#1

ehas always bothered me — not the letter, but themathematical constant. What does it really mean?

This is a companion discussion topic for the original entry at//www.i494.com/articles/an-intuitive-guide-to-exponential-functions-e/
#2

Nice graphics!:slight_smile:

#3

Thanks. I made them in PowerPoint 2007, which makes graphics pretty easy. You can see the powerpoint file here:
//www.i494.com/examples/graphics/Exponential-growth.pptx

#4

大Kalid !我认为自己是一个非常聪明的人，每天都与数字打交道，但e对我来说一直是一个不透明的主题。从高中开始它就和它的朋友对数一起玩除了嘲笑我什么都不做!我认为这篇文章将帮助我以新的方式思考e，我甚至可能会把它用到实际中去。我真的很高兴我找到了你的网站!

#5

太好了，鲍勃，我很高兴你喜欢它!是的，e和它那些讨厌的朋友，比如自然对数，曾经是我的眼中钉。It really bugs me when I use a concept withoutreallyknowing what it meant.

E got all this attention and I wanted to dig in and see what all the fuss was about:slight_smile:

I’ll be writing more on this topic as there are some really interesting ways of looking at “this constant, approximately equal to 2.71828…”

#6

不错的文章，当涉及到数学思想时，很难找到实际的解释，我很高兴能找到这篇文章。

#7

Thanks Jayson, the lack of intuitive explanations motivated me to start this site. I’m glad you are finding it useful:slight_smile:

#8

I have a question.:slight_smile:x=速率*时间e^x=增长。利用这些信息，就有可能找到允许的最小时间量，即一个时间与下一个时间之间的差距。这是否能给我们一个无穷小的值，在每个时刻之间。只是一个随机的重量。很好的指南，特别喜欢向量微积分系列。2022世界杯南美我开始担心我将不得不把数学作为第二语言。他们在微积分课本上使用的单词对我来说太大了。

#9

Hi Stephen, great question!

是的，如果你让时间单位越来越小，你会得到某一点的瞬时增长率。令人惊讶的是，这是e^x!

I want to write more about the calculus of it, but basically, when you have a certain amount of “stuff” (say, 10 units) then you aregrowingat 10 units per unit of time as well. Of course, once you grow just a little bit you have a new amount of “stuff” (10.1 units) and now you are growing at 10.1 per unit time.

It’s a bit mind-boggling, but it’s the way e works – the current instantaneous rate of growth is equal to the current amount. I’ll be writing more on this as I get a good, intuitive understanding of it:slight_smile:

#10

Holy cow…somehow despite four semesters of calculus I forgot or failed to grasp that with any calculator I can do compound interest calculations as easily as circle areas. I’ve been dependent on financial calculators for twenty years. Now I’m going to read all your math-related posts. Keep writing!

#11

That’s awesome Joe! I know what you mean, I had forgotten about being able to do compound interest as well – it’s funny how rarely we revisit old topics we’ve learned. E had always bothered me.

I hope you enjoy the other posts, I’ll keep cranking them out:slight_smile:

#12

[…] After understanding the exponential function our next target is the natural logarithm. […]

#13

Thanks for the wonderful explanation. However, at places, confusion seems to arise due to wrong use of terminology. e.g. instead of ‘e is the fundamental rate of change shared by all continually growing processes.’ it should be ‘e is the fundamental net growth in all continually growing processes (in a unit time that would account for 100% simple growth)’, if I have understood correctly. Elsewhere, you do say growth=e^rt, where r is the rate of growth. So e is the total growth & not rate of growth.

#14

Hi Dr. Jani, that’s a great point – the current description of e is confusing. As you say, e^1 = e = the amount of continuous growth after 1 unit of time (assuming growing at a rate of 100% simple growth per unit time).

E ^rt可以计算任意速率和时间下的净增长率。我将更新这篇文章，使这更清楚，我感谢反馈!12强赛积分榜最新

#15

Wow thank you so much for explaining e. The wikipedia article completely confused me about it

#16

Thanks David. Yeah, it really bugs me when math topics are explained in a complicated way, I’m glad you liked it.

#17

Kalid - thank you (and your contributors for their comments) for this site. I hope that your blog will be even more popular than Wiki and be the “goto” place for teachers and their students!

#18

Thanks for the encouragement Sophie, you must have read my mind! I’m hoping this blog evolves into a place to think about new topics in a fresh, intuitive way, and share those “a-ha” moments from everyone.

维基百科是一个很好的参考，但百科全书倾向于关注事实和理解。两者都有一席之地:slight_smile:

#19

This is a great discussion. You should do the fundamental theorem of calculus, also.

#20

Thanks Jobie, appreciate the comment. Yep, the Fundamental Theorem of Calculus is definitely on my list of upcoming topics:slight_smile: