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The information you shared is something maybe 1 out of 50.000 bachelor's graduates worldwide know, despite almost all being exposed to calculus or college algebra in their lifetime. I am assuming that even a very big percentage of college math teachers also won't have the intuition within this video. In ALL classes they teach e as a random universal number that we found from continuous interest and eventually found in other places, and NOT what it really was all along. Thank you so much, videos like these should put even the famous universities to shame. Just 20 minutes and such a beautiful chunk of info
You saved me a lot of time I could have wasted with this video listening to a mind-numbing discussion of compound interest, which topic has a perverse capacity to compound boredom.
*_e_** is the eigenvector of the Differential Operator.* For those who might not grasp what that statement means... It is saying that if you use this as your input, it will PASS THROUGH essentially unchanged. The same applies to how ZERO is the eigenvector of the Addition Operator. And how ONE is the eigenvector of the Multiplication Operator.
@@dahawk8574 Yes, this is a more concise way of saying it than mine, which is: "e is the base of the exponential function that is the solution of the differential equation, y'(x) = y(x)." [This being one of the simplest non-trivial differential equations.] Fred
I worked for a bank where savings account interest was compounded continuously. I used both Euler’s Number e and natural log ln quite frequently. Your explanation made sense to me. Next time you are talking to a banker ask for a proof of how your savings interest (Annual Percentage Yield) was determined. I bet no one in the bank can do the calculation let alone explain it.
I remember once a financial journalist, might have been the BBC don't recall, took the terms of interest to a maths profession at a top university, might have been Oxford, and asked him (or her) to evaluate it. It took him four hours. Could have been three or five. I'm guessing he was just presented with insufficient information but still, it obviously wasn't straightforward. A mere mortal wouldn't stand a chance. Also, I accept that my hazy recollection, plus this being a RUclips comment, is pretty much the definition of apocryphal.
I remember once a financial journalist, might have been the BBC don't recall, took the terms of interest to a maths profession at a top university, might have been Oxford, and asked him (or her) to evaluate it. It took him four hours. Could have been three or five. I'm guessing he was just presented with insufficient information but still, it obviously wasn't straightforward. A mere mortal wouldn't stand a chance. Also, I accept that my hazy recollection, plus this being a RUclips comment, is pretty much the definition of apocryphal.
The reason we choose 100%(100% = 100/100 = 1) is because 1 is the multiplicative identity over the real numbers(and all of the numbers you will ever use)
Every exponential function F has pair of functions of which one of them has smaller factor (base) and it represents instantaneous growth rate of the function F, and the other one has bigger base (factor) which represents continuous growth factor of function F. Thus function's instantaneous growth rate is it's instantaneous counterpart's discrete growth rate i.e: 1.051278^x have instantaneous growth rate of 5 %, the discrete rate of (1.05^x) And any 2 exponential functions' instantaneous growth rates are proportional to the exponential ratio of their factors: f(x) = a^x g(x) = b^x b = a^r r = log(b)/log(a) = ln(b) if instantaneous growth rate of f(x) is p: then for g(x) it's equal to r*p. Explanation with derivatives: b = a^r then g(x) = b^x = a^rx = (a^x)^r = (f(x))^r (f(x) to the pow r). And instantaneous growth rate is actually a constant that occures in the derivative of exponential function which is for f(x) is simply f'(x) = k1*f(x), for g'(x) acrrording to chain rule is: g'(x) = g(x) * k2 ((f(x))^r)' => r*f(x)^(r-1)*f'(x) = r*f(x)^r/f(x)*f(x)*k1 => r*g(x)*k1 = g(x) * k2 => r*k1 = k2
If U want to why growth rate is actually derivative for exponential functions watch 3Blue1Brown's video (3:57): ruclips.net/video/m2MIpDrF7Es/видео.html I could'n yet figured out proportionality relationship between discrete and continuous growth rates of functions. This trio of functions ..... I(x) D(x) C(x) ...... chained in both direction infinitely. Every single one of them is instantaneous equivalent of the one on it's right and continuous equivalent of the one on it's left. In other words, every single function has instantaneous equivalent on it's left and continuous equivalent on it's right.
I've got a one-sentence description for "e". It goes as: "The number e helps quantify the decay rate for simple dissipative processes in nature". There are countless examples for these so-called "simple dissipative processes", which behave in an asymptotic fashion. - The time-rate temperature decrease of a given volume of hot liquid (a cup of coffee). - The time-rate level decrease of water in a bathtub after the plug is removed from the drain. - The time-rate amplitude decrease in a mechanical oscillation system (mass-spring, pendulum, a swing, vibrating string under tension, etc.).
The most popular definitions for e: The one in this video: e = lim(n to inf) of (1 + 1/n)^n The infinite sum of 1/n! starting at 0 (n! = 1*2*3*4*.....*n) e is base for the natural logarithm e is the base for the exponential function exp(x) where the derivative of that function is also exp(x) so f(x) = f'(x)
Up until your discovery of constant-variable lambda I felt I was with you. From that point on I would need much more graduation. Very unfortunately I lost you from that point on. David Lixenberg
There's a slight problem with your opening screen. Four constants are shown, all but the third of which are pure numbers. Planck's constant, h, is not a number; it's an angular momentum. So unlike i, π, and e, its numeric value depends on the choice of physical units, which from the point of view of physics, is arbitrary. The value given here is in the "mks" system, and so, needs to be 'finished' with "kg•m²/s." Also, at the 20-min mark, eˣ isn't for x%, but for 100x% growth rate. Fred
I haven't watched it yet, but I just feel like it would have more impact if he at least would have started the one simple sentence at the beginning. I mean, that's the only way to gauge if he succeeded or not, right?
Summarizing: e connects discrete to continuous growth in self-reflecting growing systems. In a period where discrete growth leads to doubling of the amount or number (factor 2), e would be the factor to be obtained if growth was completely continuous, in that same period of time.
@19:59 There's a mistake: It's "100 times x percent", not x percent! Biggest trip-hazard with percentages that there is: forgetting that it's a hundredth-factor!
You seem to have accidentally given one of the great argument for why τ (the ratio of a circle's circumference to its radius) is more practical than π (τ/2), right at 18:31 !
@@fullfungo You need for a^x to be defined for all x, and since 0^0 is undefined (I know taking the limit implies it should be 1, but that still requires a limit) you can immediately rule out 0 as a possible value for a.
This is anything BUT a simple easy to follow analysis of the constant e. I mean WOW!!!!! Not many of us with a healthy curiosity of math concepts will be to able to roll with this. I finally threw in the towel at 23.17.
Hot Tip: (1.0000000001)^⁹ = e This is the smallest number taken to the exponent of its decimal place which is e, or 2.718.... Just add more zeros between the ones and take it to a higher exponent to get "e" with more decimal places.
Of two integers, no…but there’s no guarantee that the circumference and diameter of a circle will both be integers (therefore it is still a ratio of two numbers). In fact, pi being irrational tells us that that’s impossible!
@@michaelcolbourn6719 Doesn't even have to be two irrational numbers. In fact most of the time it's one irrational and one rational. It just can't be two integers!
scalability of large numbers by a factor, decimal converter? Used highly in diff eq, the visual helps, thanks, now how to upscale control theory into life?
Eulers formula is one of the least intuitive imo…I can’t think of a good way to describe it. The “why” of your question can be answered most easily probably using Taylor series-if you look up the Taylor expansions for e^x, cos x, and sin x, you can see why e^ix = cos x + isin x. Though that’s just a mathematical derivation, not an intuitive explanation
I'm sorry, but to ignore calculus in a video trying to give an intuition for e is a somewhat baffling move, since what makes e significant is its use in calculus (specifically for taking the derivative or anti-derivative of an exponential). Walking through how solving f'(x) = f(x) naturally leads to an exponential, and further showing that the base of that exponential is an exact constant, also gives an intuition on why it matters in chemistry and physics, after perhaps explaining why exponential equations are used to model how certain kinds of systems evolve (and how the past of them can be predicted) based only on some constants of the system and the current state.
*_e_** is the eigenvector of the Differential Operator.* For those who might not grasp what that statement means... It is saying that if you use this as your input, it will PASS THROUGH essentially unchanged. The same applies to how ZERO is the eigenvector of the Addition Operator. And how ONE is the eigenvector of the Multiplication Operator.
Your "h" -- yeah, I don't feel like hunting down the actual symbol either -- would be a lot more well known if Heisenburg's Uncertainty principle weren't one of the two most misunderstood principle in science. I usually say the the other is second law of thermodynamics, but after arguing with a bunch of people that yes, light does slow down in mediums, I might just have to up my number to three.
Your equation at (13:25..13:32): (lambda^(x) = 1.05, finding x) does not make sense at all. Lambda is per definition a limit value of an infinity series lambda = lim (1 + r/n)^n = 1.05 ..............n->oo So, for n=100,000 ; 10,000 ; 1,000 ; 100 we got: r= 0.048790176 ; 0.048790283 ; 0.04879135 ; 0.0488020
What I hate about you tube is that is seems to universally try to make things simple and intuitive. The truth is that some things are better understood in their complexity and better seen by letting go of your intuitions. Simple isn’t always better, intuitive isn’t always truer.
So we should make things more complex to understand them better? That doesn't sound very intuitive to me. Could you give us a nice, easy example to illustrate your hypothesis?
It is also unfortunate that you took the bacterium reproducing as an example for "self-referential growth". Because, other than "compound interest rates" (8:38...9:41) bacteria are counted with whole numbers: 1->2->4->8->16..., while interest rates are represented, as real numbers. There are serious mathematical implications: we can not integrate, differentiate find the maximum/minimum, and importantly find the limes as n-->oo for the discrete numbers of bacteria, like it is nonsense, when 2 bacteria would reproduce pi-number of bacteria, or infinite number of bacteria in between!! The foolish chemists usually make the same mistake, when they counted their atoms/molecules with whole numbers and go ahead to apply the maths for real numbers!
I don't think there is any reason at all to bring the approximation into it. The other stuff is great. But I think I can do better. I'm going to try and slap something together.
your grip is wrong on the pen. if u held it correctly both you and the audience wouldn't have the view blocked by your thumb and the other finger. it is annoying to watch.
Yet another rehashed bad way to teach e. Not knocking Foolish Chemist. It’s a good math video. We tend to teach math by backing into concepts from an historical perspective. Instead we should relegate the arduous path to discovery of things like e and i to a math history class. Now that we have a better understanding of these concepts we should start with a modern perspective.
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e is the sum of all original angles. In one sentence. What is it really? Really, it's 0.11111111111111...
The information you shared is something maybe 1 out of 50.000 bachelor's graduates worldwide know, despite almost all being exposed to calculus or college algebra in their lifetime. I am assuming that even a very big percentage of college math teachers also won't have the intuition within this video. In ALL classes they teach e as a random universal number that we found from continuous interest and eventually found in other places, and NOT what it really was all along. Thank you so much, videos like these should put even the famous universities to shame. Just 20 minutes and such a beautiful chunk of info
But from 10 of the suspected 500,000 may 1 come out and save us all from the miseries we are in right now! Let's hope and have faith!
"It's the constant of self-similar change" is my best pre-video attempt at distilling e in one sentence.
had this exact thought as well
You saved me a lot of time I could have wasted with this video listening to a mind-numbing discussion of compound interest, which topic has a perverse capacity to compound boredom.
Very precise!
*_e_** is the eigenvector of the Differential Operator.*
For those who might not grasp what that statement means...
It is saying that if you use this as your input, it will PASS THROUGH essentially unchanged.
The same applies to how ZERO is the eigenvector of the Addition Operator.
And how ONE is the eigenvector of the Multiplication Operator.
@@dahawk8574 Yes, this is a more concise way of saying it than mine, which is:
"e is the base of the exponential function that is the solution of the differential equation, y'(x) = y(x)."
[This being one of the simplest non-trivial differential equations.]
Fred
I worked for a bank where savings account interest was compounded continuously. I used both Euler’s Number e and natural log ln quite frequently. Your explanation made sense to me. Next time you are talking to a banker ask for a proof of how your savings interest (Annual Percentage Yield) was determined. I bet no one in the bank can do the calculation let alone explain it.
I remember once a financial journalist, might have been the BBC don't recall, took the terms of interest to a maths profession at a top university, might have been Oxford, and asked him (or her) to evaluate it. It took him four hours. Could have been three or five.
I'm guessing he was just presented with insufficient information but still, it obviously wasn't straightforward. A mere mortal wouldn't stand a chance.
Also, I accept that my hazy recollection, plus this being a RUclips comment, is pretty much the definition of apocryphal.
I remember once a financial journalist, might have been the BBC don't recall, took the terms of interest to a maths profession at a top university, might have been Oxford, and asked him (or her) to evaluate it. It took him four hours. Could have been three or five.
I'm guessing he was just presented with insufficient information but still, it obviously wasn't straightforward. A mere mortal wouldn't stand a chance.
Also, I accept that my hazy recollection, plus this being a RUclips comment, is pretty much the definition of apocryphal.
These people are barely educated. There’s no point showing off to them.
The reason we choose 100%(100% = 100/100 = 1) is because 1 is the multiplicative identity over the real numbers(and all of the numbers you will ever use)
doodoo kaka
@Yunahsky wtb modular arithmetic, for example 6 is a multiplicative identity mod 5
@paridhaxholli 6 = 1 mod 5.
Excellent explanation - thanks!
Minor one at 19:59: e^x is not self-growth at x % but x*100 %.
Every exponential function F has pair of functions of which one of them has smaller factor (base) and it represents instantaneous growth rate of the function F, and the other one has bigger base (factor) which represents continuous growth factor of function F. Thus function's instantaneous growth rate is it's instantaneous counterpart's discrete growth rate i.e:
1.051278^x have instantaneous growth rate of 5 %, the discrete rate of (1.05^x)
And any 2 exponential functions' instantaneous growth rates are proportional to the exponential ratio of their factors:
f(x) = a^x
g(x) = b^x
b = a^r
r = log(b)/log(a) = ln(b)
if instantaneous growth rate of f(x) is p:
then for g(x) it's equal to r*p.
Explanation with derivatives: b = a^r then g(x) = b^x = a^rx = (a^x)^r = (f(x))^r (f(x) to the pow r). And instantaneous growth rate is actually a constant that occures in the derivative of exponential function which is for f(x) is simply f'(x) = k1*f(x), for g'(x) acrrording to chain rule is:
g'(x) = g(x) * k2
((f(x))^r)' =>
r*f(x)^(r-1)*f'(x) = r*f(x)^r/f(x)*f(x)*k1 =>
r*g(x)*k1 = g(x) * k2 =>
r*k1 = k2
If U want to why growth rate is actually derivative for exponential functions watch 3Blue1Brown's video (3:57): ruclips.net/video/m2MIpDrF7Es/видео.html
I could'n yet figured out proportionality relationship between discrete and continuous growth rates of functions.
This trio of functions
..... I(x) D(x) C(x) ......
chained in both direction infinitely. Every single one of them is instantaneous equivalent of the one on it's right and continuous equivalent of the one on it's left. In other words, every single function has instantaneous equivalent on it's left and continuous equivalent on it's right.
Well done.
This is a better explanation for me than any other explanation Ive run into.
Thank you
I've got a one-sentence description for "e". It goes as:
"The number e helps quantify the decay rate for simple dissipative processes in nature".
There are countless examples for these so-called "simple dissipative processes", which behave in an asymptotic fashion.
- The time-rate temperature decrease of a given volume of hot liquid (a cup of coffee).
- The time-rate level decrease of water in a bathtub after the plug is removed from the drain.
- The time-rate amplitude decrease in a mechanical oscillation system (mass-spring, pendulum, a
swing, vibrating string under tension, etc.).
Excellent, Excellent, EXCELLENT video!!!
The most popular definitions for e:
The one in this video: e = lim(n to inf) of (1 + 1/n)^n
The infinite sum of 1/n! starting at 0 (n! = 1*2*3*4*.....*n)
e is base for the natural logarithm
e is the base for the exponential function exp(x) where the derivative of that function is also exp(x) so f(x) = f'(x)
or the solution to the ivp y'=y, y(0)=1
you can also change that limit to the form e=lim u->1 of u^(1/u-1), which is a lot easier to compute
Outstanding! Thank you. As a EE I’ve worked with e for many years.
Thank you for making this
Bravo. It’s always nice to learn another way of understanding something (that you thought you understood).
19:59 no that is not true. Otherwise you wouldn’t have wrote 200% at e^2. 2 isn’t the same as 200 is it? percent literally means devided by 100
Up until your discovery of constant-variable lambda I felt I was with you. From that point on I would need much more graduation. Very unfortunately I lost you from that point on.
David Lixenberg
Thanks for making this video sir, really helpful
Great explanation.
I noticed you started saying _zero_ and _point_ instead of _o_ and _dot,_ congratulations!
There's a slight problem with your opening screen. Four constants are shown, all but the third of which are pure numbers. Planck's constant, h, is not a number; it's an angular momentum. So unlike i, π, and e, its numeric value depends on the choice of physical units, which from the point of view of physics, is arbitrary. The value given here is in the "mks" system, and so, needs to be 'finished' with "kg•m²/s."
Also, at the 20-min mark, eˣ isn't for x%, but for 100x% growth rate.
Fred
Fantastic explanation.
ime impressed you are something else 😊😊👍👍
I'm NOT more confused! And... now I'm subscribed!! Great video!!
Nice! This explains why the exponential map appears even when you’re not working with numbers anymore
Great video! What tablet device do you use to write on in videos?
iPad!
Nicely explained though It was difficult to understand why you were applying the lambda exponentials.
The Goat is back🎉
I haven't watched it yet, but I just feel like it would have more impact if he at least would have started the one simple sentence at the beginning. I mean, that's the only way to gauge if he succeeded or not, right?
Great video explaining e. I never thought e can be describe as "the self referential growth or decay" of itself.
Nice!!!
Summarizing: e connects discrete to continuous growth in self-reflecting growing systems. In a period where discrete growth leads to doubling of the amount or number (factor 2), e would be the factor to be obtained if growth was completely continuous, in that same period of time.
Love to see you do this sort of thing with phi. Maybe compare Fibanochi and e.
You should wear the microphone instead of holding it in your hand
awsome explanation
to help understanding concepts
@19:59 There's a mistake: It's "100 times x percent", not x percent!
Biggest trip-hazard with percentages that there is: forgetting that it's a hundredth-factor!
You seem to have accidentally given one of the great argument for why τ (the ratio of a circle's circumference to its radius) is more practical than π (τ/2), right at 18:31 !
Ahhh, a good point! I didn’t even think about this lol
It was a useful explanation that added to my knowledge, thank you for such educational videos, it helped to improve human knowledge and science.
Thank you !!!!
What iPad app are you using?
Why at the start of the video do you list pi as ~3.1415 when it is actually far closer to 3.1416, since the actual digit stream goes to 3.14159265?
Here is e in one sentence:
The only value where the derivative of a^x = a^x
a=0
@@fullfungo except 0
@@fullfungo You need for a^x to be defined for all x, and since 0^0 is undefined (I know taking the limit implies it should be 1, but that still requires a limit) you can immediately rule out 0 as a possible value for a.
This is anything BUT a simple easy to follow analysis of the constant e. I mean WOW!!!!! Not many of us with a healthy curiosity of math concepts will be to able to roll with this. I finally threw in the towel at 23.17.
i want to know the app you were writing in, if it is an app
This is exactly how my tutor explained e
Since when does Exponential imply growing really fast?
I oddly couldn't take the guy seriously because he's holding a clip-on mic in his hand for some reason
That bothered me too!
Good explanation but you can also just say that e is such a number that if f(x) = e^x then f'(x) = f(x)
Time dialation,only correct at that moment time is a continuum
So Euler's number is the same as natural log. 2.718281828459 ?
A natural logarithm is a logarithm with base e, so ln e = 1.
Hot Tip: (1.0000000001)^⁹ = e
This is the smallest number taken to the exponent of its decimal place which is e, or 2.718.... Just add more zeros between the ones and take it to a higher exponent to get "e" with more decimal places.
What would be e if a bacteria grew three times itself.
@achyutkarve it is a three point curve using e as the moderator.
1:06 nor is pi. Pi is irrational and cant be expressed as a ratio of two numbers.
Of two integers, no…but there’s no guarantee that the circumference and diameter of a circle will both be integers (therefore it is still a ratio of two numbers). In fact, pi being irrational tells us that that’s impossible!
@FoolishChemist so it can be expressed as a ratio of two irrational numbers?
@@michaelcolbourn6719 Doesn't even have to be two irrational numbers. In fact most of the time it's one irrational and one rational. It just can't be two integers!
Pi is the ratio of
Circumference:diameter
We aint talking about numbers here.
E😫
scalability of large numbers by a factor, decimal converter? Used highly in diff eq, the visual helps, thanks, now how to upscale control theory into life?
Doode ur the goat
Ok, I'll need to watch that a few more times and with a pen and paper.
Tell me, why does e come on when making a complex circle?
Eulers formula is one of the least intuitive imo…I can’t think of a good way to describe it. The “why” of your question can be answered most easily probably using Taylor series-if you look up the Taylor expansions for e^x, cos x, and sin x, you can see why e^ix = cos x + isin x. Though that’s just a mathematical derivation, not an intuitive explanation
Continuum has time dialation,only true at that exact moment new computing has up charge
π has a solution,a circle Alfa and Omega
You just made e more confusing than ever! I've seen much better videos explaining e, and even in my college classes.
Time Dialation,continuum only correct at that exact moment,,benchmark that point x,2x+5 =8')
E is a letter that represents a number of
I'm sorry, but to ignore calculus in a video trying to give an intuition for e is a somewhat baffling move, since what makes e significant is its use in calculus (specifically for taking the derivative or anti-derivative of an exponential).
Walking through how solving f'(x) = f(x) naturally leads to an exponential, and further showing that the base of that exponential is an exact constant, also gives an intuition on why it matters in chemistry and physics, after perhaps explaining why exponential equations are used to model how certain kinds of systems evolve (and how the past of them can be predicted) based only on some constants of the system and the current state.
*_e_** is the eigenvector of the Differential Operator.*
For those who might not grasp what that statement means...
It is saying that if you use this as your input, it will PASS THROUGH essentially unchanged.
The same applies to how ZERO is the eigenvector of the Addition Operator.
And how ONE is the eigenvector of the Multiplication Operator.
Your "h" -- yeah, I don't feel like hunting down the actual symbol either -- would be a lot more well known if Heisenburg's Uncertainty principle weren't one of the two most misunderstood principle in science. I usually say the the other is second law of thermodynamics, but after arguing with a bunch of people that yes, light does slow down in mediums, I might just have to up my number to three.
Trancedental number like pi and square root of not perfect square
The easiest way to describe pi: It’s half tau.
21:38 you bet!
Your equation at (13:25..13:32): (lambda^(x) = 1.05, finding x) does not make sense at all.
Lambda is per definition a limit value of an infinity series
lambda = lim (1 + r/n)^n = 1.05
..............n->oo
So, for n=100,000 ; 10,000 ; 1,000 ; 100 we got:
r= 0.048790176 ; 0.048790283 ; 0.04879135 ; 0.0488020
You could call it simply _natural_ growth. Note the name _natural_ logarithm.
22:00 IF 100% means doubling, it would make more 'intuitive' sense to say that; and if not, I'm not finding any of this intuitive.
e just a number no a curve please explain to each other
Nice video. Could be brilliant if you can proof it with calculus.
What I hate about you tube is that is seems to universally try to make things simple and intuitive. The truth is that some things are better understood in their complexity and better seen by letting go of your intuitions. Simple isn’t always better, intuitive isn’t always truer.
So we should make things more complex to understand them better? That doesn't sound very intuitive to me. Could you give us a nice, easy example to illustrate your hypothesis?
Thank you for making it so clear and simple. Appreciated.
It is also unfortunate that you took the bacterium reproducing as an example for "self-referential growth". Because, other than "compound interest rates" (8:38...9:41) bacteria are counted with whole numbers: 1->2->4->8->16..., while interest rates are represented, as real numbers. There are serious mathematical implications: we can not integrate, differentiate find the maximum/minimum, and importantly find the limes as n-->oo for the discrete numbers of bacteria, like it is nonsense, when 2 bacteria would reproduce pi-number of bacteria, or infinite number of bacteria in between!!
The foolish chemists usually make the same mistake, when they counted their atoms/molecules with whole numbers and go ahead to apply the maths for real numbers!
I don't think there is any reason at all to bring the approximation into it. The other stuff is great. But I think I can do better. I'm going to try and slap something together.
Good video on euler's number
I tried to educate 3m to multiply the money in my bank account but they did not have the necessary algebra skills to get the right answer.
It’s the greatest number ever
Two-point-seven Jackson Jackson...
😞 Someday, I'd be smart enough to understand this.
If you don't understand, then it's not taught right for you.
X,2x+5=8')
E
E
*Everytime*
Fail.
Tipical american: all about money.
It started in Babylonia.
The fact, that the language of the Math is not comprehensive to me, makes me sad.
your grip is wrong on the pen. if u held it correctly both you and the audience wouldn't have the view blocked by your thumb and the other finger. it is annoying to watch.
🙃 Euler’s identity e^(i*π)+1=0 comes about as close as possible to proving that God exists.
Yet another rehashed bad way to teach e. Not knocking Foolish Chemist. It’s a good math video. We tend to teach math by backing into concepts from an historical perspective. Instead we should relegate the arduous path to discovery of things like e and i to a math history class. Now that we have a better understanding of these concepts we should start with a modern perspective.
There is a book about e, and is full of history.
wtf is h
Plancks constant!
Too much complicated
e = 1/0! + 1/1! + 1/2! .............
The way you hold the pen is just awful. It doesn't let to see what you are writing
Yeah why is his thumb sticking out😭