Understanding the number e | BetterExplained

[betterexplained]

+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.

[waterdaisy]

first time in life ....I can understand e intuitively....thank you

[emfkv]

This is a great explanation. Should be mandatory viewing for every math teacher in the world.
...er for the students too.

[Gergaferg]

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.

[mireazma]

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!

[Captain_Rhodes]

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

[xanderx]

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!

[Mojoman]

I feel like I arrive JUST IN TIME to take classes that these fresh, new, more useful videos to be uploaded!

[cachah]

Thank you so much!! I was confused about e for a year until I finally saw this! Way better than my math teacher!

[JustAwks]

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!

[climatereview]

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.

[MrKilltastic]

this is the first video by you that i have every seen, and it alone has earned my subscription.

[michaelmag]

This explanation is better than the 14 videos I watch previously.

[SubhodeepMukhopadhyay]

Thanks. Very intuitive. This is exactly what I was looking for!

[jamescobalt]

I watched four videos on e and THIS was the one where I went Thank you!

[sumkin]

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 !

[betterexplained]

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.

[samuelbeaubien]

Wow! I don't think there's a better way to explain this! Thank you!

[cerinobrien]

Thank you so much. I have to explain this to my class for a project and now I won't completely humiliate myself!!

[bjdarras]

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?