Daniel Sullivan e itself has tons of utility but e^x being its own derivative...well I mean there are some practical uses for it but the man was all prepared for his great presentation on irrationality, not to show some odd physics and what not.
Why e is useful: When you are solving differential equations (which wind up describing an awful lot of things when you look carefully) you get lots of situations where a rate of change is related to the value of a thing. (ex: The rate of bunny births/deaths is related to how many bunnies there are.) When you find a solution, or even an approximation, for these sorts of things, e pops up all over the place. Compound interest, radioactive decay, population modeling, temperature change over time - all involve e.
@@Manish_Kumar_Singh Not so. Curve fitting is the fashioning of a "curve", or in other words a formula, to match a finite dataset. It is also known as interpolating, and, while generally looked down upon as an "inelegant" method, it has many a time provided essential insight in physical and mathematical matters. However, the fact that e pops up in most our differential equations is solely a consequence of the exponential function being the eigenfunction of the differential. In other words, e appears naturally around the differential simply because of what we have defined the differential to mean. If we were to look at equations based on other operators, other eigenfunctions (and, as a consequence, constants) would emerge.
@@NoriMori1992 yeah, the thumbnail is the only thing that lets you know it isn’t one of the ancients, and even the ‘e’ in the thumbnail looks oldschool
I reckon it is a bit old! We haven’t seen any videos with Ed recently which makes me think it been sitting unedited for a while. Also Brady’s camerawork is SO pre-pandemic.
@@godfreypigott OK, OK, I'll come in again. Nooooobody expects the self-derivative! Our chief weapons are the trivial function, e^x, ke^x and ke^{x+c}.
14:28 "We know that R has to be positive, because its a sum of positive terms". The irony of this being said by the same guy who did the "1+2+3+4+... = -1/12" video.
@@santhosh_se5476 still, saying that if you are adding positive numbers inifinite times leads to a negative number, without clarifying that you mean a different version of addition, is pretty sloppy
@ゴゴ Joji Joestar ゴゴ yeah, but its not the standard sum. Using standard addition would mean that the series of partial sums diverges to infinity. It doesnt converge to -1/12
Btw if anyone’s curious about how he got that series expansion e^x = 1 + x + 1/2x^2 + 1/3! x^3 ... , a really easy way to verify that this makes sense is to use the property that e^x = d/dx e^x. If you take a derivative of each of the terms in the infinite series, they all kind of “shuffle” down. 1 -> 0 so it disappears, x -> 1, 1/2x^2 -> x, etc! (One of the reasons I think this expansion is so neat is it’s another visual way to see why e^(i pi) + 1 = 0
@@beeble2003 The proof for the finite case was essentially rigorous. In the infinite case, he asserts that if | _x_ | < 1, _xⁿ_ → 0 as _n_ → ∞, which technically requires a definition of a limit to prove, but which is certainly true. If you are at the point where you know enough math to even define _e_ , you probably have no trouble understanding or proving that fact. The biggest handwave might have been factoring 1/(q+1) out of the series, but that is also perfectly valid and something you would presumably know by the time you know the McLaurin series for _e_ .
3:26 I am surprised he didn't say the obvious reason: that property lets us solve a whole bunch of differential equations that model physical and non-physical dynamics.
@@SlightlyAsync Any simple separable differential equation where there is a relationship between a function and itself will end up needing e as it represents a base case of, say, dx/dt = x which depending on the problem can then be scaled, transformed etc. This then is useful in things like radioactive decay, SIR epidemiological models, pharmacokinetics, ecological models etc etc. In fact ALL exponentials have the property that their derivative is n^xln(n) - so in fact by a scale quantity related to e (the natural log) all exponentials have this behaviour and e is in some sense the 'base case' for exponential growth that is then scaled/manipulated according to the needs. Exponential growth or decay are everywhere in nature due to the fact that many phenomena are multiplicative, more of one thing causes more of something else. And that is why e appears everywhere. Why is it that number is as fruitless a question as asking why the radius and circumference are 3.14, or why the ratios of right angle triangles follow the trigonometric functions, or the a/b = a + b/a is the Golden ratio. Nature is just that way.
@@SlightlyAsyncit is simplification, but the sequence is roughly like this: 1. we seek the number n which satisfies this property of function f(x) = n^x such that df(x) / dx = f(x) (fixed point of differentiation ). From this we see that relative growth in function df(x)/f(x) with x is equal to increase in x, which is characteristic of exponential n^x, now n happens to be 2.71828..., we name it e because it is important, 2. we typically substitute a function which contains e^x with some additives into equations of models (they contain derivatives or integrals) and dividing by it, we get algebraic equations (i.e. in numbers, not functions) which are easier to solve. That's why e is so important
The number e has applications in finance, economics, growth rates, statistics, and tons of other stuff. Surprised he didn't know any applications outside of pure math
Ha! Yes! At the very least, e is theoretically important because it is "natural," in the sense that it answers that fundamental question of f'(x)=f(x). (In particular, of the general answers for this question, e^x is the one with multiplicative and additive identities as "choices" in the appropriate places.)
@@tinyanisu1927 Yes! Used to study an object for which its instantaneous rate of growth/decay is proportional to its value, so to speak. The "natural" proportion being 1. This is why f(x)=e^x with f'(x)=1*e^x=1*f(x) is more "natural" than g(x)=e^(2x) with g'(x)=2*e^(2x)=2*g(x). And considering all h(x)=a*e^(bx+c) with h'(x)=a*b*e^(bx+c), it is most "natural" to use a=b=1 (mult. id.) and c=0 (add. id.). [I'm only adding all of the math here, now, to clarify what I meant in my first comment.] If you'd like to see a funny video about e, and why taking a=b=c=0 is actually the most "natural" choice, check out my Calc 2 (Integration Techniques and Applications) playlist on my channel - the video is called Exponential Function - How to Differentiate, How to Integrate // FUNNY/HUMOR
There is really nothing else special about e than the fact that e^x = D(e^x). All exponential functions (including all laws of nature etc) could be written with any other number as the base, differentiating those equations is just easier when using e, thats why the convention to use e exists.
@@Arduu123 Yes, I agree (although the first sentence is a touch subjective). What you're highlighting is why the adjective "natural" is appended to the particular exponential function f(x)=e^x.
3:14 Brady is a brilliant interviewer. I love how he's able to ask "normal human" questions and how those are the ones that experts trip over and make them think. I bet Brady could have asked all kinds of very in depth detail questions about some obscure technicality and Prof Copeland would have had a quick answer but a simple "Why is that useful?" is not a thing that he has thought about :-D
"Why is that useful?" To me it is very useful when solving differential equations, a lot of the methods for solving them involve e in some way. Since differential equations describe a lot things in nature, e becomes a really important function.
That moment you have to explain the function that's so important and used in basically everywhere, that you have no idea where to start with, then you simply say, "I don't really know."
I think it was more of a why is that function literally everywhere, why do physical processes behave in ways where this one number pops up everywhere. And that's what he doesn't know.
'Why is that important?' - Consider you have some generic continuous function, but only know statements about its various derivatives; this is common in physical systems, where different physical quantities based on position, velocity, acceleration, mass, etc, all have to relate to each other. A complicated function can often be written as a combination of simpler functions, and as mentioned, the e^x function never 'goes away' no matter how you differentiate or integrate it, while other functions kind of 'shrivel away'. So if you can write your mystery unknown function in terms of some e^x part and some non-e^x part that combine in some way, you can often end up with a bunch of stuff that can factor out, since all those e^x parts are going to hang around when you plug them in. This often ends up making finding the 'other part' a lot easier. Put another way, having a function that self-generates under differentiation gives you a 'stable spot' from which to look for other parts of the answer to large classes of problems. e^x on the reals contains exponential growth, which is common in systems. On the complex numbers, e^x contains oscillations around a central value, which is also common in systems. So you end up with a single function that can codify two very common behaviours, *and* which is self-stable to the kinds of equations you often have to solve to deal with physical systems.
One definition of e that I like is that it is the only base for exponentiation where the slope around 0 is one. This follows the derivative definition ((e^x)'(0) = e^0 = 1), but has a nice consequence - e^x around 0 behaves like x + 1 which is useful for establishing logarithmic units: using e as the base means that multiplying by something close to 1 (imagine adding or subtracting small percentages) can be seen through the logarithm as adding that multiplier minus 1. x + y % = x × (1 + y / 100) ≈ x × e ^ (y / 100) The logarithmic unit that is based on e is called the neper (Np). Units of percentages are analogous to centinepers (cNp) but behave in a more consistent fashion.
@3:14 My answer: Any k^x looks the same if you ignore the scale. e is the value of k where no scaling is required after you differentiate it. It's like the inflexion point or the origin for that scaling. Consequently it pops out as a sort of 'correction factor' when you make other values of k and the derivatives fit the curve.
For the use in physics : the fact that the d/dx (e^x) = e^x can rbing us easy solutions for differentials equations (equations with functions and their diffenretials). It's important because differenciate a fonction tells us about how this functions evolve in time, and if you have a relation between the state of the system you study and the way it will evolve, you have a differential equation, and we can solve some of these with the exponential fonction.
The myth about Gauss have to be developed and thought to the younger generations. That it isn't precisely true is not that important. How do you think myths about other historical figures came to be?
3:19 “why is that useful?” Because this way e^(cx) is an eigenfunction of the differential operator, which makes solving (certain) differential equations easy. For example, a linear dynamical system’s response to a (complex) exponential is always another (complex) exponential with a (complex) scale factor. This is one of the reasons why Fourier analysis is so useful for analysing linear dynamical systems.
@@WritingGeekNL he got caught up a bit, instead of defining e, he described an interesting property of e. Explaining why that property is important is a little harder than describing why e is important.
I would say that the reason that it is useful is due to the fact that it is the solution of all linear ODE’s which are ubiquitous in physics and maths.
There is a mistake at 11:18 in the equation presented. The top already factored out the q+1, so it is supposed to be q+2 and q+3. Sorry. I loved the video! Ed is great!
It's the first time I'm this early to a numberphile video. Also, we came across this problem in our real analysis course a few days back. What a coincidence!
Ok my 12 yr old daughter asked me what "e" was last night and I needed to look this up because I couldn't remember it's significance. Funny but I was talking to her about tangents, rate of change, acceleration etc but stopped because I thought I was digging myself too deep into this stuff at her age. She's a super sharp kid who's in advanced classes at school. Now I'm thinking I could use this channel to push her even further ahead as she picks up concepts very quickly. I'm going to get her to start watching this channel with me! Wonderful stuff!
(hey wait this gives me an idea :3 ) a fangirl fantasizes about someone and then meets them and they become friends irl > so your friendship started out imaginary but now it's also real! >> yeah, you might say our relationship is.._complex_ >> 8^)
Pi talking to i on March 15... Pi: Why was e^x so lonely at my birthday party yesterday? i: Because every time he tried to integrate, he ended up with himself. Pi: Well, he would have ended up with himself and a constant, and by integrating further he could end up with any polynomial he wanted. i: Nah, he wouldn't befriend a constant. He has limits...(-inf to x)
3:13 "Why is that useful?" You have an analytic method for computing derivatives of similar functions with a base other than e that is simple to calculate: e.g. f(x) = 2^x 2^x = e ^ (x ln2) then using substitution (chain rule) : p = e ^ x --> f(p) = p ^ (ln2) df/dx = (dp/dx) . (df/dp) = [e ^ x] . [(ln2) p ^ (ln2 - 1)] = ln2 . [e ^ x] . [e ^ x(ln2 - 1)] = ln2 . e^ (x ln 2) = ln2 . 2 ^ x e can be thought of as a 'natural base' in the same way 'ln' can be thought of as the 'natural log' It's also why radians are the 'natural' unit for angles. Arclength / Radius = Angle if and only if you are using radians. Does this make tau more "natural" than pi? You decide.
Yo, I tried working through this exact proof yesterday; it was a Cambridge entry exam question where they guided you through it, and I could not figure out the last bit. So thanks for this video, perfect timing!
it's useful because it is a solution to the differential equation d/dx f(x) = f(x) i.e y' = y solution f(x) = e^x this differential equation shows up all the time in physics
@@MatthewOBrien314 yes, thats the definition but why is it useful and ubiquitous. What i love about mathematics is that theres always another level, the more you know the more you can appreciate what a small amount what you know really is.
One answer to the question @3:14 is that it allows us to solve certain common modelling problems (e.g., linear homogeneous ordinary differential equation with variable coefficients). However, like pi is everywhere in geometry, e is everywhere in analysis/calculus.
3:15 it's useful because it simplifies solving differential equations. That's why you see e^f(x) everywhere in analysis (or sin/cos which is e^ix in disguise)
Professor Copeland has a wonderful way of talking and teaching. He also seems like a very nice man. I wish I had him as one of my math teachers in uni.
Something I never liked in my math classes is how vague it can feel at times. I completely understand why it can be and its fine if you don’t know why something is the way it is. To me, e (I like calling it Euler’s Number) is to me the base rate at which any exponential process (growth or decay) occurs at. I’m aware of its key use in solving differential equations as well as in Euler’s Formula [e^(i*x)=i*sin(x)+cos(x)]. No matter what exponential process occurs for like if you derive the function 2^x with respect to x you get ln(2)*2^x. That natural log constant is determined by the base to the natural log of e, which to me helps describe how e is sort of a hidden factor in exponential functions. I think its more intuitive to a person who doesn’t understand calculus to give the explanation of: ‘e is the base rate of any exponentially growing or decaying processes’ rather than ‘e is the only number where you can take the function e^x and have its derivative be the same no mater how many times you differentiate it. I’m by no means necessary an expert on math. I’m happy and open to corrections of my own mistakes if I make them. I’m not even done with college as a amateur electronics engineer who likes crunching numbers and solving circuits with the tools math provides more so than is needed in my field of work. I simply love tutoring the subject and removing as many ambiguities as I can when I help others because I never want to make math seem more imposing than it is to someone who lacks the background or intuition needed to grasp the concept confidently. However I still love the visual style of explaining e here and appreciate this video as it is. Edit: typo
Great to see a proof once in a while! Especially with professor Copeland. Although there is a typo in the graphics at 11:09. In the top equation, the second term in the bracket says 1/(q+1) but it should read 1/(q+2), and similarly the third term should read 1/((q+2)(q+3)). Thanks for great content!
Whenever the amount you are going to have is related to the amount you have now, you get an e. If the rate of change is proportional to the starting amount you get and e^ax, where 1/a is the constant of proportionality.
just wanna point out an alternative (and quicker) way to finish this off: since q>1, we know that the expression in the last line at 10:03 (aka R) is at most 1/2! + 1/3! + 1/4! + ... which is just the series-representation of e (see 6:11) minus the first two terms. so 0 < R because obviously 2^k is smaller than (k+1)! for k>1). this means that 1/2! + 1/3! + 1/4! + ... < 1 which can be obtained without use of the geometric series for 1/2^k by the simple geometric "repeatedly halfening a distance of 1"-argument.
@@renyhp ...assuming that we want to figure out whether e is rational before even knowing if it's less than 3, of course. for the sake of completeness, I'll ad that bit to my original comment.
Tbh it is a hard question, mainly because there's no easy explanation of what it does. For example, pi is "the circle number", and because everyone knows about circles, everyone can understand how it can be important. But e is "the differentiation number"? That to a layman doesn't sound cool, or useful. But anyone that has ever done Calculus I or greater knows how practical and everpresent e is
If you assume q>1 in 14:50, wouldn’t you have a loose end with q = 1, and then e being an integer, and therefore rational? On the other hand, even if q = 1, you would still get R < 1, so R can’t be a positive integer anyway, so why assume q > 1?
I remember a previous Numberphile video saying that the proof that e was transcendental was that there was a whole number between 0 and 1. Now that statement makes sense.
Couple of useful things about d/dx e^x = e^x from physics: Radioactive decay is proportional to the amount of undecayed substance left, and the force a spring exerts is proportional to how stretched it is. The amount of decay is the rate of change of the amount of substance, and the force the spring produces directly moves whatever is on the end of it. You always end up needing to multiply it by something or whatever, but a function whose value and rate of change are the same is exactly the starting point you need to work with this kind of model.
To the question "why is it important"?, exponential functions are the natural representation of multiplicative growth processes, since they correspond to the rate of growth of a population at a given time being proportional to the size of the population at that time.
True. There are more answers as well, like 1. e can be used to find solutions to diff eqs because of the property of the derivative match. 2. e^(i•theta) in the complex plane can be used to relate to harmonics sin and cos. I'm sure many more.
Proof by contradiction always feels like ending a story with "and then they woke up and it was all a dream"
The best kind of story
For mathematicians
+
Nah, it's way more satisfying than that.
ههههههههه..... Nice!
Felt that
More professors and teachers should be like Ed. When you don't really know something at the moment just say "I don't really know".
Luckily at uni I have usually found that they are like that. If you are comfortable with your knowledge, you probably are okay with it
He really was stumped on the utility of e, as if he had never looked at it in that way. He had the humility to say “I don’t know”.
Daniel Sullivan e itself has tons of utility but e^x being its own derivative...well I mean there are some practical uses for it but the man was all prepared for his great presentation on irrationality, not to show some odd physics and what not.
@@stephenbeck7222 great preparation, when you are defining e as a number which does not change after diferentating xD but that number should be 0 xD
...people, not teachers only.
"R has to be positive because it's just a sum of positive terms."
Now that's rich coming from you lol
And comes the Riemann zeta function of z(-1)
Hahahaha
This made me laugh😂
Takes me back
Haaaaa
well but the series converges, which cannot be said about the other example y'all are thinking about
Why e is useful: When you are solving differential equations (which wind up describing an awful lot of things when you look carefully) you get lots of situations where a rate of change is related to the value of a thing. (ex: The rate of bunny births/deaths is related to how many bunnies there are.) When you find a solution, or even an approximation, for these sorts of things, e pops up all over the place. Compound interest, radioactive decay, population modeling, temperature change over time - all involve e.
Fellow Eddie Woo enjoyer?
Kind of a naturally occurring number.
Basically, the e^x is the eigenfunction of the derivative operator, so it's bound to crop up in equations involving derivatives.
that's just curve fitting, donsent mean it's usefull.
i say the same thing for fibonachi series, gloden number
@@Manish_Kumar_Singh Not so. Curve fitting is the fashioning of a "curve", or in other words a formula, to match a finite dataset. It is also known as interpolating, and, while generally looked down upon as an "inelegant" method, it has many a time provided essential insight in physical and mathematical matters. However, the fact that e pops up in most our differential equations is solely a consequence of the exponential function being the eigenfunction of the differential. In other words, e appears naturally around the differential simply because of what we have defined the differential to mean. If we were to look at equations based on other operators, other eigenfunctions (and, as a consequence, constants) would emerge.
This feels like an oldschool numberphile video :0
Okay, so it's not just me. It's so old-school that I was thinking to myself, "Have I seen this before? Did they reupload a really old video?" 😂
@@NoriMori1992 yeah, the thumbnail is the only thing that lets you know it isn’t one of the ancients, and even the ‘e’ in the thumbnail looks oldschool
Because the guest used to be a lot in the early videos
I reckon it is a bit old! We haven’t seen any videos with Ed recently which makes me think it been sitting unedited for a while. Also Brady’s camerawork is SO pre-pandemic.
was looking for this comment.
"e" who shall not be named...
Take those upvotes and get out!
Brilliant.
e by gum.
Actually I think he’s from Manchester?
mr. so-so?
Way underrated comment!
"e^x is the only function that differentiated it gets back to itself"
Zero function: _angry analytical noises_
OK, OK, the only non-trivial function.
ke^x: _angry analytical noises_
@@beeble2003 Still not general enough: y=k.e^(x+c)
@@godfreypigott OK, OK, I'll come in again. Nooooobody expects the self-derivative! Our chief weapons are the trivial function, e^x, ke^x and ke^{x+c}.
This is why the condition e^0 = 1 is important
@@beeble2003 Ah, I see you're a man of culture as well
"e is approximately 3"
Smells like ENGINEER in here
I still prefer the classic "Pi = 3, for small values of Pi, and/or large values of 3."
@@Varksterable the limit as x approaches π from the left is 3.
Pi = 3, and 3 is close to 5, so we can round up to 10
You seen that older Numberphile video with Dr. Padilla where Don Page wrote a paper like "yeah, e is approximately equal to 10"?
Engineer: π = e
The smiliest astrophysicist in the planet is back!
Well get him out of there already, he's got maths to do!
Hollow Earthers unite!
I love him
I went to a talk given by him a year ago safe to say it was super wholesome
Ed’s the Mister Rogers of Numberphile, for sure. I’m always happy to see him!
14:28 "We know that R has to be positive, because its a sum of positive terms". The irony of this being said by the same guy who did the "1+2+3+4+... = -1/12" video.
Haha, thought the same, can't take him serious anymore
lol yeah
haha but this series converges unlike that one ....😛
@@santhosh_se5476 still, saying that if you are adding positive numbers inifinite times leads to a negative number, without clarifying that you mean a different version of addition, is pretty sloppy
@ゴゴ Joji Joestar ゴゴ yeah, but its not the standard sum. Using standard addition would mean that the series of partial sums diverges to infinity.
It doesnt converge to -1/12
All mathematicians shoukd write "ta-dah!" At the end of their proofs instead of QED.
Tom Korner used to write "I WIN!".
I agree
I thought mathematicians draw a middle finger and address it to physicists
@@JMUDoc really?
@@coleabrahams9331 Sorry - it was Tom Korner, not Feynman.
(And I was lectured by Korner at Cambridge, so I don't know why I confused the two!)
Btw if anyone’s curious about how he got that series expansion e^x = 1 + x + 1/2x^2 + 1/3! x^3 ... , a really easy way to verify that this makes sense is to use the property that e^x = d/dx e^x. If you take a derivative of each of the terms in the infinite series, they all kind of “shuffle” down. 1 -> 0 so it disappears, x -> 1, 1/2x^2 -> x, etc!
(One of the reasons I think this expansion is so neat is it’s another visual way to see why e^(i pi) + 1 = 0
Love that it was pretty close to completely rigorous and had minimal hand waving
Including the proof of the sum!
@@forthrightgambitia1032 Well, that actually was the biggest handwave.
beeble2003 pretty much how it’s taught in high school maths already. You need some calculus to do a real proof but the concepts are all there.
@@beeble2003 The proof for the finite case was essentially rigorous. In the infinite case, he asserts that if | _x_ | < 1, _xⁿ_ → 0 as _n_ → ∞, which technically requires a definition of a limit to prove, but which is certainly true. If you are at the point where you know enough math to even define _e_ , you probably have no trouble understanding or proving that fact. The biggest handwave might have been factoring 1/(q+1) out of the series, but that is also perfectly valid and something you would presumably know by the time you know the McLaurin series for _e_ .
@@EebstertheGreat Yes, I'm not claiming anything was wrong, just that some details were handwaved away.
Prof Ed's voice is just so calming. I'm pretty sure I transcended into some dimension of e just listening to this video.
I see what you did there :^)
Man I love Ed. It’s a pleasant surprise every time he shows up
3:26 I am surprised he didn't say the obvious reason: that property lets us solve a whole bunch of differential equations that model physical and non-physical dynamics.
But how did e get into those models?
@@SlightlyAsync Any simple separable differential equation where there is a relationship between a function and itself will end up needing e as it represents a base case of, say, dx/dt = x which depending on the problem can then be scaled, transformed etc.
This then is useful in things like radioactive decay, SIR epidemiological models, pharmacokinetics, ecological models etc etc.
In fact ALL exponentials have the property that their derivative is n^xln(n) - so in fact by a scale quantity related to e (the natural log) all exponentials have this behaviour and e is in some sense the 'base case' for exponential growth that is then scaled/manipulated according to the needs. Exponential growth or decay are everywhere in nature due to the fact that many phenomena are multiplicative, more of one thing causes more of something else.
And that is why e appears everywhere. Why is it that number is as fruitless a question as asking why the radius and circumference are 3.14, or why the ratios of right angle triangles follow the trigonometric functions, or the a/b = a + b/a is the Golden ratio. Nature is just that way.
He was blind-sided by the question. "What's the use of a newborn baby?"
@@SlightlyAsyncit is simplification, but the sequence is roughly like this: 1. we seek the number n which satisfies this property of function f(x) = n^x such that df(x) / dx = f(x) (fixed point of differentiation ). From this we see that relative growth in function df(x)/f(x) with x is equal to increase in x, which is characteristic of exponential n^x, now n happens to be 2.71828..., we name it e because it is important, 2. we typically substitute a function which contains e^x with some additives into equations of models (they contain derivatives or integrals) and dividing by it, we get algebraic equations (i.e. in numbers, not functions) which are easier to solve. That's why e is so important
The number e has applications in finance, economics, growth rates, statistics, and tons of other stuff. Surprised he didn't know any applications outside of pure math
Camera man: Why is it important?
Mathematician: Wrong question!
Ha! Yes! At the very least, e is theoretically important because it is "natural," in the sense that it answers that fundamental question of f'(x)=f(x). (In particular, of the general answers for this question, e^x is the one with multiplicative and additive identities as "choices" in the appropriate places.)
@@mathanalogies9765 also important to study growth/decay of things that are proportional to their instantaneous value.
@@tinyanisu1927 Yes! Used to study an object for which its instantaneous rate of growth/decay is proportional to its value, so to speak. The "natural" proportion being 1. This is why
f(x)=e^x with f'(x)=1*e^x=1*f(x)
is more "natural" than
g(x)=e^(2x) with g'(x)=2*e^(2x)=2*g(x).
And considering all
h(x)=a*e^(bx+c) with h'(x)=a*b*e^(bx+c),
it is most "natural" to use
a=b=1 (mult. id.) and c=0 (add. id.).
[I'm only adding all of the math here, now, to clarify what I meant in my first comment.]
If you'd like to see a funny video about e, and why taking
a=b=c=0
is actually the most "natural" choice, check out my
Calc 2 (Integration Techniques and Applications)
playlist on my channel - the video is called
Exponential Function - How to Differentiate, How to Integrate // FUNNY/HUMOR
There is really nothing else special about e than the fact that e^x = D(e^x). All exponential functions (including all laws of nature etc) could be written with any other number as the base, differentiating those equations is just easier when using e, thats why the convention to use e exists.
@@Arduu123 Yes, I agree (although the first sentence is a touch subjective). What you're highlighting is why the adjective "natural" is appended to the particular exponential function f(x)=e^x.
3:14 Brady is a brilliant interviewer. I love how he's able to ask "normal human" questions and how those are the ones that experts trip over and make them think. I bet Brady could have asked all kinds of very in depth detail questions about some obscure technicality and Prof Copeland would have had a quick answer but a simple "Why is that useful?" is not a thing that he has thought about :-D
Prof Copeland's voice is ASMR in the world of math and physics. Could listen to him for hours.
4:53 The moment you understand the choice of thumbnail
His handwriting is so neat!
Yes, but why would you learn to write "x" as 2 curves? Is it due to some "don't cross the lines" philosophy?
@@Einyen i think it comes from the way cursive was taught
@@Einyen idk the exact reason but it helps with not confusing "x" the variable with the multipication symbol
@@abdullahenaya that's why no one uses an x for multiplication anymore after they learn basic algebra
The talk about the derivatives was a bit of a tangent...
A wild tangent 😁
underrated comment
GET OUT
I don't know, I found it pretty integral to understanding why e is important XD
@@TheAlps36 It's his area of expertise
Ed has such a welcoming and warm smile
"Why is that useful?"
To me it is very useful when solving differential equations, a lot of the methods for solving them involve e in some way. Since differential equations describe a lot things in nature, e becomes a really important function.
plus logarithmic differentiation makes it so easy to deal a^x functions
My love for Ed has exponentially grown!
yaaaaaaay another cuber that watches numberphile
@@dilemmacubing hey, sup?
After being primed by the 15 minutes of the proof, I read this comment as "My love for Ed has exponentially grown-factorial". :D
@@TrondReitan7000 haha nice
Yeah but what Was the accelleration of the growth?
That proof felt like setting up a lot of dominoes and then watching them all fall really quickly.
This would be a nice visualisation of a proof by contradiction.
The "tada" got me in stitches. Bravo on the presentation.
It did? Really? Hm
"Tada" must be the official pronunciation of ∎.
That moment you have to explain the function that's so important and used in basically everywhere, that you have no idea where to start with, then you simply say, "I don't really know."
I think it was more of a why is that function literally everywhere, why do physical processes behave in ways where this one number pops up everywhere. And that's what he doesn't know.
0:39 what is e? „o“
Other Vowels: I see how it is
'Why is that important?' - Consider you have some generic continuous function, but only know statements about its various derivatives; this is common in physical systems, where different physical quantities based on position, velocity, acceleration, mass, etc, all have to relate to each other. A complicated function can often be written as a combination of simpler functions, and as mentioned, the e^x function never 'goes away' no matter how you differentiate or integrate it, while other functions kind of 'shrivel away'. So if you can write your mystery unknown function in terms of some e^x part and some non-e^x part that combine in some way, you can often end up with a bunch of stuff that can factor out, since all those e^x parts are going to hang around when you plug them in. This often ends up making finding the 'other part' a lot easier.
Put another way, having a function that self-generates under differentiation gives you a 'stable spot' from which to look for other parts of the answer to large classes of problems. e^x on the reals contains exponential growth, which is common in systems. On the complex numbers, e^x contains oscillations around a central value, which is also common in systems. So you end up with a single function that can codify two very common behaviours, *and* which is self-stable to the kinds of equations you often have to solve to deal with physical systems.
One definition of e that I like is that it is the only base for exponentiation where the slope around 0 is one. This follows the derivative definition ((e^x)'(0) = e^0 = 1), but has a nice consequence - e^x around 0 behaves like x + 1 which is useful for establishing logarithmic units: using e as the base means that multiplying by something close to 1 (imagine adding or subtracting small percentages) can be seen through the logarithm as adding that multiplier minus 1.
x + y % = x × (1 + y / 100) ≈ x × e ^ (y / 100)
The logarithmic unit that is based on e is called the neper (Np). Units of percentages are analogous to centinepers (cNp) but behave in a more consistent fashion.
Why do I feel like the mathematics nerds are all just so humble people? I love it! Great to see not all of humanity is bad :)
Feel free to see Stephen Wolfram for a counterexample lol
@@CodyEthanJordan don't tell me he's the guy who made wolfram alpha
@@vladimirjosh6575 Yea he's the founder of Wolfram
Physics people too tend to be pretty humble. This guy is a physicist, though of course he's also a math nerd 😊
You dont get to see the bickering and backstabbing
15:05 "tah-dah!"
Actually, mathematicians call that "qed".
Mathematicians don't say "I love you", they say "$\blacksquare$", and I think that's beautiful
@@YaamFel $\hfill \square$ please
@@cobracrystal_ did you work for E-systems?
Q.E.D. is pronounced "tah-dah"
@3:14 My answer: Any k^x looks the same if you ignore the scale. e is the value of k where no scaling is required after you differentiate it. It's like the inflexion point or the origin for that scaling. Consequently it pops out as a sort of 'correction factor' when you make other values of k and the derivatives fit the curve.
3:13 "why is that useful?" because it makes learning calculus just a tad easier Brady
For the use in physics : the fact that the d/dx (e^x) = e^x can rbing us easy solutions for differentials equations (equations with functions and their diffenretials). It's important because differenciate a fonction tells us about how this functions evolve in time, and if you have a relation between the state of the system you study and the way it will evolve, you have a differential equation, and we can solve some of these with the exponential fonction.
14:29
-1/12 enters the chat
The level of knowledge being laid out so deep the camera is having trouble focusing
And my brain comprehending 😅
11:25 Gauss apparently did it when he was about, 3 Alright so...
I think he’s joking. The only thing I remember Gauss doing when he was 3 was checking his father’s books to make sure everything added up correctly.
I think the "about three" from e itself may have had something to do with it as well.
He did it when he was approximately e years old.
The myth about Gauss have to be developed and thought to the younger generations.
That it isn't precisely true is not that important. How do you think myths about other historical figures came to be?
Well then, I was wondering what on Earth does Voldemort have to do with e's irrationality. Now I know :D
He is irrationally made as the antagonist in Harry potter I assume
@@mysticalpie4695 and covers the tale of his transcendence of the mortal coil.
This definitely brings back the old-school Numberphile vibe.
3:19 “why is that useful?”
Because this way e^(cx) is an eigenfunction of the differential operator, which makes solving (certain) differential equations easy. For example, a linear dynamical system’s response to a (complex) exponential is always another (complex) exponential with a (complex) scale factor. This is one of the reasons why Fourier analysis is so useful for analysing linear dynamical systems.
I wish more people wrote the initial 1 as 1/(0!)
How about x^0/0! ?
But that's 1/0, which is definitely not 1
@@llamallama4536 0! is 1
@@llamallama4536 1/0! is not 1/0. 0! = 1.
@@llamallama4536 no
"Why is that useful?" "...I'm not sure."
I feel like a lot of mathematics is this. And it's part of why it's so much fun.
The number e is actually the most useful number in Applied Mathematics, so I'm not sure why he said that.
@@WritingGeekNL it's a hard question to answer on the spot like that
You and the physicist community think that. Mathematics is mental master...
@@WritingGeekNL he got caught up a bit, instead of defining e, he described an interesting property of e. Explaining why that property is important is a little harder than describing why e is important.
@@WritingGeekNL the question wasn't why e is useful, it was why that specific property is.
I like how they discuss the problem the whole video instead of just solve it
I would say that the reason that it is useful is due to the fact that it is the solution of all linear ODE’s which are ubiquitous in physics and maths.
When he said: this is going to be just like proving sqrt(2) is irrational, i was like ok nice this will be easy.
It wasn't...
There is a mistake at 11:18 in the equation presented. The top already factored out the q+1, so it is supposed to be q+2 and q+3.
Sorry. I loved the video! Ed is great!
Super appreciate the detail of Professor Copeland and also the graphics!!! Thank you very much
e π and phi are always lurking around the corner
It's the first time I'm this early to a numberphile video. Also, we came across this problem in our real analysis course a few days back. What a coincidence!
Ok my 12 yr old daughter asked me what "e" was last night and I needed to look this up because I couldn't remember it's significance. Funny but I was talking to her about tangents, rate of change, acceleration etc but stopped because I thought I was digging myself too deep into this stuff at her age. She's a super sharp kid who's in advanced classes at school. Now I'm thinking I could use this channel to push her even further ahead as she picks up concepts very quickly. I'm going to get her to start watching this channel with me! Wonderful stuff!
- "ive been to his grave.."
- "have you?"
You two are real friends arent you?
as opposed to imaginary friends :')
(hey wait this gives me an idea :3 )
a fangirl fantasizes about someone and then meets them and they become friends irl
> so your friendship started out imaginary but now it's also real!
>> yeah, you might say our relationship is.._complex_
>> 8^)
The graph y=e^x isn't the only graph where the differential is the identity. y=0 has that property too.
Ed: It's the only number where if I differentiate it [meaning e^x] I get back the same number.
Zero: Am I a joke to you?
Ce^x : I know, right?
Yes he forgot about zero
Yeah, but the joke about integration doesn't work with zero, whereas e^x is bidirectional...
Zero: am I nothing to you?
Pi talking to i on March 15...
Pi: Why was e^x so lonely at my birthday party yesterday?
i: Because every time he tried to integrate, he ended up with himself.
Pi: Well, he would have ended up with himself and a constant, and by integrating further he could end up with any polynomial he wanted.
i: Nah, he wouldn't befriend a constant. He has limits...(-inf to x)
ed has the most soothing presentation style and voice
3:13
"Why is that useful?"
You have an analytic method for computing derivatives of similar functions with a base other than e that is simple to calculate:
e.g. f(x) = 2^x
2^x = e ^ (x ln2)
then using substitution (chain rule) :
p = e ^ x
--> f(p) = p ^ (ln2)
df/dx = (dp/dx) . (df/dp)
= [e ^ x] . [(ln2) p ^ (ln2 - 1)]
= ln2 . [e ^ x] . [e ^ x(ln2 - 1)]
= ln2 . e^ (x ln 2)
= ln2 . 2 ^ x
e can be thought of as a 'natural base'
in the same way 'ln' can be thought of as the 'natural log'
It's also why radians are the 'natural' unit for angles.
Arclength / Radius = Angle
if and only if you are using radians.
Does this make tau more "natural" than pi? You decide.
That is one of the clearest, most detailed explanations I have never understood in my life
Yo, I tried working through this exact proof yesterday; it was a Cambridge entry exam question where they guided you through it, and I could not figure out the last bit. So thanks for this video, perfect timing!
"Why is that useful?"
".
.
.
.
.
.
.
I don't know?"
it's useful because it is a solution to the differential equation
d/dx f(x) = f(x)
i.e
y' = y
solution f(x) = e^x
this differential equation shows up all the time in physics
@@MatthewOBrien314 yes, thats the definition but why is it useful and ubiquitous. What i love about mathematics is that theres always another level, the more you know the more you can appreciate what a small amount what you know really is.
I saw Ed's face in the first frame and shouted YES! So happy to see a new video with Ed!
It's such a simple proof but I would never in a million years figure it out
I remember doing this STEP question and it was one of the most beautiful yet surprisingly simple proofs I have come across!
Me: Sees Voldemort on thumbnail
*So after the deathly hallows he retired and became a mathematician*
Friendly reminder that the author who created Voldemort is transphobic.
@@imveryangryitsnotbutter Passive aggressiveness I see. I still like Rowling, I just ignore their twitter and the transphobic stuff
Voldemort is not the first villain I associate with E
@@imveryangryitsnotbutter 👏
Mathemagician.
I love how e shows up when you calculate compound interest of 100% with infinitesimally small time slices.
I'm not sure but isn't that how Euler discovered e? Or was it something else?
@@sadkritx6200 According to wikipedia it was Jacob Bernoulli, but yes.
Euler is cited as naming the constant "e".
@@HotelPapa100 Ouu did not know that. Thanks 👍
I love the math in this one - it's so elegant, but man, the autofocus continually hunting was killing my eyes.
Who else just loves the professor's calm voice on everything?
Professor Copeland!!! We’ve missed you.
One answer to the question @3:14 is that it allows us to solve certain common modelling problems (e.g., linear homogeneous ordinary differential equation with variable coefficients). However, like pi is everywhere in geometry, e is everywhere in analysis/calculus.
15:05 Professor stole the spell, Lord Voldemort does not look happy.
Brilliant! Cheers! I like the calm, elegant and friendly way you are talking and being so keen on what your are doing.
James Grime: we're gonna talk about e!!
Ed Copeland: we're gonna *prove it*
3:15 it's useful because it simplifies solving differential equations. That's why you see e^f(x) everywhere in analysis (or sin/cos which is e^ix in disguise)
This was fun. Thanks.
Professor Copeland has a wonderful way of talking and teaching. He also seems like a very nice man. I wish I had him as one of my math teachers in uni.
That’s the craziest plot twist of all times
3:18 I like that humble answer!
i: [Talking to e] Be rational.
e: Be real.
I literally got a hoodie with π and I having this conversation hahahaha
Pi comes along and says: I can solve both your problems if we work together.
@@josh11735 when they get together:
1: be positive.
Something I never liked in my math classes is how vague it can feel at times. I completely understand why it can be and its fine if you don’t know why something is the way it is. To me, e (I like calling it Euler’s Number) is to me the base rate at which any exponential process (growth or decay) occurs at. I’m aware of its key use in solving differential equations as well as in Euler’s Formula [e^(i*x)=i*sin(x)+cos(x)]. No matter what exponential process occurs for like if you derive the function 2^x with respect to x you get ln(2)*2^x. That natural log constant is determined by the base to the natural log of e, which to me helps describe how e is sort of a hidden factor in exponential functions. I think its more intuitive to a person who doesn’t understand calculus to give the explanation of: ‘e is the base rate of any exponentially growing or decaying processes’ rather than ‘e is the only number where you can take the function e^x and have its derivative be the same no mater how many times you differentiate it.
I’m by no means necessary an expert on math. I’m happy and open to corrections of my own mistakes if I make them. I’m not even done with college as a amateur electronics engineer who likes crunching numbers and solving circuits with the tools math provides more so than is needed in my field of work. I simply love tutoring the subject and removing as many ambiguities as I can when I help others because I never want to make math seem more imposing than it is to someone who lacks the background or intuition needed to grasp the concept confidently. However I still love the visual style of explaining e here and appreciate this video as it is.
Edit: typo
0:36 "What actually is e?" "O"
Great to see a proof once in a while! Especially with professor Copeland.
Although there is a typo in the graphics at 11:09. In the top equation, the second term in the bracket says 1/(q+1) but it should read 1/(q+2), and similarly the third term should read 1/((q+2)(q+3)).
Thanks for great content!
Love this video - just straight into some nice proofs!
Such a pleasant voice and the manner of speech! It is a pure joy to listen.
I though that nowadays everyone is trying to keep R < 1 🤔
Whenever the amount you are going to have is related to the amount you have now, you get an e. If the rate of change is proportional to the starting amount you get and e^ax, where 1/a is the constant of proportionality.
That’s not right.
In 11:09, shouldn't the equation above say 1/q+2 instead of 1/q+1?
just wanna point out an alternative (and quicker) way to finish this off:
since q>1, we know that the expression in the last line at 10:03 (aka R) is at most 1/2! + 1/3! + 1/4! + ... which is just the series-representation of e (see 6:11) minus the first two terms.
so 0 < R because obviously 2^k is smaller than (k+1)! for k>1).
this means that 1/2! + 1/3! + 1/4! + ... < 1 which can be obtained without use of the geometric series for 1/2^k by the simple geometric "repeatedly halfening a distance of 1"-argument.
Nice! However to complete your proof you need to prove that e
@@renyhp ...assuming that we want to figure out whether e is rational before even knowing if it's less than 3, of course.
for the sake of completeness, I'll ad that bit to my original comment.
Never ask a mathematician why that is useful, enjoying it for its beauty is ok.
never ask why something mathematical is useful lol
@@JamesSpeiser It's okay to ask, just don't ask a mathematician 😂
Tbh it is a hard question, mainly because there's no easy explanation of what it does. For example, pi is "the circle number", and because everyone knows about circles, everyone can understand how it can be important. But e is "the differentiation number"? That to a layman doesn't sound cool, or useful. But anyone that has ever done Calculus I or greater knows how practical and everpresent e is
I thank you all for the compound interest on this :-)
@@orionmartoridouriet6834 Then call it the "infinite interest number" and people will love it
2:46 y=e^x is not the only function. y=a*e^x with any number "a", including the function y=0*e^x=0 holds that.
*Smooth Brains:* doesn't know what _e_ is.
*Big brain:* proves _e_ is an irrational number.
*Galaxy Brains:* Proves that _e_ is in fact a letter.
lol
*Supercluster Brains*: Proves that e is in fact, a symbol.
Multiverse Brain: *e* is the trancendental signified
@@gandelfy What is a "signified"?
I only understood -1/12% of this, but I'm happy just to see Ed again. Hi Ed!
If you assume q>1 in 14:50, wouldn’t you have a loose end with q = 1, and then e being an integer, and therefore rational? On the other hand, even if q = 1, you would still get R < 1, so R can’t be a positive integer anyway, so why assume q > 1?
I remember a previous Numberphile video saying that the proof that e was transcendental was that there was a whole number between 0 and 1. Now that statement makes sense.
So basically for e to be a rational number there would have to be an integer that exists between 0 and 1? Am I understanding that right
Couple of useful things about d/dx e^x = e^x from physics:
Radioactive decay is proportional to the amount of undecayed substance left, and the force a spring exerts is proportional to how stretched it is. The amount of decay is the rate of change of the amount of substance, and the force the spring produces directly moves whatever is on the end of it. You always end up needing to multiply it by something or whatever, but a function whose value and rate of change are the same is exactly the starting point you need to work with this kind of model.
'why is that useful?'
best question ever.
It's useful because this is how interest compounds.
"why is this useful?" ".......I don't know"
is such a mathematician answer and I love it
1:16 "e is roughly three" All the engineers scream "I REST MY CASE!!!" lol
To the question "why is it important"?, exponential functions are the natural representation of multiplicative growth processes, since they correspond to the rate of growth of a population at a given time being proportional to the size of the population at that time.
True. There are more answers as well, like 1. e can be used to find solutions to diff eqs because of the property of the derivative match.
2. e^(i•theta) in the complex plane can be used to relate to harmonics sin and cos.
I'm sure many more.
I love Euler the mathematician, because hes a genius
Being a genius is a prerequisite for becoming a famous mathematician
@@BritishBeachcomber true to be Honest
@@BritishBeachcomber (Matt Parker coughs and starts heading for the door)
XD