This link has been bookmarked by 109 people . It was first bookmarked on 03 Aug 2007, by someone privately.
-
18 Apr 12
-
03 Nov 11
-
15 Oct 11
-
21 Sep 11
-
16 Jul 11
-
13 Jul 11
-
11 Jul 11
Julian Qian文章废话太多，只有借钱那部分表明在固定期限同样收益的前提下固定利率和活期利率之间的差距和e有一定的关系。我觉得e的更特别的地方是e^x的导数还是e^x，就是说它的变化的速度和本身是一样的。RT @xiaolai: e是什么 http://t.co/y1iFoEt
-
08 Jul 11
-
07 Jul 11
-
J Me has always bothered me — not the letter, but the mathematical constant. What does it really mean?
Math books and even my beloved Wikipedia describe e using obtuse jargon:
The mathematical constant e is the base of the natural logarithm.
And when you look up natural logarithm you get:
The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2.718281828459.
Nice circular reference there. It’s like a dictionary that defines labyrinthine with Byzantine: it’s correct but not helpful. What’s wrong with everyday words like “complicated”?
I’m not picking on Wikipedia — many math explanations are dry and formal in their quest for “rigor”. But this doesn’t help beginners trying to get a handle on a subject (and we were all a beginner at one point).
No more! Today I’m sharing my intuitive, high-level insights about what e is and why it rocks. Save your “rigorous” math book for another time.
e is NOT Just a Number
Describing e as “a constant approximately 2.71828…” is like calling pi “an irrational number, approximately equal to 3.1415…”. Sure, it’s true, but you completely missed the point.
Pi is the ratio between circumference and diameter shared by all circles. It is a fundamental ratio inherent in all circles and therefore impacts any calculation of circumference, area, volume, and surface area for circles, spheres, cylinders, and so on. Pi is important and shows all circles are related, not to mention the trigonometric functions derived from circles (sin, cos, tan).
e is the base rate of growth shared by all continually growing processes. e lets you take a simple growth rate (where all change happens at the end of the year) and find the impact of compound, continuous growth, where every nanosecond (or faster) you are growing just a little bit.
e shows up whenever systems grow exponentially and continuously: population, radioactive decay, interest calculations, and more. Even jagged systems that don’t grow smoothly can be approximated by e.
Just like every number can be considered a “scaled” version of 1 (the base unit), every circle can be considered a “scaled” version of the unit circle (radius 1), and every rate of growth can be considered a “scaled” version of e (the “unit” rate of growth).
So e is not an obscure, seemingly random number. e represents the idea that all continually growing systems are scaled versions of a common rate. -
04 Jul 11
-
30 Jun 11
-
e is the base rate of growth shared by all continually growing processes. e lets you take a simple growth rate (where all change happens at the end of the year) and find the impact of compound, continuous growth, where every nanosecond (or faster) you are growing just a little bit.
-
-
27 Apr 11
-
26 Apr 11
-
Pi is the ratio between circumference and diameter shared by all circles
-
e is the base rate of growth shared by all continually growing processes
-
In geeky math terms, e is defined to be that rate of growth if we continually compound 100% return on smaller and smaller time periods:
-
-
14 Mar 11
-
e is the base rate of growth shared by all continually growing processes. e lets you take a simple growth rate (where all change happens at the end of the year) and find the impact of compound, continuous growth, where every nanosecond (or faster) you are growing just a little bit.
e shows up whenever systems grow exponentially and continuously: population, radioactive decay, interest calculations, and more. Even jagged systems that don’t grow smoothly can be approximated by e.
Just like every number can be considered a “scaled” version of 1 (the base unit), every circle can be considered a “scaled” version of the unit circle (radius 1), and every rate of growth can be considered a “scaled” version of e (the “unit” rate of growth).
So e is not an obscure, seemingly random number. e represents the idea that all continually growing systems are scaled versions of a common rate.
-
-
12 Mar 11
-
10 Feb 11
-
continually compound 100% return
-
The number e (2.718…) is the maximum possible result when compounding 100% growth for one time period. Sure, you started out expecting to grow from 1 to 2 (that’s a 100% increase, right?). But with each tiny step forward you create a little “dividend” that starts growing on its own. When all is said and done, you end up with e (2.718…) at the end of 1 time period, not 2. e is the maximum, what happens when we compound 100% as much as possible.
-
e is like a speed limit (like c, the speed of light) saying how fast you can possibly grow using a continuous process. You might not always reach the speed limit, but it’s a reference point: you can write every rate of growth in terms of this universal constant.
-
Let me explain. When dealing with compound growth, 10 years of 3% growth has the same overall impact as 1 year of 30% growth (and no growth afterward).
-
Our rate is 5%, and we’re lucky enough to compound continuously. After 10 years, we get $120 * e^(.05 * 10) = $197.85. Of course, most banks aren’t nice enough to give you the best possible rate. The difference between your actual return and the continuous one is how much they don’t like you.
-
Of course, most banks aren’t nice enough to give you the best possible rate. The difference between your actual return and the continuous one is how much they don’t like you.
-
-
02 Jan 11
-
If we have the total growth rate and want the rate of a single crystal, we work backwards and use the natural log.
-
-
28 Sep 10
-
06 Sep 10
-
07 Aug 10
Thomas Jamese has always bothered me — not the letter, but the mathematical constant. What does it really mean?
Math books and even my beloved wikipedia describe e using obtuse jargon:
The mathematical constant e is the base of the natural logarithm.
And when you le Eulersnumber maths pi numbers reference resources learning education
-
03 Aug 10
-
10 Apr 10
-
31 Jan 10
-
19 Nov 09
-
01 Jul 09
-
22 Apr 09
-
e is the base amount of growth shared by all continually growing processes.
-
e shows up whenever systems grow exponentially and continuously: population, radioactive decay, interest calculations, and more. Even jagged systems that don’t grow smoothly can be approximated by e.
-
e represents the idea that all continually growing systems are scaled versions of a common rate.
-
-
11 Jan 09
-
09 Jan 09
-
07 Aug 08
-
07 Jul 08
-
01 Jul 08
-
10 Jun 08
-
31 Mar 08
-
16 Mar 08
Jay Duggere is the base amount of growth shared by all continually growing processes. e lets you take a simple growth rate (where all change happens at the end) and find the impact of compound, continuous growth, where every instant you are growing just a bit.
-
13 Mar 08
-
23 Dec 07
-
30 Nov 07
-
25 Nov 07
-
24 Nov 07
-
23 Nov 07
-
22 Nov 07
-
Mert Nuhoglue has always bothered me — not the letter, but the mathematical constant. What does it really mean?
-
05 Oct 07
Alan Deane has always bothered me — not the letter, but the mathematical constant. What does it really mean?
-
02 Aug 07
-
01 May 07
Would you like to comment?
Join Diigo for a free account, or sign in if you are already a member.