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Pi和E

2018年11月9日 / 7次阅读
老男孩学数学

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Pi和E是两个著名的无理数,Pi在计算圆周面积时使用,E就相对神秘了一点,本文找到一些英文资料来初略感受一下。常常看到学习数学,要培养一种对数学的intuitive sense,这可能就是我们小时候学习缺乏的吧。

Describing e as “a constant approximately 2.71828182…” is like calling pi “an irrational number, approximately equal to 3.14159265…”. 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 (unit growth, perfectly compounded).

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.

 

来一点有intuitive sense的公式:

$$e^x$$

$$x = rate \cdot time$$

$$growth = e^x = e^{rt}$$

本文链接:http://www.maixj.net/misc/pi-e-19316
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  • 麦新杰

    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. [ ]


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