometric functions infinite series? Great video btw.
Those trig functions as infinite series come from the Taylor Series Expansion. You can learn about it on Wikipedia, and it even shows Sin x as an example. en.wikipedia.org/wiki/Taylor_series
It's actually really similar to the derivative of e.
As sin takes all the odd powers/factorials and cos all the evens and the derivative of a summand is the summand left to it, it's quite obvious than sin turns into cos and cos into sin. Also as the signs are alternating, sin turns into cos, cos into -sin, -sin into -cos and -cos into sin. Hard to put into words why that way around and not sin turning into -cose, but if you look at it, you should be able to figure it out :)
Thanks
anything besides sine and cosine are useless
palmer.wellesley.edu/~ivolic/pdf/Classes/Handouts/CalcHandoutsDrills/TaylorSeries.pdf
No arguments, no questions. just one word: awesome!
I also heard that e is the global maximum to the function x root x. Is that true?
Robin Powell Yes. Proof x^(1/x)=e^(ln x/x) .The derivative of ln x/x is , by the product rule of ln and 1/x, (ln x/x)'=( ln)'*1/x+(1/x)'*ln(x)=1/x^2-(ln(x)/x^2)=(1-ln(x))/x^2.For x
ustinian Constantinescu Wow that's some serious mathematics. Thanks a lot!
I've watched quite a few of the Mathologer series and they are consistently well done, ... and without a doubt one of the best at explaining complex mathematical concepts. Wish that these had been around when I was in school.
12:30 Do we have to worry about the approximation being exact so that the error is 0?
well the approximation of e will not be e so the error should be greater than zero.
This crossed my mind too, but ... e is equal to the *infinite* sum given, and every term in that sum is *positive* (e.g., 1/100! >0). This means that if you truncate the infinite sum, such as at the 8th term as he did in the video, you MUST get something *smaller* than e, and therefore there is a POSITIVE error -- not zero.
others have touched on it, but error can never be zero. That's what irrational means. If error was zero, you'd be saying you found a discrete value for e and this violates the definition of irrational. If you did this calculation and found your error to be zero, you probably ought to recheck your work and then take your best suit or dress to the cleaners for a fresh pressing because a Nobel Prize in mathematics is headed your way and you'll want to look sharp for the cameras
@Hortonius Imperialus: What you say is true about irrationals, but at this point in the video, Mathologer had not yet established that e is irrational; that's exactly what he was trying to prove!
true, i suppose. He had yet to test for it but it seems to me that a quantity defined by an infinite sum is by its nature irrational.. what I have an issue with and need to go back to a much earlier math text than i might expect, perhaps as far back as algebra to see by what identity he pulls off what he does at about the 5:00 mark..
I actually proved this to myself earlier this month. I even went a few steps further and found that b^ix = cos(ln(b)*x)+sin(ln(b)*x)*i.
AND using this identity: b^(n+m) = b^n * b^m, found that b^(x+yi) = b^x(cos(ln(b)*y)+sin(ln(b)*y)*i)
Using that formula one can raise any real number to any complex power!
now i have two numbers to calculate with python tomorrow
Your explanations are super clear, and combined with the animations and colors it's just great , thank you !
e is totally rational when I use my special integer between 1 and 2. :)
Golden ratio
The derivative proof was shockingly simple and elegant.
How did he get 22/7 to work in his T-shirt equation?
Check out this wiki page about this pretty identity en.wikipedia.org/wiki/Proof_that_22/7_exceeds_π
It's a famous putnam test problem. Try it out its easier than it looks.
Spreading math keeping the formalism and didactics in the flawless level.
This is a flawless victory!
going full circle to prove the equality?
I see what you did there! 😂
"The Mathologer": Our Mr. Clean of Mathematics!
at 8:32 you should really have said "less than".
Great video though! Thanks for making this :-)
I'm glad I found your comment before watching that part again. I thought I was completely missing something
This explanation was extremely cool! I knew nearly all of the facts presented, but the way they were put forward in a straightforward way from the start of the video (defining e) to the end (progressing from infinite sums to derivatives and integrals to get back to where you started) put all my knowledge of the concept of into one succinct packet! Great video!
Love this. A detail: Around 8:44 you say “equal to 7 factorial,” when you meant “less than 7!”
Correct, except for the timestamp. He says it at about 8:33, and in fact it would be better to watch from 8:24 to hear the full sentence.
These explanations are the best I've ever seen, I think the visual aspect helps a lot in learning the concepts in your videos
?/b!
Just think about it, the error is always positive :)
Yes, it took me like 10 seconds to realize it, but it's not really obvious.
You should have show that e-(the partial sum) is always (obviously) positive, and than your proof would be much easier to understand.
Also stumbled there.
Other than that video is pretty clear :)
Spent an hour on it, not obvious it is positive ... in general that is.
Stduhpf duh Plop an absolute value there :)
I cant replicate what you have shown after watching this video but I did understand everything you did.
Great video. I will definitely watch it again
I always read factorials as someone talking about the number excitedly.
One of my teachers told us a story about how he convinced his girlfriend that factorials are to be pronounced as "boy, oh boy", with an excited tone :)
Did you know that ln(pi^e) ~ pi
And logpi(e^pi) ~ e
Therefore pi ~ e/logpi(e)
And e ~ pi/ln(pi)
Pretty amazing
9:15 You might want to go for 1,000,001 zeros at the beginning of the error estimate because if you just had a million zeros, and the next digit was something large (like a 9 for example) then it would be very possible that your last digit was off by 1.
Simple Example: 2.720 +/- 0.009 (three zeros and only two correct digits of e)
+Jay Doherty If you needed to shoot for 1,000,001 zeros in 1/N! to ensure 1,000,000 accurate digits, that would be the reason. But you don't need 1,000,001 zeros in 1/N!, because 1/N! is so much larger than the actual error, you'll have zeros to spare.
The exponential function derivative argument really blew my mind!
What did the librarian say to the TH-camr?
Read more
AMMMM jealous much. PS I know your going to copy and past this comment.
•Nostalgia• this comment is old by now. it works better if the read more button is clickable though
LOL. MAKE THIS THE TOP COMMENT!
It may have been a tiny bit useful to remind viewers that 19/7 - (stuff) is positive because (stuff + positive error) = e.
Also at 12:52 you accidentally forgot an "a/b - " at the beginning of your expression
My TH-cam language settings is German! The joke did not work =( "Mehr anzeigen"
When my kids ge to this stage in school math, I'll make sure they watch these videos; so much better and clearer explanations than anything I've found in the text books... BIG THANKS. again.
10 minutes late, Dough!
D'oh!*
I always love your videos, but I just want to say I'm so glad you went into the "why" for the Taylor Remainder Theorem (essentially, even though you didn't quote it outright). I wish I'd have had you as a calculus professor, I never really (really) learned this stuff until much later.
maclaurin series -> taylor maclaurin series
look at this e number that I just found
Thank you for the video. I think your videos are always improving and that this was your best one yet. Very clear and to the point. I would love to hear you explain what transcendental numbers are and why e is one, so I look forward to your next vid. I would also like to see you explain why 1 and 0 are unusual, exceptional numbers that are often treated separately from all other numbers. One thing I don't ever hear mentioned is that e raised to the power phi (the Golden Ratio), that is, 2.71828...^1.61803..., is approximately 5. This might not seem important, but 5 is an important number, not least because it is the number of digits on human hands. The number 5 is also important in other contexts. Anyway, thanks for your videos. I will keep watching!
I already know the math, so this does not serve as an explanation. Nevertheless, I enjoyed it (and many of your other videos). Good job!
DitDede So did I!!
So... finally, multiple years after highschool someone explained to me why e^x is the derivative of itself in a way that I could understand. Thank you!
The irony is that really smart people fail to be able to explain things these things to the layman. It's cute when they think they can though. Nice try mathologer, but you threw e^ipi too fast at me.
I am afraid these videos do require a little bit of fluency in basic algebra and for the latter part a bit of calculus. Otherwise there is not much hope :) Also, if you have problems the first time you watch a video like this you can always watch it a second time, right?
Currently studying for an AP calculus exam and seeing an algebraic proof of the Maclaurin series for e^x is very, very helpful. Great video as always. Thanks!
I had seen e^(i*pi) = -1 back before I properly understood complex numbers, but upon thinking about them for several months straight while employed in a sometimes undemanding manual labour job (potato processing), I accidentally visualized the meaning of imaginary powers (rotations on a logarithmic tube of angle vs. magnitude) and realized that imaginary powers of e were radians of rotation, and then the identity made perfect sense.
Taylor series, on the other hand, have mostly just been my bane, because of situations where I've been silently expected to use them on assignments repeatedly showing up and passing me by without realizing that this was at all the intended solution (partly because of my very literal mind with regards to the use of the "=" sign in assignment questions). I don't find them especially difficult to understand, just aggravating to follow when theoretical physics teachers use them without mention.
Taylor series and other converging limits never felt as intuitive an explanation with regards to this identity, anyhow.
Takes me back to my youth and also to some OU maths courses done 15+ years ago. Thank you!
I have been a subscriber of your channer for 2 years and i can really confirm, this last 3 videos were best and each better than the last one. i really enjoy watching your videos i totally understood everything ! i have great fun doing what i love, math
You started at the right place for me: I have thought of the exponential function as growing out of (the limiting expression) for 'compound interest' -- that is, the rate of increase is proportional to the value of the function.
You gave a nice proof that e is an irrational number -- it is "too close" to any rational number (19/7 was a bonus).
I learned more from this video than I ever did in high school trig and calculus! excellent!
Brilliant video. Loved it.
Great explanation of irrationality of e. Remembering it being super tedious in calculus course. Can't wait the video on transcendental numbers!
Loved the proof of e being irrational. That in itself (followed by some simple steps and of course some assumptions) is proof enough for me of "why" Pi is transcendental.
Love your videos, thank you so much for making these.
Binge watching these :)
Well if that isn't Mr. "making maths entertaining again!" :-)
The way you explain the math of e feels like eating the spinich of the famous Popey the Sea-Man!
You create the desire to also want to go deep down into other (hopefully still hidden) secrets of the quenn of sciences! It feels very motivating because you empower/reactivate the inborn intuition of people, where Einstein told us: "Thge only thing that really matters is intuition!" You make math feel come alive again and making it usefull for millions of people! Thank you for that valuable contribution to society!
I'm actually a colleague of yours but not so much into software. If you tell me which software to use, then, once I become successfull with it, I'll mention you as the man who gave me the oportunity and connect with you, so we can both help even more people to inspire their lives with maths! :)
Brilliant explanation. It can also be pushed further to explain how natural logarithms (and then other logarithms) can be calculated be hand.
thanks for this great video, studying higher mathematics can be very abstract and complicated, watching videos like these renew my love for the subject.
This was a VERY engaging video, rich with interesting facts about e. The explanation of the e-irrational proof was quite easy to follow, and I never knew that estimating the error of approximations of e could be so simple! Also, I have never seen that particular proof that e^x is its own derivative. This video solidifies my choice of e as my favorite number!
Thank you so much for your great contribution to the world of math and science and your great favor to the mathematics students through out the world.
I who know calculus learned from school inside out that the derivative of lnx is 1/x, now I know why. Thanks mathologer, great video!!
I really enjoyed these explanations. I took calculus in college but this video really made it come alive for me. Thanks for making these clips!
Absolutely awesome explanation. Everything that we just took for granted regarding e has been explained so very well here. Thanks for this lovely video.
I think I'm going to start back at the beginning of your playlist and watch everything again but this time with a pencil and paper to follow along. In fact, you might encourage viewers to do that- break out a pencil and paper and follow along. I never really understood e until now. I mean, I kinda got it but no one ever really could break down the nuts and bolts. In high school, he said you'll have to wait till you get to calculus. When I got to calculus, the guy rambled for 15 minutes about natural functions, irrational numbers, scribbled a few lines on the board that really lead nowhere before finally giving up and just rewriting the identity at the bottom of the board.. it was kinda frustrating.. It makes much more sense now.
Definitely very good advice :)
How well did these explanations work for me? Absolutely stunning, Burkard. Thank you!
:)