+Dana de Jong Great feedback, this is something I wanted to clarify in the video. 2^x and e^x are both continuous, and can have fractional values of x. But 2^x is normally meant to imply 100% growth at the *end* of an interval (1, 2, 4, 8), and not, for example, "69.3% growth compounded continuously". How many possibilities do you have after x coin flips? 2^x. Can you imagine fractional numbers of coin flips? Sure. But if you were really concerned with instantaneous fractional growth, you would have written your curve as e^(r * t) instead of 2^x. Technically, any exponential (like 2^x) can be considered as some type of continuous growth, but 2^x is most convenient for discrete values of x, to model discrete changes that don't compound. This is something I wanted to make more clear in the video, thanks for the question.
I'm a full time machine learning scientist, and I was looking at some problems and I realized I had completely lost my understanding of e. I put down my ego and went back to look at what log and e really mean. This video was fantastic at explaining that. For any of you kids in high school, college, or middle school, or wherever, just know that learning is a lifelong process. Sometimes you forget things, sometimes you don't remember what you learned, that's okay. Just go back and look at it again with a fresh perspective. It's never too late to learn, and you never have to be 'above' a certain level of learning. In some ways, the way we learning things in life is a bit like the number e (please dont use this on your fucking college essay haha), the first time we learn a concept it's that principal deposit in our brain. As we learn more about the little parts it compounds our understanding, building upon interest.
You're one of my favorite humans on the World Wide Web. I'm glad I can watch lectures from you. And this is not because you're in the line of science that I come across. You still were even if your education/culture would had taken you in areas like oil painting. Ppl watching this vid may not know you for who/what you really are. Oh, and it's not your natural pedagogical skill either. World, behold one of your man and award him what he is entitled to!
but why did you consider f(2^x) to have a stair case shape when plotted? what if we substituted with x= {1.1, 1.2, 1.3, ...} or even infinitely smaller numbers between 1 & 2, wouldn't that show that f(2^x) is a smooth curve just like what we were looking for in the first place?
Even if you plotted f(2^x) using infinitely small time steps, on an infinitely small time scale you would have steps where f(2^t) becomes f(2^([t+0.00000000....0.1]), compared to ln(t) which you will not have steps for at any time scale because ln(t+T) is always the natural exponential growth of the previous time step. (I know this comment is old, but in case anyone else is curious!)
Awesome vid. Thanks. I'm a slow learner. At 40, I have finally understand e. I used it at school ad infinitum, but never was told in class (or was probably day dreaming at the time) what it really represented and why it was so significant in Math.
when i see a video like this it makes me think - maths textbooks are deliberately designed to complicate things so that normal people feel alienated and elitist intellectuals can preserve their hold on knowledge. Thanks a lot. i realy appreciate you approach to explaining things, IE with common sense and designed to tech people, not to demonstrate how clever you are. nuff respect
This video is awesome and so is your article. The concept of e has confused me ever since I first encountered it in high school, but now I'm finally starting to understand why it is used in the way that it's used. Thanks!
Aaand after years in school and now in uni, trying to solve things by heart it finally makes sense and I understand what e is useful for. Thank you Sir, keep it up !
x^n = e^kn x = e^k So every exponential curve can be represented with e. What makes e special is the slope of a line tangent to the curve: y = e^x is e^x for all x. The y value and the slope are the same at every x value. Also, the area under the curve measured from 0 to some x value and bounded by the x-axis is also e^x for all x values. e is cool!
Good point, let me clarify. When people think "Double every year" they might imagine 2^x with integer values (1, 2, 4, 8, 16, etc.). But this assumes all growth happens at the end of the interval, and no interest was earned & compounded during the year. If you mean you have 2^x, including compounding, then you're saying e^ln(2)*x = 69.3% continuous growth. This isn't what most people mean by "doubling growth" though, and I should have made that clear. Thanks for the comment.
Why would 2^x have new quantities all of a sudden if the idea is that y=x and x includes every integer on the number line? It's not like the function only calculates for the whole numbers?
The quantities in between that number are all in a straight line. That doesn't work for monetary interest which deals with compounding over the primary over and over.
It took me a while to find it again, but I did. It is in the iTunes U section. The course is called "College Algebra" by HACC. The podcast is numbered as 4.2 and called "The Story of e Ex". I really liked it, the teacher explains it really good. However, there are a few versions of this course, I'm talking about the one with a black background and some graph on it on the cover. It was published on Jul 17, 2012. To find the course just search for "College Algebra" in Tunes.
This is great! One thing I wish you had made explicit is that all the little chunks at the end (black, red, green, brown, gray) are exactly the terms you get when you expand P(1+r/n)^n.
great explanation, thank you! only small addition to the video is needed - subtitles. The automatic ones are absolutely useless, so the non-native speaker agonizes in getting back again and again in his struggle to understand the fast speech. The subtitles would help even more fluent nonnative speakers like myself - in my own experience the amount of understanding is much bigger when both speech and subtitles are there. Thank you again for your labor to spread understandable knowledge!
wow - this is amazing - it is the kind of thing that just really clicks right into what feels like i knew all along - but of course i didn't know it until you told me...
Wow didn't know you do videos now Kalid. Been a fan of yours for the longest time. Keep producing if you can, always awesome! And guys who think this is the best explanation of e, should really check out the full length article on his website.
I've studied every category of math including calculus. But I have to say the number e is the hardest concept to understand I've encountered. This video gives one of the best explanations I've seen. But I still don't understand it as in depth as I would prefer. I understand everything about it except how do you know when you've reached the limit? I mean, you could conceivably go out an infinite number of decimal places to the right like, 2.71828... and just keep on going.
I've never heard this explanation before, and I'm in university. How great it would be if math classes were about explaining things, and not just dumping the curriculum on people.
You know what, in Thailand, the 'e' is just a kind of magic number. The teacher decided to treat 'e' as a normal constant. What a shame... I kind of get it now after 6+ years. Thanks!
It seems your principal argument is that e is a special universal growth constant because f(x) = e^ax is always growing throughout every interval of x, rather than in a discrete nature as f(x) = 2^x does. But the input, x, could include any number of fractional and irrational values, so f(x) = 2^x (as well as for infinitely more base values) is also always growing in partial, non-discrete amounts. I'm not seeing how the whole "infinite amount of growth happening" thing is special to e.
Very nice approach; thanks for the video. Yet, it would be nice to have also a contextual, historical approach, on how the euler number was discovered by Euler, and why did he need it.
Thanks for this video. I've not had much to do with maths but I came across this concept of 'Euler's number' recently and this is the clearest explanation I've had so far. A real worked example would be great though. It's confusing when people just say "well then it is e to the power x"!!! Make it a little more concrete for us beginners please.
Good Job - I like to think sqrt 2 is also an underappreciaated irrational number built into every square just like Pi is in every circle and e in anything that grows
Great point -- I should have drawn the staircase better. Ideally, each step is larger than the one before (i.e. doubling), so you get some exponential curve. If each step is exactly the same, then it means your growth is not compounding [and e won't be able to model it].
Sorry if the words were unclear, I meant continuous compounding. If someone says "I am earning 100% interest per year" then 2^x is the amount they have it they compound every year, and e^x is the amount they have if they compound continuously (as fast as possible). 2^x is still a continuous function in a calculus sense, yep.
This is exactly what I wanted to know, not taking limit to infinity, not 1+ 100% interest bullshit, these things i already know cuz its mentioned in every other videos on e. But i wanted to just confirm my intuition. I kinda knew that why e is natural growth bit not completely, i knew flexibility part, we can reduce or increase it according to our needs, but i didnt quite knew what property exactly makes it a natural Growth. Time, 11:41pm
think about a flower blooming, it grows slowly over time and those growths add as well. or acid corroding something, it starts slowly then continues to add and add.
Exactly. When you compound continuously, you can merge rate and time into a single variable "x". It's hard to write in a comment, but check out the linked article in the description for more details.
thnx a lot..after struggling lot on e i finally got a perfect idea of it...willl really appreciate if u give me the link where a get the simmillllar type of help for understanding Standard deviation...
Please note: At the point where you started showing the red cell propagating the green cells, you forgot to indicate to the viewers that the black cell (which is still there) STILL CONTINUES TO PARTICIPATE in the growth process. In other words, the black cell must necessarily indicate it is propagating YET ANOTHER new cell AT THE SAME TIME the now-new red cell is propagating a new green cell. In a nutshell...At any given instant, all that exists in the population is BY DEFINITION pitching in to new growth. This is the VERY ESSENCE of natural growth.
I find it confusing when you say that 2^x is a stairstep function. It's a continuous function and when you plug in fractional values, you will get a continuous curve.
Well done...continuous growth results in Eulers number, well explained...wonder why Math teachers do not explain this so clearly....suspect they do not know.
+Dana de Jong Great feedback, this is something I wanted to clarify in the video. 2^x and e^x are both continuous, and can have fractional values of x. But 2^x is normally meant to imply 100% growth at the *end* of an interval (1, 2, 4, 8), and not, for example, "69.3% growth compounded continuously".
How many possibilities do you have after x coin flips? 2^x. Can you imagine fractional numbers of coin flips? Sure. But if you were really concerned with instantaneous fractional growth, you would have written your curve as e^(r * t) instead of 2^x.
Technically, any exponential (like 2^x) can be considered as some type of continuous growth, but 2^x is most convenient for discrete values of x, to model discrete changes that don't compound.
This is something I wanted to make more clear in the video, thanks for the question.
Man, you absolutely have no idea why e is special. please stop misleading people on youtube and spreading misconceptions.
he evidently doesnt understand fractional powers of 2 very well either.
Sami A
Kurosh Kaviani dont be Jealous he did an Excellent explanation
+Jim Keller And how is that suppled to feed his narcissistic dopamine spike, Mr. Smart Guy? /s
This is a great explanation. Should be mandatory viewing for every math teacher in the world.
...er for the students too.
first time in life ....I can understand e intuitively....thank you
I'm a full time machine learning scientist, and I was looking at some problems and I realized I had completely lost my understanding of e. I put down my ego and went back to look at what log and e really mean. This video was fantastic at explaining that. For any of you kids in high school, college, or middle school, or wherever, just know that learning is a lifelong process. Sometimes you forget things, sometimes you don't remember what you learned, that's okay. Just go back and look at it again with a fresh perspective. It's never too late to learn, and you never have to be 'above' a certain level of learning. In some ways, the way we learning things in life is a bit like the number e (please dont use this on your fucking college essay haha), the first time we learn a concept it's that principal deposit in our brain. As we learn more about the little parts it compounds our understanding, building upon interest.
I'm a machine learning student and all I can say is that this video is full of mistakes and misconceptions.
I wish I could like this twice.
@@YolcuYolunda-kc2zo very kind of you to make a bold statement like that and then proceed to not clear them up or even identify what they are!
Dude, you are awesome! I have been wondering about the meaning of e for years since I cannot simply comprehend sh*t for granted. More of such videos!
You're one of my favorite humans on the World Wide Web. I'm glad I can watch lectures from you. And this is not because you're in the line of science that I come across. You still were even if your education/culture would had taken you in areas like oil painting. Ppl watching this vid may not know you for who/what you really are.
Oh, and it's not your natural pedagogical skill either.
World, behold one of your man and award him what he is entitled to!
Did you knew him personally? Who is he to you? 🙂
but why did you consider f(2^x) to have a stair case shape when plotted? what if we substituted with x= {1.1, 1.2, 1.3, ...} or even infinitely smaller numbers between 1 & 2, wouldn't that show that f(2^x) is a smooth curve just like what we were looking for in the first place?
Even if you plotted f(2^x) using infinitely small time steps, on an infinitely small time scale you would have steps where f(2^t) becomes f(2^([t+0.00000000....0.1]), compared to ln(t) which you will not have steps for at any time scale because ln(t+T) is always the natural exponential growth of the previous time step. (I know this comment is old, but in case anyone else is curious!)
Awesome vid. Thanks.
I'm a slow learner. At 40, I have finally understand e. I used it at school ad infinitum, but never was told in class (or was probably day dreaming at the time) what it really represented and why it was so significant in Math.
when i see a video like this it makes me think - maths textbooks are deliberately designed to complicate things so that normal people feel alienated and elitist intellectuals can preserve their hold on knowledge. Thanks a lot. i realy appreciate you approach to explaining things, IE with common sense and designed to tech people, not to demonstrate how clever you are. nuff respect
this is the first video by you that i have every seen, and it alone has earned my subscription.
Thank you so much!! I was confused about e for a year until I finally saw this! Way better than my math teacher!
This video is awesome and so is your article. The concept of e has confused me ever since I first encountered it in high school, but now I'm finally starting to understand why it is used in the way that it's used. Thanks!
Aaand after years in school and now in uni, trying to solve things by heart it finally makes sense and I understand what e is useful for. Thank you Sir, keep it up !
Thanks. Very intuitive. This is exactly what I was looking for!
This is my favorite video on this topic because it addressed the flexibility of e^rt. THANK YOU!
I feel like I arrive JUST IN TIME to take classes that these fresh, new, more useful videos to be uploaded!
This explanation is better than the 14 videos I watch previously.
x^n = e^kn
x = e^k
So every exponential curve can be represented with e.
What makes e special is the slope of a line tangent to the curve:
y = e^x
is e^x for all x. The y value and the slope are the same at every x value.
Also, the area under the curve measured from 0 to some x value and bounded by the x-axis is also e^x for all x values.
e is cool!
Good point, let me clarify. When people think "Double every year" they might imagine 2^x with integer values (1, 2, 4, 8, 16, etc.). But this assumes all growth happens at the end of the interval, and no interest was earned & compounded during the year.
If you mean you have 2^x, including compounding, then you're saying e^ln(2)*x = 69.3% continuous growth. This isn't what most people mean by "doubling growth" though, and I should have made that clear. Thanks for the comment.
I watched four videos on e and THIS was the one where I went OHHHHHHHHHHHHHH. Thank you!
Thank you so much. Your video solved my question. I couldn`t continue the chapter without understanding the number e. You are a great teacher.
Wow! I don't think there's a better way to explain this! Thank you!
Why would 2^x have new quantities all of a sudden if the idea is that y=x and x includes every integer on the number line? It's not like the function only calculates for the whole numbers?
The quantities in between that number are all in a straight line. That doesn't work for monetary interest which deals with compounding over the primary over and over.
It took me a while to find it again, but I did. It is in the iTunes U section. The course is called "College Algebra" by HACC. The podcast is numbered as 4.2 and called "The Story of e Ex". I really liked it, the teacher explains it really good. However, there are a few versions of this course, I'm talking about the one with a black background and some graph on it on the cover. It was published on Jul 17, 2012. To find the course just search for "College Algebra" in Tunes.
This is great! One thing I wish you had made explicit is that all the little chunks at the end (black, red, green, brown, gray) are exactly the terms you get when you expand P(1+r/n)^n.
great explanation, thank you!
only small addition to the video is needed - subtitles. The automatic ones are absolutely useless, so the non-native speaker agonizes in getting back again and again in his struggle to understand the fast speech. The subtitles would help even more fluent nonnative speakers like myself - in my own experience the amount of understanding is much bigger when both speech and subtitles are there. Thank you again for your labor to spread understandable knowledge!
Awesome! Totally helpful. I sorta understood e but didn't know how to explain it to my friend who needed help. I just sent him the link to the video.
Thank you so much. I have to explain this to my class for a project and now I won't completely humiliate myself!!
This is amazing -- I finally get it! You seriously got a gift for explaining concepts :) thank you so much!
thank you so much you're the only one who can properly explain e
Just used this to help me with CFA...thanks so much!
wow - this is amazing - it is the kind of thing that just really clicks right into what feels like i knew all along - but of course i didn't know it until you told me...
Wow didn't know you do videos now Kalid. Been a fan of yours for the longest time. Keep producing if you can, always awesome!
And guys who think this is the best explanation of e, should really check out the full length article on his website.
I've studied every category of math including calculus. But I have to say the number e is the hardest concept to understand I've encountered. This video gives one of the best explanations I've seen. But I still don't understand it as in depth as I would prefer. I understand everything about it except how do you know when you've reached the limit? I mean, you could conceivably go out an infinite number of decimal places to the right like, 2.71828... and just keep on going.
This is a great video! It reminds me of learning about the unit circle in relation to trig for the first time.
This is the best explanation of e I have ever found well done!!!
Best explaination of e I have yet seen!
this is the best explanation of e that I have found. thanks.
Omg this was helpful! Thanks. The interest having interest is what I needed to understand e
Great explanation of the number e and great job helping my students visualize the idea of interest growing interest as well as continuous growth.
Excellent. As you also mentioned, I wish they had given me this sort of explanation when I first started calculus.
i watched this video 5 years ago and i still think about it almost everyday
That was awesome. No one has ever explained math to me in this way. Thank you.
I've never heard this explanation before, and I'm in university. How great it would be if math classes were about explaining things, and not just dumping the curriculum on people.
a semester's worth of information in 9 mins.. I love the internet!
That was an amazing explanation. The best I’ve seen.
Khan Academy should have a color switcher like that.
betterexplained should stop confusing kids. Talk for 10 min that 2^x is descrete.
You should really carry on with these, your very talented and I think your website is a fantastic idea.. please don't give up
such a clear explanation that I'm about to show it to my maths students in 10 minutes time
Beautifully explained !. e^x is the only function that is always analytical. It is no wonder that nature should choose it to represent growth !.
You know what, in Thailand, the 'e' is just a kind of magic number.
The teacher decided to treat 'e' as a normal constant. What a shame...
I kind of get it now after 6+ years. Thanks!
It seems your principal argument is that e is a special universal growth constant because f(x) = e^ax is always growing throughout every interval of x, rather than in a discrete nature as f(x) = 2^x does. But the input, x, could include any number of fractional and irrational values, so f(x) = 2^x (as well as for infinitely more base values) is also always growing in partial, non-discrete amounts. I'm not seeing how the whole "infinite amount of growth happening" thing is special to e.
Sweet! e ever made sense to me before.
I hate being told to just use the magic number, or "just punch it into the calculator."
Terrific video! I never really understood the "power of e" before. Awesome!
Excellent, the only explanation that makes sense. THANK YOU!
Very nice approach; thanks for the video. Yet, it would be nice to have also a contextual, historical approach, on how the euler number was discovered by Euler, and why did he need it.
That's the best explanation I came across
Thanks for this video. I've not had much to do with maths but I came across this concept of 'Euler's number' recently and this is the clearest explanation I've had so far. A real worked example would be great though. It's confusing when people just say "well then it is e to the power x"!!! Make it a little more concrete for us beginners please.
Good Job - I like to think sqrt 2 is also an underappreciaated irrational number built into every square just like Pi is in every circle and e in anything that grows
Amazing explanation, you have saved me before a quiz. :)
Dude thats really good, u deserve waaay more subs
Great point -- I should have drawn the staircase better. Ideally, each step is larger than the one before (i.e. doubling), so you get some exponential curve. If each step is exactly the same, then it means your growth is not compounding [and e won't be able to model it].
Sorry if the words were unclear, I meant continuous compounding. If someone says "I am earning 100% interest per year" then 2^x is the amount they have it they compound every year, and e^x is the amount they have if they compound continuously (as fast as possible). 2^x is still a continuous function in a calculus sense, yep.
thank you for this video. I always forgot everything about e because i could understand it. Now i understand
Wonderfully explained. Thank you!
I needed this 4 years ago. Still glad I to know it not.
Appreciate your work mahn.
My god, I've learned about e years ago, yet this is the first time I actually understand it.
12 years later… thanks for this!
You are a great teacher.
Best teacher ever
This is exactly what I wanted to know, not taking limit to infinity, not 1+ 100% interest bullshit, these things i already know cuz its mentioned in every other videos on e. But i wanted to just confirm my intuition. I kinda knew that why e is natural growth bit not completely, i knew flexibility part, we can reduce or increase it according to our needs, but i didnt quite knew what property exactly makes it a natural Growth.
Time, 11:41pm
think about a flower blooming, it grows slowly over time and those growths add as well. or acid corroding something, it starts slowly then continues to add and add.
great! do you have any books published so that I can sit and read leisurely.
Exactly. When you compound continuously, you can merge rate and time into a single variable "x". It's hard to write in a comment, but check out the linked article in the description for more details.
Great video! This clarifies e for me, finally. Thanks!
This was very useful for me. Big tnx!
Thanks a lot. I think I finally understand the concept of 'e' :D
Very neat explanation. Awesome. Thanks
Very clear and helpful. Thanks!
Thanks for the explanation. very neat and well concepted.
Thank you. Your explanation rocks!
Halleluiah! Thanks for an explanation I could follow.
MAN THANK YOU FROM THE BOTTOM OF MY HEART
Thank you so much! This made things a lot more clear
thnx a lot..after struggling lot on e i finally got a perfect idea of it...willl really appreciate if u give me the link where a get the simmillllar type of help for understanding Standard deviation...
Brilliant explanation!
Thanks! Hoping to crank out more videos down the road, appreciate the encouragement :).
Great explanation that earned my instant subscription. Keep it up, please! :)
Amazing explanation. e is pretty cool.
Thanks! Great explanation. It really helped.
Awesome video! e makes much more sense to me now! thanks
Excellent explanation.Thank you sir
Please note: At the point where you started showing the red cell propagating the green cells, you forgot to indicate to the viewers that the black cell (which is still there) STILL CONTINUES TO PARTICIPATE in the growth process.
In other words, the black cell must necessarily indicate it is propagating YET ANOTHER new cell AT THE SAME TIME the now-new red cell is propagating a new green cell.
In a nutshell...At any given instant, all that exists in the population is BY DEFINITION pitching in to new growth. This is the VERY ESSENCE of natural growth.
I find it confusing when you say that 2^x is a stairstep function. It's a continuous function and when you plug in fractional values, you will get a continuous curve.
thanks so much for your explanations dude
You're a bloody genius
Great video. It helped me out!
Well done...continuous growth results in Eulers number, well explained...wonder why Math teachers do not explain this so clearly....suspect they do not know.
Thanks, you are correct. I should have started with x=0 (2^0 = 1).
excellent explanation. Thanks!